Generalized trace formula and asymptotics of the averaged Turan
Journal of Approximation Theory 141 (2006) 70 – 97 www.elsevier.com/locate/jat
Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product夡 Boris P. Osilenker∗ Department of Mathematics, Moscow State Civil Engineering University, Yaroslavskoe Shosse 26, Moscow 129337, Russia Received 7 April 2004; accepted 7 November 2005 Communicated by Leonid Golinskii Available online 15 May 2006
Abstract Let be a finite positive Borel measure supported on [−1, 1] and introduce the discrete Sobolev-type inner product f, g =
1 −1
f (x)g(x) d (x) +
Nk K
Mk,i f (i) (ak )g (i) (ak ),
k=1 i=0
where the mass points ak belong to [−1, 1], and Mk,i > 0(i = 0, 1, . . . , Nk ). In this paper, we obtain generalized trace formula and asymptotics of the averaged Turan determinant for the Sobolev-type orthogonal polynomials. Asymptotics of the recurrence coefficients for symmetric Gegenbauer–Sobolev orthogonal polynomials is obtained. Trace formula and asymptotics of Turan’s determinant for Gegenbauer–Sobolev orthogonal polynomials are also given. © 2006 Elsevier Inc. All rights reserved. MSC: 42C05 Keywords: Sobolev-type orthogonal polynomials; Generalized trace formula; Asymptotics of the averaged Turan determinant; Generalized Jacobi matrix; Gegenbauer–Sobolev orthogonal polynomials
夡 The
author was supported by the Russian Foundation for Basic Research (Grant 05-01-00192).
∗ ul.Verkhnyia Pervomayskaya dom 59/35, korp. 2, kv.10, Moscow 105264, Russia.
E-mail address: [email protected]. 0021-9045/$ - see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2005.11.019
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
71
1. Generalized trace formula and asymptotics of the averaged Turan determinant Let be a finite positive Borel measure supported on the interval (−1, 1) with infinitely many points at the support, and let ak (k = 1, 2, . . . , K) be real numbers such that ak ∈ [−1, 1]. For f and g in L2 (−1, 1) such that there exist the derivatives in ak , we introduce a discrete Sobolev-type inner product 1 Nk K (1.1) Mk,i f (i) (ak )g (i) (ak ), f, g = f (x)g(x) d(x) + −1
k=1 i=0
where Mk,i > 0 (i = 0, 1, . . . , Nk , k = 1, 2, . . . , K). As it is well known, this inner product (and corresponding orthogonal systems) is used in some problems of functional analysis, function theory and mathematical physics [7,11,14,18,19]. On the other hand, if we investigate the oscillation of a string loading with masses Mk at the points ak and use the Fourier method for the corresponding Sturm–Liouville boundary value problem associated with the second-order partial differential equation, then the eigenvectors are orthogonal with respect to the inner product: 1 N f, g = f (x)g(x) dx + Mk f (ak )g(ak ). −1
k=1
If we study the oscillation of girder, we get a fourth-order partial differential equation. The corresponding eigenfunctions are orthogonal with respect to the inner product involving derivatives. This problem also closely related to important type of combinations of manifolds (elastic substructures) of various dimensions. The best-known example, which pertains to the equilibrium theory of plates strengthened by rods, was considered for the first time by S.P. Timoshenko as early as 1915 (he was a famous specialist in elastic theory). k }(k ∈ Z+ = {0, 1, 2, . . .}) be the sequence of orthonormal polynomials with respect to Let {B the inner product (1.1), i.e. n , B k = n,k (n, k ∈ Z+ ). B Let Nk∗ be the positive integer number defined by Nk + 1 if Nk is odd, ∗ Nk = Nk + 2 if Nk is even, and let wN (x) =
K
∗
(x − ak )Nk ,
(1.2)
k=1
where N =
K
∗ k=1 Nk .
n satisfy the recurrence relation: Lemma 1.1 (Rocha et al. [31]). The polynomials B N N n+j (x) + n−j (x), n (x) = wN (x)B n+j,j B n,j B j =0
j =1
−j (x) = 0, j = 1, 2, . . . ; n,s = 0, n = 0, 1, . . . , s − 1), (n ∈ Z+ ; B
(1.3)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
furthermore if (x) > 0 a.e. then lim n,j = j
(j = 0, 1, 2, . . . , N),
n→∞
(1.4)
where wN (x) = 0 + 2
N
j Tj (x),
(1.5)
j =1
and are given by (N−j )
(0) 1 j = N 2N 2 (N − j )!
(j = 0, 1, 2, . . . , N )
(1.6)
with 2N () =
K
∗
(2 − 2ak + 1)Nk ,
(1.7)
k=1
where Ts (x) (s = 0, 1, . . .) is the Chebyshev polynomial of the first kind and degree s. Lemma 1.2. Let = {n , n ∈ R1 , n = 0, n ∈ Z+ ; −n = 0, n = 1, 2, . . .} be an arbitrary sequence. Then the following formula n k (t)B k+N (x) [wN (t) − wN (x)] k B k=0
=
n N
k (t)B k−j +N (x) [k−j k,j − k k+N,j ]B
j =0 k=0
+
N n
k (t)B k+j +N (x) [k+j k+j,j − k k+N+j,j ]B
j =1 k=0
+
N n+j
k (t)B k−j +N (x) − k−j k,j B
j =1 k=n+1
N n+j
k−j (t)B k+N (x) k k,j B
j =1 k=n+1
holds for all t,x and n ∈ Z+ . Proof. By (1.3) one obtains n k (t)B k+N (x) [wN (t) − wN (x)] k B k=0
=
n N
k+j (t)B k+N (x) + k k+j,j B
j =0 k=0
−
n N j =0 k=0
n N
k−j (t)B k+N (x) k k,j B
j =1 k=0
k (t)B k+N+j (x) − k k+N+j,j B
n N j =1 k=0
k (t)B k+N−j (x). k k+N,j B
(1.8)
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73
By Abel’s transform and using the initial conditions (1.3), one gets [wN (t) − wN (x)]
n
k (t)B k+N (x) k B
k=0 n
=
k (t)B k+N (x) k [k,0 − k+N,0 ]B
k=0
+
N n
k+j (t)B k+N (x) − k k+j,j B
N n
j =1 k=0
+
j =1 k=0
N n
k−j (t)B k+N (x) − k k,j B
j =1 k=0 n
=
n N
k (t)B k+N+j (x)] k k+N+j,j B
j =1 k=0
k (t)B k+N (x) + k [k,0 − k+N,0 ]B
N n+j
k (t)B k+N−j (x) k−j k,j B
j =1 k=n+1
k=0
+
k (t)B k−j +N (x) k k+N,j B
n N
k (t)B k+N−j (x) (k−j k,j − k k+N,j )B
j =1 k=0
−
N
n
k (t)B k+N+j (x) k+j k+j,j B
j =1 k=n−j +1
+
n N
k (t)B k+N+j (x), (k+j k+j,j − k k+N+j,j )B
j =1 k=0
which proves Lemma 1.2.
We introduce the averaged Turan -determinant: G(N) n (x; ) :=
N n+j
k (x)B k−j (x)B k+N−j (x) − k B k+N (x)]. k,j [k−j B
(1.9)
j =1 k=n+1
Putting t = x in Lemma 1.2, one has Corollary 1.3. For all t,x and n ∈ Z+ , the following relation G(N) n (x; ) =
n N
k (x)B k−j +N (x) (k k+N,j − k−j k,j )B
j =0 k=0
+
n N j =1 k=0
holds.
k (x)B k+j +N (x) (k k+N+j,j − k+j k+j,j )B
(1.10)
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Lemma 1.4 (Rocha et al. [31]). If (x) > 0 a.e., then for every function f continuous in [−1, 1] and for every positive k, one has 1 n (x)B n+k (x) d(x) lim f (x)B n→∞ −1 1 1 1 f (x)Tk (x) √ dx (k ∈ Z+ ). (1.11) = −1 1 − x2 Now we get a weak-type asymptotics of the averaged Turan -determinant. Theorem 1. If f ∈ C([−1, 1]), (x) > 0 a.e. and for sequence (1.8) relation lim n = = 0
(1.12)
n→∞
holds, then
1
lim
n→∞ −1
f (x)G(N) n (x; ) d(x) =
1
−1
f (x)UN−1 (x)wN (x) 1 − x 2 dx,
(1.13)
where UN−1 (x) is the Chebyshev polynomial of the second kind and degree (N − 1), and wN (x) is defined by (1.2). Proof. By (1.4), (1.6), (1.7), (1.9), (1.11), (1.12), one obtains 1 lim f (x)G(N) n (x; ) d(x) n→∞ −1 N
=
j =1
−
n+j
lim
n→∞
N j =1
k,j k−j
−1
k=n+1
lim
n→∞
n+j
k,j k
k=n+1
1
1 −1
k (x)B k+N−j (x) d(x) f (x)B k−j (x)B k+N (x) d(x) f (x)B
1 N 1 = j j f (x)[TN−j (x) − TN+j (x)] √ dx 1 − x2 −1 j =1
2 = j j N
j =1
1 −1
f (x) 1 − x 2 UN−1 (x)Uj −1 (x) dx,
where Us (x) =
sin(s + 1) arccos x sin(arccos x)
(−1 x 1; s ∈ Z+ )
is the Chebyshev polynomial of the second kind and degree s. Combining relations lim
1
n→∞ −1
f (x)G(N) n (x; ) d(x)=
2
1
−1
f (x)
n j =1
j j Uj −1 (x)UN−1 (x) 1 − x 2 dx
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
75
and Tj (x) = j Uj −1 (x)
(j = 1, 2, . . .)
with (1.5), one obtains (1.13). Theorem 1 is proved.
Let EK := (−1, 1)\
K
{ak }.
(1.14)
k=1
Using (1.10) and (1.13) by standard methods [13,22,24–27,35,36], one can obtain the main result.
Theorem 2. Assume that sequence (1.8) satisfies (1.12), and the estimate N ∞
|k − k+j | < ∞
(1.15)
j =0 k=0
n }(n ∈ Z+ ) satisfies the following condiholds. Suppose the orthonormal polynomial system {B tions: (a) The recurrence coefficients (1.3) are of N-bounded variation, i.e. ∞ N
|k,j − k+N,j | < ∞.
j =0 k=0
(b) There exists a continuous function h(x) on EK (see (1.14)) such that n (x)| h(x) |B
(x ∈ EK ).
(c) The measure is absolutely continuous in EK , and (x) = (x) is strictly positive and continuous on EK . Then the following assertions are valid: (1) At every x ∈ EK and uniformly on every compact subset of EK , the asymptotics of the averaged Turan -determinant √ 1 − x2 (N) lim Gn (x; ) = UN−1 (x)wN (x) n→∞ (x) holds; an upper bound for the uniform error of approximation for x on a closed subset of EK is given by √ 1 − x 2 (N) Gn (x; ) − UN−1 (x)wN (x) (x) C
∞ N j =0 k=n+1
|k − k+j | + C
∞ N
|k,j − k+N,j |,
j =0 k=n+1
where C’s are positive constants independent of n ∈ Z+ ;
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(2) At every x ∈ EK and uniformly on every compact subset of EK , the following generalized trace formula N ∞
k+N−j (x) k (x)B (k k+N,j − k−j k,j )B
j =0 k=0
+
N ∞
k (x)B k+N+j (x) (k k+j +N,j − k+j k+j,j )B
j =1 k=0
√ 1 − x2 = UN−1 (x)wN (x) (x)
holds. Put k = 1(k ∈ Z+ ), then we get the following analog of Turan’s determinant [12,13,16,22,24,26,32,35,36]: (N) G(N) n (x) = Gn (x, 1) =
N n+j
k (x)B k+N−j (x) − B k−j (x)B k+N (x)]. k,j [B
j =1 k=n+1
Corollary 1.5. Under conditions (a)–(c) of Theorem 2, the following assertions are valid at every x ∈ EK and uniformly on every compact subset of EK : √ 1 − x2 1 (N) (a) lim Gn (x) = UN−1 (x)wN (x) ; n→∞ (x) (b)
∞ N
k (x)B k+N−j (x) (k+N,j − k,j )B
j =0 k=0
+
∞ N k=1 k=0
(k+N+j,j
√ 1 − x2 1 − k+j,j )Bk (x)Bk+N+j (x) = UN−1 (x)wN (x) . (x)
Remarks. (1) This result partially have been announced in [30]. (2) For N = 1 we obtain a new approach to the proof of the remarkable formulas of papers [8,22]. Let {pn }(n ∈ Z+ ) be a sequence of polynomials pn (x) = k(pn )x n + r(pn )x n−1 + · · · , such that 1 −1
pn (x)pm (x) d(x) = m,n
k(pn ) > 0
(n ∈ Z+ ),
(m, n ∈ Z+ ).
These orthonormal polynomials satisfy a three-term recurrence relation: xpn (x) = an+1 pn+1 (x) + bn pn (x) + an pn−1 (x)
(n ∈ Z+ ; p−1 (x) = 0),
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
77
where an+1 =
k(pn ) > 0, k(pn+1 )
bn =
r(pn ) r(pn+1 ) − k(pn ) k(pn+1 )
(n ∈ Z+ ).
We define the Turan determinant as follows: 2 Gn (x) = pn+1 (x) − pn (x)pn+2 (x)
(n ∈ Z+ ).
If the recurrence coefficients converge lim an = 21 ,
n→∞
lim bn = 0,
n→∞
and ∞
(|ak+1 − ak+2 | + |bk − bk+1 |) < ∞,
k=0
then uniformly on every closed subset of (−1, 1), the following formulas are valid: (1) √ 2 1 − x2 lim Gn (x) = ; n→∞ (x) (2) Trace formula: ∞
2 [an+1
− an2 ]pn2 (x) + an+1 (bn
n=0
√ 1 1 − x2 . − bn+1 )pn+1 pn (x) = 2 (x)
(3) One can apply this result to recover the spectral measure from the Jacobi matrix [23,13,36]. “. . .highly effective method for recovering the spectral measure from the Jacobi matrix is the one using Turan determinant” [23, p. 457]. For analogous remark on use of Trace formula see [23, p. 460]. “It is evident that Legendre projection is poor near the end-points, whereas the Sobolev projection displays reasonably good behavior throughout the interval” [15, p. 124]. 2. Symmetric Gegenbauer–Sobolev orthogonal polynomials We consider a nonstandard inner product: 1 f, g = f g d + M[f (1)g(1) + f (−1)g(−1)] −1
+N [f (1)g (1) + f (−1)g (−1)], where M > 0, N > 0 and d (x) =
(2 + 2) 22+1 2 ( + 1)
(1 − x 2 ) dx
a probability Gegenbauer measure.
( > −1)
(2.1)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97 ()
Define the Gegenbauer polynomial Rn (x) by the Rodrigues formula Rn() (x) = Then
1
−1
(−1)n ( + 1) (1 − x 2 )− ((1 − x 2 )n+ )(n) 2n (n + + 1)
() Rm (x)Rn() (x)(1 − x 2 ) dx = 0
( > −1).
(m = n; m, n ∈ Z+ ),
()
moreover Rn (1) = 1(n ∈ Z+ ). Lemma 2.1 (Szegö [33]). For polynomial coefficients Rn() (x) = k(Rn() )x n + l(Rn() )x n−2 + m(Rn() )x n−4 + · · ·
(2.2)
the following formulas are valid: k(Rn() ) =
( + 1)(2n + 2 + 1) 2n (n + + 1)(n + 2 + 1)
l(Rn() ) = −
(n ∈ Z+ )
( + 1) (n − 1)n(2n + 2 − 1) 2n (n + )(n + 2 + 1)
(n 2)
(2.3) (2.4)
and m(Rn() ) =
( + 1) (2n + 2 − 3)(n − 3)4 2n+1 (n + − 1)(n + 2 + 1)
(n 4),
(2.5)
where (c)n is the shifted factorial defined by (c)n := c(c + 1) . . . (c + n − 1) =
(n + c) (c)
(n = 1, 2, . . .), (c)0 = 1.
Corollary 2.2.
(n + 2 + 1)(n + 2 + 2) 1 2 + 1 −2 = = 1+ + O(n ) , () (2n + 2 + 1)(2n + 2 + 3) 4 n k(Rn+2 ) ()
k(Rn ) ()
(2.6)
n(n − 1) 2(2n + 2 − 1)
2 + 1 1 42 − 1 1 n + =− 1− 4 2 n 4 n2 2 (1 − 4 )(2 − 1) 1 −4 + + O(n ) , (2.7) 8 n3 () (n − 3)4 m(Rn ) 1 = () 8 (2n + 2 − 1)(2n + 2 − 3) k(Rn )
2( + 2) 3(2 + 1)(2 + 3) 1 83 + 122 − 2 − 3 1 n2 1− + − = 2 32 n 4 n 2 n3 +O(n−4 ) . (2.8) l(Rn )
() k(Rn )
=−
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97 ()
79
()
Bavinck and Meijer [3,4] have introduced polynomials {Bn (x) = Bn (x; M, N )}(n ∈ Z+ ), orthogonal with respect to the inner product (2.1), as () Bm (x; M, N ), Bn() (x; M, N ) = 0
(m = n; m, n ∈ Z+ )
and have proved the following representation (see also [10]): Bn() (x) = an (1 − x 2 )2 {Rn() (x)}(4) + bn (1 − x 2 ){Rn() (x)}(2) + cn Rn() (x),
(2.9)
where an = MN +N bn = −N −M
(n − 1)n(n + + 2) () 2 1 (n ) ( + 1)3 ( + 2) n + 2 + 1 1 n() 2( + 1)3 n
(2.10)
1 (n − 2)n(n + 2 + 3)(n) 2( + 1)( + 3) 2 () n + 2 + 1 n
cn = 1 − N
(n) =
(n ∈ Z+ ),
(n ∈ Z+ ),
(2.11)
1 (n − 2)3 (n + 2 + 2)2 (n) 2( + 1)3
(n = 3, 4, . . . ; c0 = c1 = c2 = 1),
(2.12)
(n + 2 + 2) (2 + 3)n!
(2.13)
(n ∈ Z+ ).
n() (x; M, N )}(x ∈ [−1, 1], n ∈ Z+ ) be the corresponding sequence of orn() (x) = B Let {B thonormal polynomials (symmetric Gegenbauer–Sobolev orthonormal polynomials): 1 (2 + 2) () m n() (x)(1 − x 2 ) dx B (x)B 22+1 2 ( + 1) −1 () () n() (1) + B m n() (−1)] m (1)B (−1)B +M[B () () m n() } (1) + {B m n() } (−1)] = m,n +N [{B } (1){B } (−1){B
(m, n ∈ Z+ ; M > 0, N > 0). Then () n() (x) = (n) B n Bn (x),
(2.14)
where −2 ( (n) n )
=
−2 ( (n) n ())
(2 + 2) = 2 2 +1 2 ( + 1)
1 −1
[Bn() (x)]2 (1 − x 2 ) dx
+2M[Bn() (1)]2 + 2N [{Bn() } (1)]2 .
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
Lemma 2.3 (Osilenker [28,29], (=0, M=0), (N =0)). The following representation is valid: −2 ( (n) = Bn() (x; M, N ), Bn() (x; M, N ) n ) (2 + 2) n! () =
() (M, N ) n+2 (M, N ) 2n + 2 + 1 (n + 2 + 1) n
(n ∈ Z+ ); (2.15)
in addition
(n) (M, N ) = MN n() + N (n) + M (n) + 1
(n ∈ Z+ ),
(2.16)
where (i) (n) =
1 (n − 3)(n − 2) ( + 1)3 ( + 2)2 (2 + 3) (n + 2 + 1)(n + 2 + 3) × , [(n − 2)!]2
(2.17)
(ii) (n) =
(n + 2 + 2) [( + 2)n2 + ( + 2)(2 − 1)n 2( + 1)3 (2 + 3)(n − 3)! −2( + 1)(2 + 3)],
(2.18)
(iii)
(n) =
2 (n + 2 + 1) . (2 + 3) (n − 2)!
(2.19)
Proof. First, we consider case n = 4, 5, . . . . By formula (3.9) from [10], one has
(n + 2 + 2)n(n − 1) () 2 n(n + 2 + 1) −2 2 + M ) = 2Mc + 2N ( (n) n n n 2( + 1) ( + 1)( + 2) 1 n + 2 + 1 1 + [(n − 3)4 (n + 2 + 1)4 an2 2( + 1) 2n + 2 + 1 (n) +n(n − 1)(n + 2 + 1)(n + 2 + 2)bn2 −2(n − 3)4 (n + 2 + 1)(n + 2 + 2)an bn +cn2 − 2(n − 1)nbn cn + 2(n − 3)4 an cn ],
(2.20)
where an , bn and cn are defined by (2.10)–(2.13). Since (see (2.13)) 1 n + 2 + 1 1 (n + 1) (2 + 2) , = ( ) 2( + 1) 2n + 2 + 1 n 2n + 2 + 1 (n + 2 + 1) we get −2 = ( (n) n )
(2 + 2) (n + 1) {N 2 [An M 2 + Bn M + Cn ] 2n + 2 + 1 (n + 2 + 1) +N[Dn M 2 + En M + Fn ] + Kn M 2 + Ln M + Rn }.
Our aim is to find all coefficients in (2.21).
(2.21)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
81
If we substitute an , bn , cn (see formulas (2.10)–(2.13)) in (2.20), then we obtain the coefficients in (2.21): An = Bn =
[(n) ]4
(n − 3)4 (n − 1)2 n2 (n + 2 + 2)2 , n + 2 + 1 ( + 1)23 ( + 2)2 [(n) ]3 2( + 1)23 ( + 2) 4
(2.22)
1 (n − 2)(n − 1)2 n2 (n + 2 + 2)2 (n + 2 + 3) n + 2 + 1
×[( + 2)n + 2( + 2)(2 + 1)n3 + (63 + 152 − 14)n2 +(44 + 83 − 152 − 41 − 16)n −2( + 1)( + 2)(2 + 1)(2 + 3)]
(2.23)
and Cn =
[(n) ]2 4( + 1)23
(n − 2)(n − 1)n2 (n + 2 + 2)(n + 2 + 3)
×[( + 2)n4 + 2( + 2)(2 + 1)n3 + ( + 2)(43 + 82 − 5 − 14)n2 +2( + 2)(43 + 162 + 23 + 8)n + 4( + 1)(2 + 3)].
(2.24)
Taking into account (2.17) and (2.18), from (2.22) to (2.24), one gets ()
An = (n) n+2 , ()
()
Bn = (n) n+2 + n+2 (n) , ()
Cn = (n) n+2 . So expression at N 2 is the following:
() () () () N 2 (n) n+2 M 2 + [ (n) n+2 + n+2 (n) ]M + n n+2 .
(2.25)
Find coefficient at N. One gets from (2.20) that
4 (2n + 2 + 1)(n + 2 + 2)2 n2 (n − 1)2 Dn = [(n) ]3 ( + 1)( + 2)2 n + 2 + 1 4 (n − 3)(n − 2)(n − 1)2 n2 (n + 2 + 2)2 + n + 2 + 1 ( + 1)3 ( + 2) () 3 2 2 (n − 1) n (n + 2 + 2)2 2 4[n ] [n + (2 + 1)n + (22 + 7 + 9)], = ( + 1)3 ( + 2) n + 2 + 1 By definitions (2.17) and (2.19), one has ()
()
Dn = (n) n+2 + n+2 (n) . By a similar way En =
2[(n) ]2 n(n − 1)(n + 2 + 2) 2 [( + 4 + 5)n4 ( + 1)3 ( + 2) n + 2 + 1 +2(23 + 92 + 14 + 5)n3 + (44 + 163 + 312 + 40 + 31)n2 +(−84 + 163 + 262 + 68 + 26)n + 4( + 1)( + 2)(2 + 1)(2 + 3)],
(2.26)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
and by (2.17)–(2.19), one obtains ()
()
()
En = n() n+2 + n+2 n() + (n) + n+2 .
(2.27)
As before, from (2.20), we get the following: n() Fn = n [( + 2)( + 3)n(2n + 2 + 1)(n + 2 + 1) ( + 1)3 −(n − 2)(n − 1)(2n + 2 + 2)(2n + 2 + 3) +( + 2)(n − 2)3 (n + 2 + 3) + (n − 3)4 ], and by straightforward calculation, we obtain 1 n() [( + 2)n4 + 2( + 2)(2 + 1)n3 Fn = ( + 1)3 n +(63 + 232 + 34 + 22)n2 + (2 + 1)(23 + 112 + 25 + 20)n −4( + 1)(2 + 3)]. Using notation (2.18), one gets ()
Fn = n() + n+2 .
(2.28)
Taking into account (2.26)–(2.28), we get the following representation for the expression in (2.21) containing N: () () () N [ n() n+2 + n+2 (n) ]M 2 + [ (n) n+2
() () () + n+2 (n) + (n) + n+2 ]M + [ (n) + n+2 ] . (2.29) We find the remaining coefficients in (2.21). Using (2.19) from (2.20), one obtains Kn =
4 2 (2 + 3)
(n + 2 + 1) (n + 2 + 3) () = (n) n+2 (n − 1) (n + 1)
and 4 (n + 2 + 1) 2 [n + (2 + 1)n + ( + 1)(2 + 1)] (2 + 3) n+1 () = (n) + n+2 ,
Ln =
Rn = 1. By the last three formulas, we get the following representation for the expression in (2.21), which does not contain N: ()
()
(n) n+2 M 2 + [ (n) + n+2 ]M + 1.
(2.30)
Finally, substituting expressions (2.25), (2.29) and (2.30) in (2.21), one obtains (n + 1) (2 + 2) () () () −2 {[ () M 2 + ( (n) n+2 + n+2 (n) )M = ( (n) n ) 2n + 2 + 1 (n + 2 + 1) n n+2 ()
()
()
()
()
+ (n) n+2 ]N 2 + [( (n) n+2 + n+2 (n) )M 2 + ( (n) n+2 + n+2 (n) ()
()
+ (n) + n+2 )M + (n) + n+2 ]N ()
()
+ (n) n+2 M 2 + ( (n) + n+2 )M + 1},
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
83
and this coincides with (2.15) and (2.16). We prove Lemma 2.3 in the case n = 4, 5, . . . Now we prove formulas (2.15)–(2.19) in the case, n = 0, 1, 2, 3. In fact, it follows from (2.16) to (2.19) that ()
()
()
0 (M, N ) = 1 (M, N ) = 1;
2 (M, N ) = 2M + 1,
()
3 (M, N ) = 2(2 + 3)(M + N ) + 1, ()
4 (M, N ) =
()
5 (M, N ) =
2(2 + 5)(2 + 3)2 4(2 + 3)2 (2 + 5) MN + N +1 +1 +2( + 2)(2 + 3)M + 1, 4 (2 + 3)2 (2 + 5)2 (2 + 7) MN 3 +1 (2 + 7) 1 (2 + 6) + M + 1. (32 + 15 + 17)N + 3 (2 + 3) 2( + 1)3 (2 + 3)
On the other hand, it follows from (2.9) to (2.13) that ()
B0 (x) = 1,
()
B1 (x) = x,
1 {(2 + 3)(2M + 1)x 2 − [2(2 + 3)M + 1]}, 2( + 1)
(2 + 3)(2 + 5) (2 + 3)(2 + 5) 2 + 5 () B3 (x) = M+ N+ x3 +1 +1 2( + 1)
3(2 + 3)(2 + 5) 3 (2 + 3)(2 + 5) M+ N+ x. − +1 +1 2( + 1) ()
B2 (x) =
By definition of inner product (2.1), one has ()
()
()
()
()
()
B0 (x; M, N ), B0 (x; M, N ) = 2M + 1 = 0 (M, N ) 2 (M, N ), B1 (x; M, N ), B1 (x; M, N ) = 2(M + N ) + = ()
1 2 + 3
1 () ()
(M, N ) 3 (M, N ), 2 + 3 1
()
B2 (x; M, N ), B2 (x; M, N ) =
8(2 + 3)2 2(2 + 3)2 4( + 2)(2 + 3) 2 M (M 2 N + MN) + N+ 2 ( + 1) ( + 1)2 ( + 1)(2 + 5)
2(22 + 7 + 7) 1 M+ ( + 1)(2 + 5) ( + 1)(2 + 5) 2(2 + 2) () () =
(M, N ) 4 (M, N ), (2 + 5)(2 + 3) 2 +
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and ()
()
B3 (x; M, N ), B3 (x; M, N ) 8(2 + 3)2 (2 + 5)2 = (MN 2 + M 2 N ) ( + 1)2 8(2 + 3)(2 + 5)(72 + 36 + 44) 4( + 2)(2 + 3)(2 + 5) 2 + MN + M ( + 1)2 (2 + 7) ( + 1)(2 + 7) 6 (63 + 452 + 110 + 86)N + 2 ( + 1) (2 + 7) 2(22 + 9 + 13) 3 + M+ ( + 1)(2 + 3)(2 + 7) ( + 1)(2 + 7) 6(2 + 2) () () =
(M, N ) 5 (M, N ). (2 + 7)(2 + 5) 3 Lemma 2.3 is completely proved.
()
()
Lemma 2.4. For coefficients of polynomial Bn (x) = Bn (x; M, N )(n ∈ Z+ ) Bn() (x) = k(Bn() )x n + l(Bn() )x n−2 + m(Bn() )x n−4 + · · ·
(2.31)
the following representations are valid: (i) k(Bn ) = (n) (M, N )k(Rn() ) (n ∈ Z+ ), (ii) l(Bn ) =
8 (n − 3)(n − 2) ( + 1)3 2 (2 + 3) (n + 2 + 1)(n + 2 + 3) × n!(n − 2)! 2(n − 2)(n + 2 − 1) (n + 2 + 2) +N ( + 1)(2 + 3) (n − 1)! 4 (n + 2 + 1) +M l(Rn() ) (n2), (2 + 2) n!
(2.32)
(n) (M, N ) + MN
(2.33)
and (iii)
m(Bn() ) = (n) (M, N ) + MN
16 ( + 1)3 2 (2 + 3) (n + 2 + 1)(n + 2 + 3) 2 [n − 5n + 2( + 6)] × (n)!(n − 2)! 4(n + 2 + 2) +N [n2 + (2 − 3)n − 2(2 − 3)] ( + 1)(2 + 3)(n − 1)! (n + 2 + 1) 8 m(Rn() ) (n4), +M (2 + 2) (n)!
()
()
()
(2.34)
where k(Rn ), l(Rn ), m(Rn ), (n) (M, N ) are defined in (2.3)–(2.5) and (2.16)–(2.19).
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
85
Proof. We substitute (2.2) and (2.31) in (2.9) k(Bn() )x n + l(Bn() )x n−2 + m(Bn() )x n−4 + · · · = an (x 4 − 2x 2 + 1)[(n − 3)4 k(Rn() )x n−4 + (n − 5)4 l(Rn() )x n−6 +(n − 7)4 m(Rn() )x n−8 + · · ·] +bn (1 − x 2 )[(n − 1)nk(Rn() )x n−2 + (n − 3)(n − 2)l(Rn() )x n−4 +(n − 5)(n − 4)m(Rn() )x n−6 + · · ·] +cn [k(Rn() )x n + l(Rn() )x n−2 + m(Rn() )x n−4 + · · ·]. The comparison of the coefficients of
xn
(2.35)
in the above relation yields
k(Bn() ) = [(n − 3)4 an − (n − 1)nbn + cn ]k(Rn() ) and substituting an , bn , cn for (2.10)–(2.13), we get (n − 3)4 an − (n − 1)nbn + cn 1 (n − 3)4 (n + 2 + 1)(n + 2 + 3) = MN 2 (n − 2)!n! ( + 1)3 ( + 2) (2 + 3) (n − 3)4 (n + 2 + 2) 1 +N (n − 1)! 2( + 1)3 (2 + 3) 1 (n − 1)n(n − 2)(n + 2 + 3)(n + 2 + 2) +N 2( + 1)( + 3)(2 + 3) (n − 1)! 1 (n + 2 + 4) −N 2( + 1)3 (2 + 3) (n − 3)! 2 (n − 1)n(n + 2 + 1) +M + 1, (2 + 3) n! It follows from (2.16) to (2.19) that (n − 3)4 an − (n − 1)nbn + cn = (n) (M, N ), and we get relation (2.32). By (2.3) and (2.4) it is easily to see that k(Rn() ) = −
2(2n + 2 − 1) l(Rn() ). (n − 1)n
Thus, comparing the coefficients at x n−2 (n2) in relation (2.35), one obtains l(Bn() ) = an [(n − 5)4 l(Rn() ) − 2(n − 3)4 k(Rn() )] +bn [(n − 1)nk(Rn() ) − (n − 3)(n − 2)l(Rn() )] + cn l(Rn() ) = l(Rn() ){[(n − 5)4 + 4(n − 3)(n − 2)(2n + 2 − 1)]an −[(n − 3)(n − 2) + 2(2n + 2 − 1)]bn + cn } = {[(n − 3)4 an − (n − 1)nbn + cn ] + 8( + 2)(n − 3)(n − 2)an −4( + 1)bn }l(Rn() ).
(2.36)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
By (2.36), we get l(Bn() ) = [ (n) (M, N ) + 8( + 2)(n − 3)(n − 2)an − 4( + 1)bn ]l(Rn() ).
(2.37)
A consequence of (2.10), (2.11) and (2.13) is the following: 8( + 2)(n − 3)(n − 2)an − 4( + 1)bn 8(n − 3)(n − 2) (n + 2 + 1)(n + 2 + 3) = MN (n − 2)!n! ( + 1)3 2 (2 + 3) (n + 2 + 2) 2( + 2)( + 3) (n − 2)(n + 2 − 1) +N ( + 1)3 (2 + 3) (n − 1)! 8( + 1) (n + 2 + 1) . +M n! (2 + 3) Substituting this expression in (2.37) we obtain (2.33). () For finding the coefficient m(Bn ), we compare coefficients at x n−4 (n 4) in (2.35): m(Bn() ) = (n − 7)4 an m(Rn() ) − 2(n − 5)4 an l(Rn() ) + (n − 3)4 an k(Rn() )
+(n − 3)(n − 2)bn l(Rn() ) − (n − 5)(n − 4)bn m(Rn() ) + cn m(Rn() ).
Note by definition (n − 7)4 = 0 From (2.3) to (2.5), one has ()
k(Rn ) () m(Rn ) () l(Rn ) () m(Rn )
=
(n = 4, 5, 6, 7) and (n − 5)4 = 0
(n = 4, 5).
8(2n + 2 − 1)(2n + 2 − 3) , (n − 3)4
=−
4(2n + 2 − 3) (n − 3)(n − 2)
(n 4).
Consequently, m(Bn() ) = {[(n − 7)4 + 8(2n + 2 − 3)(n2 − 7n + 2 + 19)]an −[n2 − n + 8( + 1)]bn + cn }m(Rn() ).
(2.38)
We calculate the expression in brackets. As above, by (2.10)–(2.13) and (2.36), one gets (n − 7)4 + 8(2n + 2 − 3)(n2 − 7n + 2 + 19)an − [n2 − n + 8( + 1)]bn + cn = [(n − 3)4 an − (n − 1)nbn + cn ] + 16( + 2)[n2 − 5n + 2( + 6)]an − 8( + 1)bn n2 − 5n + 2( + 6) = (n) (M, N ) + 16MN( + 2) ( + 1)3 ( + 2)2 (2 + 3) (n + 2 + 1)(n + 2 + 3) × (n − 2)!n! n2 − 5n + 2( + 6) (n + 2 + 2) +16N( + 2) 2( + 1)3 (2 + 3) (n − 1)! 1 (n − 2)(n + 2 + 3)(n + 2 + 2) +8( + 1)N 2( + 1)( + 3)(2 + 3) (n − 1)! (n + 2 + 1) 2 . +8( + 1)M (2 + 3) n!
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
87
From the last relation, we deduce (n − 7)4 + 8(2n + 2 − 3)(n2 − 7n + 2 + 19)an − [n2 − n + 8( + 1)]bn + cn 16 [n2 − 5n + 2( + 6)] = (n) (M, N ) + MN ( + 1)3 2 (2 + 3) 4 (n + 2 + 1)(n + 2 + 3) +N × ( + 1)(2 + 3) (n − 2)!(n)! (n + 2 + 2) ×[n2 + (2 − 3)n − 2(2 − 3)] (n − 1)! 16( + 1) (n + 2 + 1) +M , (2 + 3) n! Substituting this relation in (2.38) one obtains (2.34). Lemma 2.4 is completely proved. In the next section we use the following assertion. Lemma 2.5 (Abramowitz and Stegun [1], Copson [6], Tricomi and Erdelyi [34]). For n large enough, the following asymptotic equality:
(n + ) ( − )( + − 1) 1 − 1+ =n 2 n (n + ) 1 3( + − 1)2 − + − 1 + ( − )( − − 1) 24 n2 1 . (2.39) +O n3 holds. Using representation (2.9)–(2.13) and Lemmas 2.3 and 2.5 one can get another proof of the following proposition. Lemma 2.6 (Marcellan and Osilenker [21], Foulquie Moreno et al. [10]). (case M>0, N >0). The following properties for symmetric Gegenbauer–Sobolev polynomials hold: (i) −3− 2 (n) , n n 15
(ii) n() (x)| C (1 − x 2 )− 2 − 41 , − 1 ; |B 2 1 n() (x)| C −1 < < − |B (n ∈ Z+ , −1 < x < 1), 2 (iii) max
−1 x 1
n() (x)| C (n + 1) 2 (n ∈ Z+ ), |B 1
(iv) n() } (1)| n−− 2 , n() (1)| n−− 2 , |{B |B 3
7
88
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
where the constants C , are independent of x ∈ (−1, 1), and n ∈ Z+ (in formula ii) and n ∈ Z+ (in formula iii). n() (x) have Remark. It should be noted that the Gegenbauer–Sobolev orthonormal polynomials B n() (x) (see, some properties other than the corresponding Gegenbauer orthonormal polynomials R for example [2–5,9,10,15,17,18,20,21,28–31]). We mention the following properties only. 1. The inner product (2.1) cannot be obtained by a weight function since x, x = 1, x 2 . This implies that well-known results for classical orthogonal polynomials, depending on the existence of a weight function do not longer hold. The well-known three-term recurrence relation for orthogonal polynomials does not hold in this case, but has to be replaced by a seven-term relation [5], in addition, this order is minimal [9]. n() (x) lying For n large enough, there exists exactly one pair of real zeros −n and n of B outside (−1, 1) [2,5]. n() (x) are eigenfunctions of a class of linear differential operators, usually of in2. Polynomials B finite order. In the case that is a nonnegative integer, this class contains a differential operator of finite order. This order is 4 + 10 [17,3]). n() (x) are eigenfunctions of differential operator of second We recall that polynomials R order [33], 3. As well-known [33] n() (±1) n+ 21 , {R n } (±1) n+ 25 . R These asymptotics are different from (iv). 3. Asymptotics of the averaged Turan determinant and generalized trace formula for symmetric Gegenbauer–Sobolev orthogonal polynomials As known [5], symmetric Gegenbauer–Sobolev orthonormal polynomials satisfy a nine-term recurrence relation (note that a degree N in (1.2) is even) ()
()
()
()
()
() () () n() (x) = B (x 2 − 1)2 B n+4 n+4 (x) + n+2 Bn+2 (x) + n Bn (x) + n Bn−2 (x) () () (x)(n ∈ Z+ ; B −s (x) = 0, s = 1, 2, . . . ; s() = 0, +(n) B n−4 ()
()
s = 0, 1, 2, 3; 0 = 1 = 0).
(3.1)
Our main aim is to calculate the recurrence coefficients and to prove that they are bounded variation. Let n() (x) = k(B n() )x n + l(B n() )x n−2 + m(B n() )x n−4 + · · · . B
(3.2)
By (2.14) and (2.31) one gets (n) (n) () () () n() ) = (n) () () k(B n k(Bn ), l(Bn ) = n l(Bn ), m(Bn ) = n m(Bn ),
where (n) n are defined in Lemma 2.3.
(3.3)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
89
Comparing coefficients at x n+4 , x n+2 , x n on both sides of (3.1) and using (3.2), (3.3), we obtain the following. Lemma 3.1. For the recursion coefficients of the symmetric band matrix associated with symmetn() }(n ∈ Z+ ), the following representations ric Gegenbauer–Sobolev orthonormal polynomials {B are valid: (i) ()
(n) n
k(Bn )
n+4
(n+4)
k(Bn+4 )
(n) n
k(Bn )
(n+2)
k(Bn+2 )
()
n+4 =
(3.4)
,
()
(ii) () n+2
=
()
n+2
()
()
l(Bn ) ()
k(Bn )
()
l(Bn+4 )
−
()
k(Bn+4 )
−2 ,
(3.5)
(iii) (n) =
()
m(Bn ) ()
k(Bn ) −
()
−
m(Bn+4 ) ()
k(B ) n+4 () l(Bn+2 ) l(Bn() ) ()
k(Bn+2 )
()
k(Bn )
−2
()
l(Bn ) ()
k(Bn )
+1
()
−
l(Bn+4 ) ()
k(Bn+4 )
−2 .
(3.6)
Lemma 3.2. For the sequence (n) (M, N ), defined by formulas (2.16)–(2.19), the following relations: (i) 8( + 2) 1
n() (M, N ) = 1 − , (3.7) + O 2 () n n
n+2 (M, N ) (ii) ()
n+2 (M, N )
8( + 2) =1+ +O () n
n (M, N )
1 n2
(3.8)
,
(iii)
(n) (M, N )
16( + 2) +O =1− () n
n+4 (M, N )
1 n2
,
(3.9)
(iv) ()
n+4 (M, N )
(n) (M, N )
hold.
=1+
16( + 2) +O n
1 n2
(3.10)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
We prove equality (3.8) only because the other ones can be deduced from (3.8). Taking into account (2.16), we find ()
n+2 (M, N )
n() (M, N ) =1+
()
()
()
()
MN[ n+2 − (n) ] + N[ n+2 − n ] + M[ n+2 − (n) ] ()
MN (n) + N n + M (n) + 1
(3.11)
.
It is not difficult to see that from (2.17) to (2.19), we have ()
n+2 − n() =
(n − 1)n (n + 2 + 1)(n + 2 + 3) 2 (n!)2 ( + 1)3 ( + 2) (2 + 3) 3 2 2 ×[8( + 2)n + 12(2 + 5 + 2)n +4(83 + 302 + 35 + 14)n + 4( + 1)( + 2)(2 + 1)(2 + 3)],
()
n+2 − n() =
(n + 2 + 2) 1 [4( + 2)( + 3)n3 (n − 1)! 2( + 1)3 (2 + 3) +2(63 + 332 + 51 − 2)n2 + 2(84 + 243 + 452 + 29 − 30)n +8( + 1)(2 + 3)]
and ()
n+2 − (n) =
4( + 1) (n + 2 + 1) (2n + 2 + 1). (2 + 3) n!
Putting the last three relations in (3.11) and using (2.39) we obtain (3.8). This completes the proof of Lemma 3.2. Lemma 3.3. For the ratio of the normalized multipliers, the following relations (n) 1 6 + 15 n , +O =1+ 2 (n+2) n n n+2 (n) n (n+4)
n+4
=1+
2(6 + 15) +O n
1 n2
(3.12)
(3.13)
hold. Proof. We prove relation (3.12). By (2.15), one has (n) n (n+2)
n+2 Since
=
2n + 2 + 1 2n + 2 + 5
() (M, N ) (n + 1)(n + 2) n+4 (n + 2 + 1)(n + 2 + 2) (n) (M, N )
√ 1 + x = 1 + 21 x + O(x 2 )(x → 0),
(n ∈ Z+ ).
(3.14)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
91
we see that 1 2n + 2 + 1 1 =1− +O , 2n + 2 + 5 n2 n (n + 1)(n + 2) 2 1 =1− +O (n + 2 + 1)(n + 2 + 2) n n2 and by (3.10), (3.14) () (M, N ) n+4
8( + 2) =1+ +O () n
n (M, N )
1 n2
,
we deduce (3.12). Relation (3.13) follows from (3.12). Lemma 3.3 is completely proved.
()
Lemma 3.4. For the ratio of the leading coefficients of polynomial Bn (x) (see (2.31)), the following relations are valid: (i) () ()
l(Bn ) l(Rn ) 8( + 2) () + T (M, N ) , (3.15) = 1 + n () () (n − 1)n k(Rn ) k(Bn ) where 1
|Tn() (M, N )| C
n2+4
,
()
|Tn() (M, N ) − Tn+1 (M, N )|C (ii) ()
m(Bn ) ()
k(Bn )
=
()
m(Rn ) ()
k(Rn ) (n4),
1+
1 n2+5
(3.16)
,
16( + 2)[n2 − 5n + 2( + 6)] + Wn() (M, N ) (n − 3)4 (3.17)
where |Wn() (M, N )|C
1 , n2+4
()
|Wn() (M, N ) − Wn+1 (M, N )| C
1 . n2+5
The constants in (3.16) and (3.18) are independent of n ∈ Z+ . Proof. It follows from (2.32) and (2.33) that () () ˜ n() ) l(Rn ) l(B l(Bn ) = 1 + () , () ()
n (M, N ) k(Bn ) k(Rn ) where ˜ n() ) = 8MN (n − 3)(n − 2) (n + 2 + 1)(n + 2 + 3) l(B (n − 2)!n! ( + 1)3 2 (2 + 3) 4M (n + 2 + 1) 2(n − 2)(n + 2 − 1) (n + 2 + 2) + , +N ( + 1)(2 + 3) (n − 1)! (2 + 2) n!
(3.18)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
and function n() (M, N ) is defined by (2.16)–(2.19). It is not difficult to see that ˜ n() ) l(B
(n) (M, N )
=
8( + 2) + Tn() (M, N ), (n − 1)n
where Tn() (M, N ) =
()
Sn (M, N ) (n − 1)n (n) (M, N )
with 2N (n − 2)(n − 1) (n + 2 + 2) ( + 1)( + 3)(2 + 3) (n − 1)! 2 2 ×[( + 1)n + (2 + − 1)n + 4( + 1)(2 + 3)] 8M( + 3) (n + 2 + 1) − + 8( + 2). (2 + 3) (n − 2)!
Sn() (M, N ) = −
Then, ()
Tn() (M, N ) − Tn+1 (M, N )
()
()
=
()
(n − 1)Sn+1 (M, N ) (n) (M, N ) − (n + 1)Sn (M, N ) n+1 (M, N ) ()
(n − 1)3 (n) (M, N ) n+1 (M, N )
.
()
Using representation of (n) (M, N ), Sn (M, N ) and relation (2.39), we get estimates (3.16). Taking into account (2.32) and (2.34), we obtain () () () m(Rn ) m(B ˜ n ) m(Bn ) = 1 + , ()
(n) (M, N ) k(Rn() ) k(Bn ) where m(B ˜ n() ) =
(n + 2 + 1)(n + 2 + 3) 2 16MN [n − 5n + 2( + 6)] 2 (n − 2)!n! ( + 1)3 (2 + 3) (n + 2 + 2) 4N [n2 + (2 − 3)n − 2(2 − 3)] + ( + 1)(2 + 3) (n − 1)! 8M (n + 2 + 1) + . (2 + 2) n!
Hence, ()
m(B ˜ n )
(n) (M, N )
=
16( + 2)[n2 − 5n + 2( + 6)] + Wn() (M, N ), (n − 3)4
where Wn() (M, N ) =
()
Un (M, N ) (n − 3)4 (n) (M, N )
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
93
with Un() (M, N ) 4N (n − 1)(n − 2) (n + 2 + 2) [( + 1)n4 + 2( − 1)n3 =− (n − 1)! ( + 1)( + 3)(2 + 3) −(102 + 3 + 25)n2 + (83 + 522 + 116 + 156)n −(163 + 1122 − 300 + 252)]. Thus we have ()
Wn() (M, N ) − Wn+1 (M, N ) ()
=
()
()
(n + 1)Un (M, N ) n+1 (M, N ) − (n − 3)Un+1 (M, N ) (n) (M, N ) ()
(n − 3)5 (n) (M, N ) n+1 (M, N )
,
()
and application of definition of (n) (M, N ), Un (M, N ) and formula (2.39) yields estimates (3.18). Lemma 3.4 is proved. Our main result of this section is the following ()
()
Theorem 3. For the recurrence coefficients n , (n) , n (see (3.1)) the following assertions are valid: (i) lim n() =
n→∞
1 , 16
()
1 lim (n) = − , n→∞ 4
lim n() =
n→∞
3 ; 8
()
(ii) The sequences {n }, {(n) }, {n }(n ∈ Z+ ) are bounded variation ∞
(|(n) | + |(n) | + |(n) |) < ∞.
n=0
Proof. By (2.32) ()
k(Bn ) ()
k(Bn+4 )
=
()
(n) (M, N ) k(Rn ) ()
()
n+4 (M, N ) k(Rn+4 )
(n ∈ Z+ ).
If we combine this with (2.6), (3.4), (3.9) and (3.13), then we get
1 1 () n+4 = , 1+O 16 n2 ()
and we prove our result for the sequence {n }(n ∈ Z+ ). From relations (2.7), (3.15) and (3.16), we obtain the following: () () l(Bn+4 ) l(Bn ) 1 42 + 32 + 63 +O − − 2 = −1 − 2 () () 4n n3 k(Bn ) k(B ) n+4
+En() (M, N ),
(3.19)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
with |En() (M, N )|C
1 , n2+4
(3.20)
where the constant C > 0 is independent of n ∈ Z+ . Taking into account (2.6), (2.32) and (3.7), we obtain
() () 6 + 15 1 k(Bn )
(n) (M, N ) k(Rn ) 1 1 − + O . = = () () () 4 n n2 k(B )
(M, N ) k(R ) n+2
n+2
n+2
Now, by relations (3.5) and (3.12), one gets
6 + 5 1 1 () + E˜ n() (M, N ), +O n+2 = − 1 − 4 n n2 where |E˜ n() (M, N )|C
1 , n2+4
and we obtain the assertions of Theorem 3 for the sequence {(n) }(n ∈ Z+ ). From (2.7), (3.15) and (3.16) we deduce the following:
1 1 3 − 2 42 + 32 + 63 1 =− 1+ + E¯ n() (M, N ), +O + 2 3 () 4 2n 4 n n k(Bn+2 ) ()
l(Bn+2 )
where |E¯ n() (M, N )|C
1 n2+4
,
and the constant C > 0 is independent of n ∈ Z+ . Then, using (3.19), (3.20) and last two relations, one obtains () () () l(Bn+4 ) l(Bn+2 ) l(Bn ) − − 2 () () () k(Bn ) k(Bn+4 ) k(Bn+2 )
n 3 − 2 42 + 32 + 63 1 = + Fn() (M, N ), 1+ + + O 2 4 2n 2n n3 where |Fn() (M, N )|C
1 n2+4
,
and the constant C > 0 is independent of n ∈ Z+ . Combining this with (3.15) and (3.16), we get () () () Bn+4 l(Bn+2 ) l(Bn() ) l(Bn ) − − 2 − 2 1− () () () () k(Bn+2 ) k(Bn ) k(Bn+4 ) k(Bn ) 1 n 2 − 3 + F˜n() (M, N ), +O = − (3.21) 4 8 n2
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
95
where |F˜n() (M, N )| C
1
(3.22)
n2+3
with the constant C > 0 independent of n ∈ Z+ . On the other hand, using (2.8), (3.17) and (3.18), we obtain
() () m(Bn ) m(Bn+4 ) n2 8 8 32 1 − = − + 2 − 3 +O + G(n) (M, N ), 4 () () 32 n n n n k(Bn ) k(Bn+4 ) where |Gn() (M, N )| C
1 n2+3
,
with the constant C > 0 independent of n ∈ Z+ . By (3.6), summing the last relations and (3.21), (3.22), we obtain 3 1 () ˜ (n) (M, N ), |G ˜ (n) (M, N )| C 1 , +G n = − + O 2 n n2+3 8 n with the constant C > 0 independent of n ∈ Z+ . () Thus we have the assertion of Theorem 3 for the sequence n . Theorem 3 is completely proved. Let us introduce the averaged Turan -determinant for the system of symmetric Gegenbauer– Sobolev orthonormal polynomials G(n) (x; ) =
n+4
()
()
()
(x)B (x)]2 − k B (x)]} k {k−4 [B k k−4 k+4
k=n+1 n+2
+
()
()
()
()
(x)B (x)B (x) − k B (x)]. k [k−2 B k k+2 k−2 k+4
k=n+1
Combining Theorem 2 with Lemma 2.6 and Theorem 3, we obtain the following statement. Theorem 4. At every point x ∈ (−1, 1) and uniformly on every closed subset of (−1, 1) the following statements are valid: 1. If for the sequence (1.8), conditions (1.12) and (1.15) are satisfied, then the following asymptotics lim G(n) (x; ) =
n→∞
3 22+5 ( + 1) 2 x (1 − 2x 2 )(1 − x 2 ) 2 − (2 + 2)
holds. 2. The generalized trace formula ∞ s() (x)]2 [4(2n+4 − 2n ) + 2(2n+2 − 2n )][B n=0
+
∞ n=0
() (x)B n() (x) [2n+2 (n+2 − n ) + 6(n+4 n+4 − n+2 n )]B n+2
96
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
+4
∞
()
(x)B n() (x) n+4 (n+4 − n )B n+4
n=0
+2
∞
() (x)B n() (x) (n+4 n+6 − n+6 n+2 )B n+6
n=0 22+4 ( + 1)
=
(2 + 2)
x 2 (1 − x 2 ) 2 − . 5
holds. Acknowledgments This work was partially completed (the case of the Legendre polynomials) while the author visited the Departamento de Matematicas, Universidad Carlos III de Madrid for part of academic year 2002–2003. Author is grateful to Professors J. Arvesu, F. Marcellan and G. Lopez for several helpful comments. I am thankful to F. Marcellan for seeking and obtaining financial support for this period. Author is very grateful to the referee for his valuable suggestions and constructive comments, which have significantly improved the first draft of this paper. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1977. [2] J. Arvesu, R. Alvarez-Nodarse, F. Marcellan, K. Pan, Jacobi–Sobolev-type orthogonal polynomials: second-order differential equation and zeros, J. Comput. Appl. Math. 90 (1998) 135–156. [3] H. Bavinck, Differential operators having Sobolev-type Gegenbauer polynomials as eigenfunctions, J. Comput. Appl. Math. 118 (2000) 23–42. [4] H. Bavinck, H.G. Meijer, Orthogonal polynomials with respect to a symmetric inner product involving derivatives, Appl. Anal. 33 (1989) 103–117. [5] H. Bavinck, H.G. Meijer, On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations, Indag. Math. (N.S.) 1 (1990) 7–14. [6] E.T. Copson, Asymptotic Expansions, Cambridge University Press, Cambridge, 1965. [7] R. Courant, D. Hilbert, Methoden der mathematischen Physik, Berlin, 1931. [8] J. Dombrowski, P. Nevai, Orthogonal polynomials, measures and recurrence relation, SIAM J. Math. Anal. 17 (1986) 752–759. [9] W.D. Evans, L.L. Littlejohn, F. Marcellan, C. Markett, A. Ronveaux, On recurrence relations for Sobolev orthogonal polynomials, SIAM J. Math. Anal. 26 (1995) 446–467. [10] A. Foulquie Moreno, F. Marcellan, B.P. Osilenker, Estimates for polynomials orthogonal with respect to some Gegenbauer–Sobolev inner product, J. Ineq. Appl. 3 (1999) 401–419. [11] C.T. Fulton, S. Pruess, Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions, J. Math. Anal. Appl. 71 (1979) 431–462. [12] G. Gasper, On extension of Turan’s inequality to Jacobi polynomials, Duke Math. J. 38 (1971) 415–428. [13] J. Geronimo, W. Van Assche, Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991) 341–371. [14] A.A. Gonchar, On the convergence of Pade approximants for some classes of meromorphic functions, USSR Sb. 26 (1975) 555–575. [15] A. Iserles, P.E. Koch, S.P. Norsett, J.M. Sanz-Serna, Orthogonality and approximation in a Sobolev space, in: J.C. Masson, M.G. Cox (Eds.), Algorithms for Approximations, Chapman & Hall, New York, 1990, pp. 117–124. [16] S. Karlin, G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. d’Anal. Math. 8 (19601961) 1–157. [17] R. Koekoek, Differential equations for symmetric generalized ultraspherical polynomials, Trans. Amer. Math. Soc. 345 (1994) 47–72. [18] A. Krall, Hilbert space, Boundary value problems and orthogonal polynomials, Operator Theory: Advances and Applications, vol. 133, Birkhäuser, Basel, 2002.
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[19] G. Lopez Lagomasino, Convergence of Pade approximants of Stieltjes type meromorphic functions and comparative asymptotics for orthogonal polynomials, Mat. USSR Sb. 64 (1989) 207–227. [20] F. Marcellan, M. Alfaro, M.L. Rezola, Orthogonal polynomials: old and new directions, J. Comput. Appl. Math. 48 (1993) 113–132. [21] F. Marcellan, B.P. Osilenker, Estimates for polynomials orthogonal with respect to some Legendre-Sobolev inner product, Mat. Zametki 62 (1997) 871–880 (in Russian). [22] A. Maté, P. Nevai, V. Totik, Asymptotics for orthogonal polynomials, defined by a recurrence relation, Constr. Approx. 1 (1985) 231–248. [23] P. Nevai, Research problems in orthogonal polynomials, in: C.K. Chui, L.L. Schumaker, J.D. Ward (Eds.), Approximation Theory, vol. 2, Academic Press, New York, 1989, pp. 449–489. [24] P. Nevai, Orthogonal polynomials recurrences, Jacobi matrix and measures, in: A.A. Gonchar, E.B. Saff (Eds.), Progress in Approximation Theory, Springer, Berlin, 1992, pp. 79–104. [25] B.P. Osilenker, Trace formula for orthogonal polynomials with asymptotically 2-periodic recurrence coefficients, J. Approx. Theory 92 (1998) 442–462. [26] B.P. Osilenker, Turan’s determinant for orthogonal polynomials with asymptotically periodic recurrence coefficients, Dokl. RAN 361 (1998) 318–320 (in Russian). [27] B.P. Osilenker, Generalized Trace Formula for symmetric Legendre-orthogonal polynomials, in: V.B. Priezzev, V.P. Spiridonov (Eds.), Self-Similar Systems, Dubna, 1999, pp. 332–334 (in Russian). [28] B.P. Osilenker, Legendre–Sobolev symmetric orthogonal polynomials in Pontryagin–Sobolev spaces, Math. Phys. Anal. Geom. 9 (2002) 385–393 (in Russian). [29] B.P. Osilenker, Symmetric orthogonal Gegenbauer polynomials in indefinite inner product spaces, Proceeding of the MSCIU, Moscow 10 (2003) 39–45 (in Russian). [30] B.P. Osilenker, Generalized Trace formula for polynomials orthogonal in a discrete Sobolev space, in: Contemporary problems in function theory and applications, Abstract. Saratov. 2004, 132–133 (in Russian). [31] I. Rocha, F. Marcellan, L. Salto, Relative asymptotics and Fourier Series of orthogonal polynomials with a discrete Sobolev inner product, J. Approx. Theory 121 (2003) 336–356. [32] G. Szegö, On an inequality of P. Turan concerning Legendre polynomials, Bull. Amer. Math. Soc. 54 (1948) 401–405. [33] G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquim Publications, vol. 23, fourth ed., American Mathematical Society Providence, Rhode Island, 1975. [34] F.G. Tricomi, A. Erdelyi, The asymptotics expansion of a rational Gamma function, Pacific J. Math. 1 (1951) 133–142. [35] W. Van Assche, Asymptotics for orthogonal polynomials and three-term recurrences, in: P. Nevai (Ed.), Orthogonal Polynomials: Theory and Practice, Kluwer, Dordrecht, 1990, pp. 435–462. [36] W. Van Assche, Christoffel functions and Turan determinants on several intervals, J. Comput. Appl. Math. 48 (1993) 207–223.
Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product夡 Boris P. Osilenker∗ Department of Mathematics, Moscow State Civil Engineering University, Yaroslavskoe Shosse 26, Moscow 129337, Russia Received 7 April 2004; accepted 7 November 2005 Communicated by Leonid Golinskii Available online 15 May 2006
Abstract Let be a finite positive Borel measure supported on [−1, 1] and introduce the discrete Sobolev-type inner product f, g =
1 −1
f (x)g(x) d (x) +
Nk K
Mk,i f (i) (ak )g (i) (ak ),
k=1 i=0
where the mass points ak belong to [−1, 1], and Mk,i > 0(i = 0, 1, . . . , Nk ). In this paper, we obtain generalized trace formula and asymptotics of the averaged Turan determinant for the Sobolev-type orthogonal polynomials. Asymptotics of the recurrence coefficients for symmetric Gegenbauer–Sobolev orthogonal polynomials is obtained. Trace formula and asymptotics of Turan’s determinant for Gegenbauer–Sobolev orthogonal polynomials are also given. © 2006 Elsevier Inc. All rights reserved. MSC: 42C05 Keywords: Sobolev-type orthogonal polynomials; Generalized trace formula; Asymptotics of the averaged Turan determinant; Generalized Jacobi matrix; Gegenbauer–Sobolev orthogonal polynomials
夡 The
author was supported by the Russian Foundation for Basic Research (Grant 05-01-00192).
∗ ul.Verkhnyia Pervomayskaya dom 59/35, korp. 2, kv.10, Moscow 105264, Russia.
E-mail address: [email protected]. 0021-9045/$ - see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2005.11.019
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
71
1. Generalized trace formula and asymptotics of the averaged Turan determinant Let be a finite positive Borel measure supported on the interval (−1, 1) with infinitely many points at the support, and let ak (k = 1, 2, . . . , K) be real numbers such that ak ∈ [−1, 1]. For f and g in L2 (−1, 1) such that there exist the derivatives in ak , we introduce a discrete Sobolev-type inner product 1 Nk K (1.1) Mk,i f (i) (ak )g (i) (ak ), f, g = f (x)g(x) d(x) + −1
k=1 i=0
where Mk,i > 0 (i = 0, 1, . . . , Nk , k = 1, 2, . . . , K). As it is well known, this inner product (and corresponding orthogonal systems) is used in some problems of functional analysis, function theory and mathematical physics [7,11,14,18,19]. On the other hand, if we investigate the oscillation of a string loading with masses Mk at the points ak and use the Fourier method for the corresponding Sturm–Liouville boundary value problem associated with the second-order partial differential equation, then the eigenvectors are orthogonal with respect to the inner product: 1 N f, g = f (x)g(x) dx + Mk f (ak )g(ak ). −1
k=1
If we study the oscillation of girder, we get a fourth-order partial differential equation. The corresponding eigenfunctions are orthogonal with respect to the inner product involving derivatives. This problem also closely related to important type of combinations of manifolds (elastic substructures) of various dimensions. The best-known example, which pertains to the equilibrium theory of plates strengthened by rods, was considered for the first time by S.P. Timoshenko as early as 1915 (he was a famous specialist in elastic theory). k }(k ∈ Z+ = {0, 1, 2, . . .}) be the sequence of orthonormal polynomials with respect to Let {B the inner product (1.1), i.e. n , B k = n,k (n, k ∈ Z+ ). B Let Nk∗ be the positive integer number defined by Nk + 1 if Nk is odd, ∗ Nk = Nk + 2 if Nk is even, and let wN (x) =
K
∗
(x − ak )Nk ,
(1.2)
k=1
where N =
K
∗ k=1 Nk .
n satisfy the recurrence relation: Lemma 1.1 (Rocha et al. [31]). The polynomials B N N n+j (x) + n−j (x), n (x) = wN (x)B n+j,j B n,j B j =0
j =1
−j (x) = 0, j = 1, 2, . . . ; n,s = 0, n = 0, 1, . . . , s − 1), (n ∈ Z+ ; B
(1.3)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
furthermore if (x) > 0 a.e. then lim n,j = j
(j = 0, 1, 2, . . . , N),
n→∞
(1.4)
where wN (x) = 0 + 2
N
j Tj (x),
(1.5)
j =1
and are given by (N−j )
(0) 1 j = N 2N 2 (N − j )!
(j = 0, 1, 2, . . . , N )
(1.6)
with 2N () =
K
∗
(2 − 2ak + 1)Nk ,
(1.7)
k=1
where Ts (x) (s = 0, 1, . . .) is the Chebyshev polynomial of the first kind and degree s. Lemma 1.2. Let = {n , n ∈ R1 , n = 0, n ∈ Z+ ; −n = 0, n = 1, 2, . . .} be an arbitrary sequence. Then the following formula n k (t)B k+N (x) [wN (t) − wN (x)] k B k=0
=
n N
k (t)B k−j +N (x) [k−j k,j − k k+N,j ]B
j =0 k=0
+
N n
k (t)B k+j +N (x) [k+j k+j,j − k k+N+j,j ]B
j =1 k=0
+
N n+j
k (t)B k−j +N (x) − k−j k,j B
j =1 k=n+1
N n+j
k−j (t)B k+N (x) k k,j B
j =1 k=n+1
holds for all t,x and n ∈ Z+ . Proof. By (1.3) one obtains n k (t)B k+N (x) [wN (t) − wN (x)] k B k=0
=
n N
k+j (t)B k+N (x) + k k+j,j B
j =0 k=0
−
n N j =0 k=0
n N
k−j (t)B k+N (x) k k,j B
j =1 k=0
k (t)B k+N+j (x) − k k+N+j,j B
n N j =1 k=0
k (t)B k+N−j (x). k k+N,j B
(1.8)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
73
By Abel’s transform and using the initial conditions (1.3), one gets [wN (t) − wN (x)]
n
k (t)B k+N (x) k B
k=0 n
=
k (t)B k+N (x) k [k,0 − k+N,0 ]B
k=0
+
N n
k+j (t)B k+N (x) − k k+j,j B
N n
j =1 k=0
+
j =1 k=0
N n
k−j (t)B k+N (x) − k k,j B
j =1 k=0 n
=
n N
k (t)B k+N+j (x)] k k+N+j,j B
j =1 k=0
k (t)B k+N (x) + k [k,0 − k+N,0 ]B
N n+j
k (t)B k+N−j (x) k−j k,j B
j =1 k=n+1
k=0
+
k (t)B k−j +N (x) k k+N,j B
n N
k (t)B k+N−j (x) (k−j k,j − k k+N,j )B
j =1 k=0
−
N
n
k (t)B k+N+j (x) k+j k+j,j B
j =1 k=n−j +1
+
n N
k (t)B k+N+j (x), (k+j k+j,j − k k+N+j,j )B
j =1 k=0
which proves Lemma 1.2.
We introduce the averaged Turan -determinant: G(N) n (x; ) :=
N n+j
k (x)B k−j (x)B k+N−j (x) − k B k+N (x)]. k,j [k−j B
(1.9)
j =1 k=n+1
Putting t = x in Lemma 1.2, one has Corollary 1.3. For all t,x and n ∈ Z+ , the following relation G(N) n (x; ) =
n N
k (x)B k−j +N (x) (k k+N,j − k−j k,j )B
j =0 k=0
+
n N j =1 k=0
holds.
k (x)B k+j +N (x) (k k+N+j,j − k+j k+j,j )B
(1.10)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
Lemma 1.4 (Rocha et al. [31]). If (x) > 0 a.e., then for every function f continuous in [−1, 1] and for every positive k, one has 1 n (x)B n+k (x) d(x) lim f (x)B n→∞ −1 1 1 1 f (x)Tk (x) √ dx (k ∈ Z+ ). (1.11) = −1 1 − x2 Now we get a weak-type asymptotics of the averaged Turan -determinant. Theorem 1. If f ∈ C([−1, 1]), (x) > 0 a.e. and for sequence (1.8) relation lim n = = 0
(1.12)
n→∞
holds, then
1
lim
n→∞ −1
f (x)G(N) n (x; ) d(x) =
1
−1
f (x)UN−1 (x)wN (x) 1 − x 2 dx,
(1.13)
where UN−1 (x) is the Chebyshev polynomial of the second kind and degree (N − 1), and wN (x) is defined by (1.2). Proof. By (1.4), (1.6), (1.7), (1.9), (1.11), (1.12), one obtains 1 lim f (x)G(N) n (x; ) d(x) n→∞ −1 N
=
j =1
−
n+j
lim
n→∞
N j =1
k,j k−j
−1
k=n+1
lim
n→∞
n+j
k,j k
k=n+1
1
1 −1
k (x)B k+N−j (x) d(x) f (x)B k−j (x)B k+N (x) d(x) f (x)B
1 N 1 = j j f (x)[TN−j (x) − TN+j (x)] √ dx 1 − x2 −1 j =1
2 = j j N
j =1
1 −1
f (x) 1 − x 2 UN−1 (x)Uj −1 (x) dx,
where Us (x) =
sin(s + 1) arccos x sin(arccos x)
(−1 x 1; s ∈ Z+ )
is the Chebyshev polynomial of the second kind and degree s. Combining relations lim
1
n→∞ −1
f (x)G(N) n (x; ) d(x)=
2
1
−1
f (x)
n j =1
j j Uj −1 (x)UN−1 (x) 1 − x 2 dx
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
75
and Tj (x) = j Uj −1 (x)
(j = 1, 2, . . .)
with (1.5), one obtains (1.13). Theorem 1 is proved.
Let EK := (−1, 1)\
K
{ak }.
(1.14)
k=1
Using (1.10) and (1.13) by standard methods [13,22,24–27,35,36], one can obtain the main result.
Theorem 2. Assume that sequence (1.8) satisfies (1.12), and the estimate N ∞
|k − k+j | < ∞
(1.15)
j =0 k=0
n }(n ∈ Z+ ) satisfies the following condiholds. Suppose the orthonormal polynomial system {B tions: (a) The recurrence coefficients (1.3) are of N-bounded variation, i.e. ∞ N
|k,j − k+N,j | < ∞.
j =0 k=0
(b) There exists a continuous function h(x) on EK (see (1.14)) such that n (x)| h(x) |B
(x ∈ EK ).
(c) The measure is absolutely continuous in EK , and (x) = (x) is strictly positive and continuous on EK . Then the following assertions are valid: (1) At every x ∈ EK and uniformly on every compact subset of EK , the asymptotics of the averaged Turan -determinant √ 1 − x2 (N) lim Gn (x; ) = UN−1 (x)wN (x) n→∞ (x) holds; an upper bound for the uniform error of approximation for x on a closed subset of EK is given by √ 1 − x 2 (N) Gn (x; ) − UN−1 (x)wN (x) (x) C
∞ N j =0 k=n+1
|k − k+j | + C
∞ N
|k,j − k+N,j |,
j =0 k=n+1
where C’s are positive constants independent of n ∈ Z+ ;
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
(2) At every x ∈ EK and uniformly on every compact subset of EK , the following generalized trace formula N ∞
k+N−j (x) k (x)B (k k+N,j − k−j k,j )B
j =0 k=0
+
N ∞
k (x)B k+N+j (x) (k k+j +N,j − k+j k+j,j )B
j =1 k=0
√ 1 − x2 = UN−1 (x)wN (x) (x)
holds. Put k = 1(k ∈ Z+ ), then we get the following analog of Turan’s determinant [12,13,16,22,24,26,32,35,36]: (N) G(N) n (x) = Gn (x, 1) =
N n+j
k (x)B k+N−j (x) − B k−j (x)B k+N (x)]. k,j [B
j =1 k=n+1
Corollary 1.5. Under conditions (a)–(c) of Theorem 2, the following assertions are valid at every x ∈ EK and uniformly on every compact subset of EK : √ 1 − x2 1 (N) (a) lim Gn (x) = UN−1 (x)wN (x) ; n→∞ (x) (b)
∞ N
k (x)B k+N−j (x) (k+N,j − k,j )B
j =0 k=0
+
∞ N k=1 k=0
(k+N+j,j
√ 1 − x2 1 − k+j,j )Bk (x)Bk+N+j (x) = UN−1 (x)wN (x) . (x)
Remarks. (1) This result partially have been announced in [30]. (2) For N = 1 we obtain a new approach to the proof of the remarkable formulas of papers [8,22]. Let {pn }(n ∈ Z+ ) be a sequence of polynomials pn (x) = k(pn )x n + r(pn )x n−1 + · · · , such that 1 −1
pn (x)pm (x) d(x) = m,n
k(pn ) > 0
(n ∈ Z+ ),
(m, n ∈ Z+ ).
These orthonormal polynomials satisfy a three-term recurrence relation: xpn (x) = an+1 pn+1 (x) + bn pn (x) + an pn−1 (x)
(n ∈ Z+ ; p−1 (x) = 0),
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
77
where an+1 =
k(pn ) > 0, k(pn+1 )
bn =
r(pn ) r(pn+1 ) − k(pn ) k(pn+1 )
(n ∈ Z+ ).
We define the Turan determinant as follows: 2 Gn (x) = pn+1 (x) − pn (x)pn+2 (x)
(n ∈ Z+ ).
If the recurrence coefficients converge lim an = 21 ,
n→∞
lim bn = 0,
n→∞
and ∞
(|ak+1 − ak+2 | + |bk − bk+1 |) < ∞,
k=0
then uniformly on every closed subset of (−1, 1), the following formulas are valid: (1) √ 2 1 − x2 lim Gn (x) = ; n→∞ (x) (2) Trace formula: ∞
2 [an+1
− an2 ]pn2 (x) + an+1 (bn
n=0
√ 1 1 − x2 . − bn+1 )pn+1 pn (x) = 2 (x)
(3) One can apply this result to recover the spectral measure from the Jacobi matrix [23,13,36]. “. . .highly effective method for recovering the spectral measure from the Jacobi matrix is the one using Turan determinant” [23, p. 457]. For analogous remark on use of Trace formula see [23, p. 460]. “It is evident that Legendre projection is poor near the end-points, whereas the Sobolev projection displays reasonably good behavior throughout the interval” [15, p. 124]. 2. Symmetric Gegenbauer–Sobolev orthogonal polynomials We consider a nonstandard inner product: 1 f, g = f g d + M[f (1)g(1) + f (−1)g(−1)] −1
+N [f (1)g (1) + f (−1)g (−1)], where M > 0, N > 0 and d (x) =
(2 + 2) 22+1 2 ( + 1)
(1 − x 2 ) dx
a probability Gegenbauer measure.
( > −1)
(2.1)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97 ()
Define the Gegenbauer polynomial Rn (x) by the Rodrigues formula Rn() (x) = Then
1
−1
(−1)n ( + 1) (1 − x 2 )− ((1 − x 2 )n+ )(n) 2n (n + + 1)
() Rm (x)Rn() (x)(1 − x 2 ) dx = 0
( > −1).
(m = n; m, n ∈ Z+ ),
()
moreover Rn (1) = 1(n ∈ Z+ ). Lemma 2.1 (Szegö [33]). For polynomial coefficients Rn() (x) = k(Rn() )x n + l(Rn() )x n−2 + m(Rn() )x n−4 + · · ·
(2.2)
the following formulas are valid: k(Rn() ) =
( + 1)(2n + 2 + 1) 2n (n + + 1)(n + 2 + 1)
l(Rn() ) = −
(n ∈ Z+ )
( + 1) (n − 1)n(2n + 2 − 1) 2n (n + )(n + 2 + 1)
(n 2)
(2.3) (2.4)
and m(Rn() ) =
( + 1) (2n + 2 − 3)(n − 3)4 2n+1 (n + − 1)(n + 2 + 1)
(n 4),
(2.5)
where (c)n is the shifted factorial defined by (c)n := c(c + 1) . . . (c + n − 1) =
(n + c) (c)
(n = 1, 2, . . .), (c)0 = 1.
Corollary 2.2.
(n + 2 + 1)(n + 2 + 2) 1 2 + 1 −2 = = 1+ + O(n ) , () (2n + 2 + 1)(2n + 2 + 3) 4 n k(Rn+2 ) ()
k(Rn ) ()
(2.6)
n(n − 1) 2(2n + 2 − 1)
2 + 1 1 42 − 1 1 n + =− 1− 4 2 n 4 n2 2 (1 − 4 )(2 − 1) 1 −4 + + O(n ) , (2.7) 8 n3 () (n − 3)4 m(Rn ) 1 = () 8 (2n + 2 − 1)(2n + 2 − 3) k(Rn )
2( + 2) 3(2 + 1)(2 + 3) 1 83 + 122 − 2 − 3 1 n2 1− + − = 2 32 n 4 n 2 n3 +O(n−4 ) . (2.8) l(Rn )
() k(Rn )
=−
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97 ()
79
()
Bavinck and Meijer [3,4] have introduced polynomials {Bn (x) = Bn (x; M, N )}(n ∈ Z+ ), orthogonal with respect to the inner product (2.1), as () Bm (x; M, N ), Bn() (x; M, N ) = 0
(m = n; m, n ∈ Z+ )
and have proved the following representation (see also [10]): Bn() (x) = an (1 − x 2 )2 {Rn() (x)}(4) + bn (1 − x 2 ){Rn() (x)}(2) + cn Rn() (x),
(2.9)
where an = MN +N bn = −N −M
(n − 1)n(n + + 2) () 2 1 (n ) ( + 1)3 ( + 2) n + 2 + 1 1 n() 2( + 1)3 n
(2.10)
1 (n − 2)n(n + 2 + 3)(n) 2( + 1)( + 3) 2 () n + 2 + 1 n
cn = 1 − N
(n) =
(n ∈ Z+ ),
(n ∈ Z+ ),
(2.11)
1 (n − 2)3 (n + 2 + 2)2 (n) 2( + 1)3
(n = 3, 4, . . . ; c0 = c1 = c2 = 1),
(2.12)
(n + 2 + 2) (2 + 3)n!
(2.13)
(n ∈ Z+ ).
n() (x; M, N )}(x ∈ [−1, 1], n ∈ Z+ ) be the corresponding sequence of orn() (x) = B Let {B thonormal polynomials (symmetric Gegenbauer–Sobolev orthonormal polynomials): 1 (2 + 2) () m n() (x)(1 − x 2 ) dx B (x)B 22+1 2 ( + 1) −1 () () n() (1) + B m n() (−1)] m (1)B (−1)B +M[B () () m n() } (1) + {B m n() } (−1)] = m,n +N [{B } (1){B } (−1){B
(m, n ∈ Z+ ; M > 0, N > 0). Then () n() (x) = (n) B n Bn (x),
(2.14)
where −2 ( (n) n )
=
−2 ( (n) n ())
(2 + 2) = 2 2 +1 2 ( + 1)
1 −1
[Bn() (x)]2 (1 − x 2 ) dx
+2M[Bn() (1)]2 + 2N [{Bn() } (1)]2 .
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
Lemma 2.3 (Osilenker [28,29], (=0, M=0), (N =0)). The following representation is valid: −2 ( (n) = Bn() (x; M, N ), Bn() (x; M, N ) n ) (2 + 2) n! () =
() (M, N ) n+2 (M, N ) 2n + 2 + 1 (n + 2 + 1) n
(n ∈ Z+ ); (2.15)
in addition
(n) (M, N ) = MN n() + N (n) + M (n) + 1
(n ∈ Z+ ),
(2.16)
where (i) (n) =
1 (n − 3)(n − 2) ( + 1)3 ( + 2)2 (2 + 3) (n + 2 + 1)(n + 2 + 3) × , [(n − 2)!]2
(2.17)
(ii) (n) =
(n + 2 + 2) [( + 2)n2 + ( + 2)(2 − 1)n 2( + 1)3 (2 + 3)(n − 3)! −2( + 1)(2 + 3)],
(2.18)
(iii)
(n) =
2 (n + 2 + 1) . (2 + 3) (n − 2)!
(2.19)
Proof. First, we consider case n = 4, 5, . . . . By formula (3.9) from [10], one has
(n + 2 + 2)n(n − 1) () 2 n(n + 2 + 1) −2 2 + M ) = 2Mc + 2N ( (n) n n n 2( + 1) ( + 1)( + 2) 1 n + 2 + 1 1 + [(n − 3)4 (n + 2 + 1)4 an2 2( + 1) 2n + 2 + 1 (n) +n(n − 1)(n + 2 + 1)(n + 2 + 2)bn2 −2(n − 3)4 (n + 2 + 1)(n + 2 + 2)an bn +cn2 − 2(n − 1)nbn cn + 2(n − 3)4 an cn ],
(2.20)
where an , bn and cn are defined by (2.10)–(2.13). Since (see (2.13)) 1 n + 2 + 1 1 (n + 1) (2 + 2) , = ( ) 2( + 1) 2n + 2 + 1 n 2n + 2 + 1 (n + 2 + 1) we get −2 = ( (n) n )
(2 + 2) (n + 1) {N 2 [An M 2 + Bn M + Cn ] 2n + 2 + 1 (n + 2 + 1) +N[Dn M 2 + En M + Fn ] + Kn M 2 + Ln M + Rn }.
Our aim is to find all coefficients in (2.21).
(2.21)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
81
If we substitute an , bn , cn (see formulas (2.10)–(2.13)) in (2.20), then we obtain the coefficients in (2.21): An = Bn =
[(n) ]4
(n − 3)4 (n − 1)2 n2 (n + 2 + 2)2 , n + 2 + 1 ( + 1)23 ( + 2)2 [(n) ]3 2( + 1)23 ( + 2) 4
(2.22)
1 (n − 2)(n − 1)2 n2 (n + 2 + 2)2 (n + 2 + 3) n + 2 + 1
×[( + 2)n + 2( + 2)(2 + 1)n3 + (63 + 152 − 14)n2 +(44 + 83 − 152 − 41 − 16)n −2( + 1)( + 2)(2 + 1)(2 + 3)]
(2.23)
and Cn =
[(n) ]2 4( + 1)23
(n − 2)(n − 1)n2 (n + 2 + 2)(n + 2 + 3)
×[( + 2)n4 + 2( + 2)(2 + 1)n3 + ( + 2)(43 + 82 − 5 − 14)n2 +2( + 2)(43 + 162 + 23 + 8)n + 4( + 1)(2 + 3)].
(2.24)
Taking into account (2.17) and (2.18), from (2.22) to (2.24), one gets ()
An = (n) n+2 , ()
()
Bn = (n) n+2 + n+2 (n) , ()
Cn = (n) n+2 . So expression at N 2 is the following:
() () () () N 2 (n) n+2 M 2 + [ (n) n+2 + n+2 (n) ]M + n n+2 .
(2.25)
Find coefficient at N. One gets from (2.20) that
4 (2n + 2 + 1)(n + 2 + 2)2 n2 (n − 1)2 Dn = [(n) ]3 ( + 1)( + 2)2 n + 2 + 1 4 (n − 3)(n − 2)(n − 1)2 n2 (n + 2 + 2)2 + n + 2 + 1 ( + 1)3 ( + 2) () 3 2 2 (n − 1) n (n + 2 + 2)2 2 4[n ] [n + (2 + 1)n + (22 + 7 + 9)], = ( + 1)3 ( + 2) n + 2 + 1 By definitions (2.17) and (2.19), one has ()
()
Dn = (n) n+2 + n+2 (n) . By a similar way En =
2[(n) ]2 n(n − 1)(n + 2 + 2) 2 [( + 4 + 5)n4 ( + 1)3 ( + 2) n + 2 + 1 +2(23 + 92 + 14 + 5)n3 + (44 + 163 + 312 + 40 + 31)n2 +(−84 + 163 + 262 + 68 + 26)n + 4( + 1)( + 2)(2 + 1)(2 + 3)],
(2.26)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
and by (2.17)–(2.19), one obtains ()
()
()
En = n() n+2 + n+2 n() + (n) + n+2 .
(2.27)
As before, from (2.20), we get the following: n() Fn = n [( + 2)( + 3)n(2n + 2 + 1)(n + 2 + 1) ( + 1)3 −(n − 2)(n − 1)(2n + 2 + 2)(2n + 2 + 3) +( + 2)(n − 2)3 (n + 2 + 3) + (n − 3)4 ], and by straightforward calculation, we obtain 1 n() [( + 2)n4 + 2( + 2)(2 + 1)n3 Fn = ( + 1)3 n +(63 + 232 + 34 + 22)n2 + (2 + 1)(23 + 112 + 25 + 20)n −4( + 1)(2 + 3)]. Using notation (2.18), one gets ()
Fn = n() + n+2 .
(2.28)
Taking into account (2.26)–(2.28), we get the following representation for the expression in (2.21) containing N: () () () N [ n() n+2 + n+2 (n) ]M 2 + [ (n) n+2
() () () + n+2 (n) + (n) + n+2 ]M + [ (n) + n+2 ] . (2.29) We find the remaining coefficients in (2.21). Using (2.19) from (2.20), one obtains Kn =
4 2 (2 + 3)
(n + 2 + 1) (n + 2 + 3) () = (n) n+2 (n − 1) (n + 1)
and 4 (n + 2 + 1) 2 [n + (2 + 1)n + ( + 1)(2 + 1)] (2 + 3) n+1 () = (n) + n+2 ,
Ln =
Rn = 1. By the last three formulas, we get the following representation for the expression in (2.21), which does not contain N: ()
()
(n) n+2 M 2 + [ (n) + n+2 ]M + 1.
(2.30)
Finally, substituting expressions (2.25), (2.29) and (2.30) in (2.21), one obtains (n + 1) (2 + 2) () () () −2 {[ () M 2 + ( (n) n+2 + n+2 (n) )M = ( (n) n ) 2n + 2 + 1 (n + 2 + 1) n n+2 ()
()
()
()
()
+ (n) n+2 ]N 2 + [( (n) n+2 + n+2 (n) )M 2 + ( (n) n+2 + n+2 (n) ()
()
+ (n) + n+2 )M + (n) + n+2 ]N ()
()
+ (n) n+2 M 2 + ( (n) + n+2 )M + 1},
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
83
and this coincides with (2.15) and (2.16). We prove Lemma 2.3 in the case n = 4, 5, . . . Now we prove formulas (2.15)–(2.19) in the case, n = 0, 1, 2, 3. In fact, it follows from (2.16) to (2.19) that ()
()
()
0 (M, N ) = 1 (M, N ) = 1;
2 (M, N ) = 2M + 1,
()
3 (M, N ) = 2(2 + 3)(M + N ) + 1, ()
4 (M, N ) =
()
5 (M, N ) =
2(2 + 5)(2 + 3)2 4(2 + 3)2 (2 + 5) MN + N +1 +1 +2( + 2)(2 + 3)M + 1, 4 (2 + 3)2 (2 + 5)2 (2 + 7) MN 3 +1 (2 + 7) 1 (2 + 6) + M + 1. (32 + 15 + 17)N + 3 (2 + 3) 2( + 1)3 (2 + 3)
On the other hand, it follows from (2.9) to (2.13) that ()
B0 (x) = 1,
()
B1 (x) = x,
1 {(2 + 3)(2M + 1)x 2 − [2(2 + 3)M + 1]}, 2( + 1)
(2 + 3)(2 + 5) (2 + 3)(2 + 5) 2 + 5 () B3 (x) = M+ N+ x3 +1 +1 2( + 1)
3(2 + 3)(2 + 5) 3 (2 + 3)(2 + 5) M+ N+ x. − +1 +1 2( + 1) ()
B2 (x) =
By definition of inner product (2.1), one has ()
()
()
()
()
()
B0 (x; M, N ), B0 (x; M, N ) = 2M + 1 = 0 (M, N ) 2 (M, N ), B1 (x; M, N ), B1 (x; M, N ) = 2(M + N ) + = ()
1 2 + 3
1 () ()
(M, N ) 3 (M, N ), 2 + 3 1
()
B2 (x; M, N ), B2 (x; M, N ) =
8(2 + 3)2 2(2 + 3)2 4( + 2)(2 + 3) 2 M (M 2 N + MN) + N+ 2 ( + 1) ( + 1)2 ( + 1)(2 + 5)
2(22 + 7 + 7) 1 M+ ( + 1)(2 + 5) ( + 1)(2 + 5) 2(2 + 2) () () =
(M, N ) 4 (M, N ), (2 + 5)(2 + 3) 2 +
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
and ()
()
B3 (x; M, N ), B3 (x; M, N ) 8(2 + 3)2 (2 + 5)2 = (MN 2 + M 2 N ) ( + 1)2 8(2 + 3)(2 + 5)(72 + 36 + 44) 4( + 2)(2 + 3)(2 + 5) 2 + MN + M ( + 1)2 (2 + 7) ( + 1)(2 + 7) 6 (63 + 452 + 110 + 86)N + 2 ( + 1) (2 + 7) 2(22 + 9 + 13) 3 + M+ ( + 1)(2 + 3)(2 + 7) ( + 1)(2 + 7) 6(2 + 2) () () =
(M, N ) 5 (M, N ). (2 + 7)(2 + 5) 3 Lemma 2.3 is completely proved.
()
()
Lemma 2.4. For coefficients of polynomial Bn (x) = Bn (x; M, N )(n ∈ Z+ ) Bn() (x) = k(Bn() )x n + l(Bn() )x n−2 + m(Bn() )x n−4 + · · ·
(2.31)
the following representations are valid: (i) k(Bn ) = (n) (M, N )k(Rn() ) (n ∈ Z+ ), (ii) l(Bn ) =
8 (n − 3)(n − 2) ( + 1)3 2 (2 + 3) (n + 2 + 1)(n + 2 + 3) × n!(n − 2)! 2(n − 2)(n + 2 − 1) (n + 2 + 2) +N ( + 1)(2 + 3) (n − 1)! 4 (n + 2 + 1) +M l(Rn() ) (n2), (2 + 2) n!
(2.32)
(n) (M, N ) + MN
(2.33)
and (iii)
m(Bn() ) = (n) (M, N ) + MN
16 ( + 1)3 2 (2 + 3) (n + 2 + 1)(n + 2 + 3) 2 [n − 5n + 2( + 6)] × (n)!(n − 2)! 4(n + 2 + 2) +N [n2 + (2 − 3)n − 2(2 − 3)] ( + 1)(2 + 3)(n − 1)! (n + 2 + 1) 8 m(Rn() ) (n4), +M (2 + 2) (n)!
()
()
()
(2.34)
where k(Rn ), l(Rn ), m(Rn ), (n) (M, N ) are defined in (2.3)–(2.5) and (2.16)–(2.19).
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
85
Proof. We substitute (2.2) and (2.31) in (2.9) k(Bn() )x n + l(Bn() )x n−2 + m(Bn() )x n−4 + · · · = an (x 4 − 2x 2 + 1)[(n − 3)4 k(Rn() )x n−4 + (n − 5)4 l(Rn() )x n−6 +(n − 7)4 m(Rn() )x n−8 + · · ·] +bn (1 − x 2 )[(n − 1)nk(Rn() )x n−2 + (n − 3)(n − 2)l(Rn() )x n−4 +(n − 5)(n − 4)m(Rn() )x n−6 + · · ·] +cn [k(Rn() )x n + l(Rn() )x n−2 + m(Rn() )x n−4 + · · ·]. The comparison of the coefficients of
xn
(2.35)
in the above relation yields
k(Bn() ) = [(n − 3)4 an − (n − 1)nbn + cn ]k(Rn() ) and substituting an , bn , cn for (2.10)–(2.13), we get (n − 3)4 an − (n − 1)nbn + cn 1 (n − 3)4 (n + 2 + 1)(n + 2 + 3) = MN 2 (n − 2)!n! ( + 1)3 ( + 2) (2 + 3) (n − 3)4 (n + 2 + 2) 1 +N (n − 1)! 2( + 1)3 (2 + 3) 1 (n − 1)n(n − 2)(n + 2 + 3)(n + 2 + 2) +N 2( + 1)( + 3)(2 + 3) (n − 1)! 1 (n + 2 + 4) −N 2( + 1)3 (2 + 3) (n − 3)! 2 (n − 1)n(n + 2 + 1) +M + 1, (2 + 3) n! It follows from (2.16) to (2.19) that (n − 3)4 an − (n − 1)nbn + cn = (n) (M, N ), and we get relation (2.32). By (2.3) and (2.4) it is easily to see that k(Rn() ) = −
2(2n + 2 − 1) l(Rn() ). (n − 1)n
Thus, comparing the coefficients at x n−2 (n2) in relation (2.35), one obtains l(Bn() ) = an [(n − 5)4 l(Rn() ) − 2(n − 3)4 k(Rn() )] +bn [(n − 1)nk(Rn() ) − (n − 3)(n − 2)l(Rn() )] + cn l(Rn() ) = l(Rn() ){[(n − 5)4 + 4(n − 3)(n − 2)(2n + 2 − 1)]an −[(n − 3)(n − 2) + 2(2n + 2 − 1)]bn + cn } = {[(n − 3)4 an − (n − 1)nbn + cn ] + 8( + 2)(n − 3)(n − 2)an −4( + 1)bn }l(Rn() ).
(2.36)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
By (2.36), we get l(Bn() ) = [ (n) (M, N ) + 8( + 2)(n − 3)(n − 2)an − 4( + 1)bn ]l(Rn() ).
(2.37)
A consequence of (2.10), (2.11) and (2.13) is the following: 8( + 2)(n − 3)(n − 2)an − 4( + 1)bn 8(n − 3)(n − 2) (n + 2 + 1)(n + 2 + 3) = MN (n − 2)!n! ( + 1)3 2 (2 + 3) (n + 2 + 2) 2( + 2)( + 3) (n − 2)(n + 2 − 1) +N ( + 1)3 (2 + 3) (n − 1)! 8( + 1) (n + 2 + 1) . +M n! (2 + 3) Substituting this expression in (2.37) we obtain (2.33). () For finding the coefficient m(Bn ), we compare coefficients at x n−4 (n 4) in (2.35): m(Bn() ) = (n − 7)4 an m(Rn() ) − 2(n − 5)4 an l(Rn() ) + (n − 3)4 an k(Rn() )
+(n − 3)(n − 2)bn l(Rn() ) − (n − 5)(n − 4)bn m(Rn() ) + cn m(Rn() ).
Note by definition (n − 7)4 = 0 From (2.3) to (2.5), one has ()
k(Rn ) () m(Rn ) () l(Rn ) () m(Rn )
=
(n = 4, 5, 6, 7) and (n − 5)4 = 0
(n = 4, 5).
8(2n + 2 − 1)(2n + 2 − 3) , (n − 3)4
=−
4(2n + 2 − 3) (n − 3)(n − 2)
(n 4).
Consequently, m(Bn() ) = {[(n − 7)4 + 8(2n + 2 − 3)(n2 − 7n + 2 + 19)]an −[n2 − n + 8( + 1)]bn + cn }m(Rn() ).
(2.38)
We calculate the expression in brackets. As above, by (2.10)–(2.13) and (2.36), one gets (n − 7)4 + 8(2n + 2 − 3)(n2 − 7n + 2 + 19)an − [n2 − n + 8( + 1)]bn + cn = [(n − 3)4 an − (n − 1)nbn + cn ] + 16( + 2)[n2 − 5n + 2( + 6)]an − 8( + 1)bn n2 − 5n + 2( + 6) = (n) (M, N ) + 16MN( + 2) ( + 1)3 ( + 2)2 (2 + 3) (n + 2 + 1)(n + 2 + 3) × (n − 2)!n! n2 − 5n + 2( + 6) (n + 2 + 2) +16N( + 2) 2( + 1)3 (2 + 3) (n − 1)! 1 (n − 2)(n + 2 + 3)(n + 2 + 2) +8( + 1)N 2( + 1)( + 3)(2 + 3) (n − 1)! (n + 2 + 1) 2 . +8( + 1)M (2 + 3) n!
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
87
From the last relation, we deduce (n − 7)4 + 8(2n + 2 − 3)(n2 − 7n + 2 + 19)an − [n2 − n + 8( + 1)]bn + cn 16 [n2 − 5n + 2( + 6)] = (n) (M, N ) + MN ( + 1)3 2 (2 + 3) 4 (n + 2 + 1)(n + 2 + 3) +N × ( + 1)(2 + 3) (n − 2)!(n)! (n + 2 + 2) ×[n2 + (2 − 3)n − 2(2 − 3)] (n − 1)! 16( + 1) (n + 2 + 1) +M , (2 + 3) n! Substituting this relation in (2.38) one obtains (2.34). Lemma 2.4 is completely proved. In the next section we use the following assertion. Lemma 2.5 (Abramowitz and Stegun [1], Copson [6], Tricomi and Erdelyi [34]). For n large enough, the following asymptotic equality:
(n + ) ( − )( + − 1) 1 − 1+ =n 2 n (n + ) 1 3( + − 1)2 − + − 1 + ( − )( − − 1) 24 n2 1 . (2.39) +O n3 holds. Using representation (2.9)–(2.13) and Lemmas 2.3 and 2.5 one can get another proof of the following proposition. Lemma 2.6 (Marcellan and Osilenker [21], Foulquie Moreno et al. [10]). (case M>0, N >0). The following properties for symmetric Gegenbauer–Sobolev polynomials hold: (i) −3− 2 (n) , n n 15
(ii) n() (x)| C (1 − x 2 )− 2 − 41 , − 1 ; |B 2 1 n() (x)| C −1 < < − |B (n ∈ Z+ , −1 < x < 1), 2 (iii) max
−1 x 1
n() (x)| C (n + 1) 2 (n ∈ Z+ ), |B 1
(iv) n() } (1)| n−− 2 , n() (1)| n−− 2 , |{B |B 3
7
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
where the constants C , are independent of x ∈ (−1, 1), and n ∈ Z+ (in formula ii) and n ∈ Z+ (in formula iii). n() (x) have Remark. It should be noted that the Gegenbauer–Sobolev orthonormal polynomials B n() (x) (see, some properties other than the corresponding Gegenbauer orthonormal polynomials R for example [2–5,9,10,15,17,18,20,21,28–31]). We mention the following properties only. 1. The inner product (2.1) cannot be obtained by a weight function since x, x = 1, x 2 . This implies that well-known results for classical orthogonal polynomials, depending on the existence of a weight function do not longer hold. The well-known three-term recurrence relation for orthogonal polynomials does not hold in this case, but has to be replaced by a seven-term relation [5], in addition, this order is minimal [9]. n() (x) lying For n large enough, there exists exactly one pair of real zeros −n and n of B outside (−1, 1) [2,5]. n() (x) are eigenfunctions of a class of linear differential operators, usually of in2. Polynomials B finite order. In the case that is a nonnegative integer, this class contains a differential operator of finite order. This order is 4 + 10 [17,3]). n() (x) are eigenfunctions of differential operator of second We recall that polynomials R order [33], 3. As well-known [33] n() (±1) n+ 21 , {R n } (±1) n+ 25 . R These asymptotics are different from (iv). 3. Asymptotics of the averaged Turan determinant and generalized trace formula for symmetric Gegenbauer–Sobolev orthogonal polynomials As known [5], symmetric Gegenbauer–Sobolev orthonormal polynomials satisfy a nine-term recurrence relation (note that a degree N in (1.2) is even) ()
()
()
()
()
() () () n() (x) = B (x 2 − 1)2 B n+4 n+4 (x) + n+2 Bn+2 (x) + n Bn (x) + n Bn−2 (x) () () (x)(n ∈ Z+ ; B −s (x) = 0, s = 1, 2, . . . ; s() = 0, +(n) B n−4 ()
()
s = 0, 1, 2, 3; 0 = 1 = 0).
(3.1)
Our main aim is to calculate the recurrence coefficients and to prove that they are bounded variation. Let n() (x) = k(B n() )x n + l(B n() )x n−2 + m(B n() )x n−4 + · · · . B
(3.2)
By (2.14) and (2.31) one gets (n) (n) () () () n() ) = (n) () () k(B n k(Bn ), l(Bn ) = n l(Bn ), m(Bn ) = n m(Bn ),
where (n) n are defined in Lemma 2.3.
(3.3)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
89
Comparing coefficients at x n+4 , x n+2 , x n on both sides of (3.1) and using (3.2), (3.3), we obtain the following. Lemma 3.1. For the recursion coefficients of the symmetric band matrix associated with symmetn() }(n ∈ Z+ ), the following representations ric Gegenbauer–Sobolev orthonormal polynomials {B are valid: (i) ()
(n) n
k(Bn )
n+4
(n+4)
k(Bn+4 )
(n) n
k(Bn )
(n+2)
k(Bn+2 )
()
n+4 =
(3.4)
,
()
(ii) () n+2
=
()
n+2
()
()
l(Bn ) ()
k(Bn )
()
l(Bn+4 )
−
()
k(Bn+4 )
−2 ,
(3.5)
(iii) (n) =
()
m(Bn ) ()
k(Bn ) −
()
−
m(Bn+4 ) ()
k(B ) n+4 () l(Bn+2 ) l(Bn() ) ()
k(Bn+2 )
()
k(Bn )
−2
()
l(Bn ) ()
k(Bn )
+1
()
−
l(Bn+4 ) ()
k(Bn+4 )
−2 .
(3.6)
Lemma 3.2. For the sequence (n) (M, N ), defined by formulas (2.16)–(2.19), the following relations: (i) 8( + 2) 1
n() (M, N ) = 1 − , (3.7) + O 2 () n n
n+2 (M, N ) (ii) ()
n+2 (M, N )
8( + 2) =1+ +O () n
n (M, N )
1 n2
(3.8)
,
(iii)
(n) (M, N )
16( + 2) +O =1− () n
n+4 (M, N )
1 n2
,
(3.9)
(iv) ()
n+4 (M, N )
(n) (M, N )
hold.
=1+
16( + 2) +O n
1 n2
(3.10)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
We prove equality (3.8) only because the other ones can be deduced from (3.8). Taking into account (2.16), we find ()
n+2 (M, N )
n() (M, N ) =1+
()
()
()
()
MN[ n+2 − (n) ] + N[ n+2 − n ] + M[ n+2 − (n) ] ()
MN (n) + N n + M (n) + 1
(3.11)
.
It is not difficult to see that from (2.17) to (2.19), we have ()
n+2 − n() =
(n − 1)n (n + 2 + 1)(n + 2 + 3) 2 (n!)2 ( + 1)3 ( + 2) (2 + 3) 3 2 2 ×[8( + 2)n + 12(2 + 5 + 2)n +4(83 + 302 + 35 + 14)n + 4( + 1)( + 2)(2 + 1)(2 + 3)],
()
n+2 − n() =
(n + 2 + 2) 1 [4( + 2)( + 3)n3 (n − 1)! 2( + 1)3 (2 + 3) +2(63 + 332 + 51 − 2)n2 + 2(84 + 243 + 452 + 29 − 30)n +8( + 1)(2 + 3)]
and ()
n+2 − (n) =
4( + 1) (n + 2 + 1) (2n + 2 + 1). (2 + 3) n!
Putting the last three relations in (3.11) and using (2.39) we obtain (3.8). This completes the proof of Lemma 3.2. Lemma 3.3. For the ratio of the normalized multipliers, the following relations (n) 1 6 + 15 n , +O =1+ 2 (n+2) n n n+2 (n) n (n+4)
n+4
=1+
2(6 + 15) +O n
1 n2
(3.12)
(3.13)
hold. Proof. We prove relation (3.12). By (2.15), one has (n) n (n+2)
n+2 Since
=
2n + 2 + 1 2n + 2 + 5
() (M, N ) (n + 1)(n + 2) n+4 (n + 2 + 1)(n + 2 + 2) (n) (M, N )
√ 1 + x = 1 + 21 x + O(x 2 )(x → 0),
(n ∈ Z+ ).
(3.14)
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
91
we see that 1 2n + 2 + 1 1 =1− +O , 2n + 2 + 5 n2 n (n + 1)(n + 2) 2 1 =1− +O (n + 2 + 1)(n + 2 + 2) n n2 and by (3.10), (3.14) () (M, N ) n+4
8( + 2) =1+ +O () n
n (M, N )
1 n2
,
we deduce (3.12). Relation (3.13) follows from (3.12). Lemma 3.3 is completely proved.
()
Lemma 3.4. For the ratio of the leading coefficients of polynomial Bn (x) (see (2.31)), the following relations are valid: (i) () ()
l(Bn ) l(Rn ) 8( + 2) () + T (M, N ) , (3.15) = 1 + n () () (n − 1)n k(Rn ) k(Bn ) where 1
|Tn() (M, N )| C
n2+4
,
()
|Tn() (M, N ) − Tn+1 (M, N )|C (ii) ()
m(Bn ) ()
k(Bn )
=
()
m(Rn ) ()
k(Rn ) (n4),
1+
1 n2+5
(3.16)
,
16( + 2)[n2 − 5n + 2( + 6)] + Wn() (M, N ) (n − 3)4 (3.17)
where |Wn() (M, N )|C
1 , n2+4
()
|Wn() (M, N ) − Wn+1 (M, N )| C
1 . n2+5
The constants in (3.16) and (3.18) are independent of n ∈ Z+ . Proof. It follows from (2.32) and (2.33) that () () ˜ n() ) l(Rn ) l(B l(Bn ) = 1 + () , () ()
n (M, N ) k(Bn ) k(Rn ) where ˜ n() ) = 8MN (n − 3)(n − 2) (n + 2 + 1)(n + 2 + 3) l(B (n − 2)!n! ( + 1)3 2 (2 + 3) 4M (n + 2 + 1) 2(n − 2)(n + 2 − 1) (n + 2 + 2) + , +N ( + 1)(2 + 3) (n − 1)! (2 + 2) n!
(3.18)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
and function n() (M, N ) is defined by (2.16)–(2.19). It is not difficult to see that ˜ n() ) l(B
(n) (M, N )
=
8( + 2) + Tn() (M, N ), (n − 1)n
where Tn() (M, N ) =
()
Sn (M, N ) (n − 1)n (n) (M, N )
with 2N (n − 2)(n − 1) (n + 2 + 2) ( + 1)( + 3)(2 + 3) (n − 1)! 2 2 ×[( + 1)n + (2 + − 1)n + 4( + 1)(2 + 3)] 8M( + 3) (n + 2 + 1) − + 8( + 2). (2 + 3) (n − 2)!
Sn() (M, N ) = −
Then, ()
Tn() (M, N ) − Tn+1 (M, N )
()
()
=
()
(n − 1)Sn+1 (M, N ) (n) (M, N ) − (n + 1)Sn (M, N ) n+1 (M, N ) ()
(n − 1)3 (n) (M, N ) n+1 (M, N )
.
()
Using representation of (n) (M, N ), Sn (M, N ) and relation (2.39), we get estimates (3.16). Taking into account (2.32) and (2.34), we obtain () () () m(Rn ) m(B ˜ n ) m(Bn ) = 1 + , ()
(n) (M, N ) k(Rn() ) k(Bn ) where m(B ˜ n() ) =
(n + 2 + 1)(n + 2 + 3) 2 16MN [n − 5n + 2( + 6)] 2 (n − 2)!n! ( + 1)3 (2 + 3) (n + 2 + 2) 4N [n2 + (2 − 3)n − 2(2 − 3)] + ( + 1)(2 + 3) (n − 1)! 8M (n + 2 + 1) + . (2 + 2) n!
Hence, ()
m(B ˜ n )
(n) (M, N )
=
16( + 2)[n2 − 5n + 2( + 6)] + Wn() (M, N ), (n − 3)4
where Wn() (M, N ) =
()
Un (M, N ) (n − 3)4 (n) (M, N )
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
93
with Un() (M, N ) 4N (n − 1)(n − 2) (n + 2 + 2) [( + 1)n4 + 2( − 1)n3 =− (n − 1)! ( + 1)( + 3)(2 + 3) −(102 + 3 + 25)n2 + (83 + 522 + 116 + 156)n −(163 + 1122 − 300 + 252)]. Thus we have ()
Wn() (M, N ) − Wn+1 (M, N ) ()
=
()
()
(n + 1)Un (M, N ) n+1 (M, N ) − (n − 3)Un+1 (M, N ) (n) (M, N ) ()
(n − 3)5 (n) (M, N ) n+1 (M, N )
,
()
and application of definition of (n) (M, N ), Un (M, N ) and formula (2.39) yields estimates (3.18). Lemma 3.4 is proved. Our main result of this section is the following ()
()
Theorem 3. For the recurrence coefficients n , (n) , n (see (3.1)) the following assertions are valid: (i) lim n() =
n→∞
1 , 16
()
1 lim (n) = − , n→∞ 4
lim n() =
n→∞
3 ; 8
()
(ii) The sequences {n }, {(n) }, {n }(n ∈ Z+ ) are bounded variation ∞
(|(n) | + |(n) | + |(n) |) < ∞.
n=0
Proof. By (2.32) ()
k(Bn ) ()
k(Bn+4 )
=
()
(n) (M, N ) k(Rn ) ()
()
n+4 (M, N ) k(Rn+4 )
(n ∈ Z+ ).
If we combine this with (2.6), (3.4), (3.9) and (3.13), then we get
1 1 () n+4 = , 1+O 16 n2 ()
and we prove our result for the sequence {n }(n ∈ Z+ ). From relations (2.7), (3.15) and (3.16), we obtain the following: () () l(Bn+4 ) l(Bn ) 1 42 + 32 + 63 +O − − 2 = −1 − 2 () () 4n n3 k(Bn ) k(B ) n+4
+En() (M, N ),
(3.19)
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
with |En() (M, N )|C
1 , n2+4
(3.20)
where the constant C > 0 is independent of n ∈ Z+ . Taking into account (2.6), (2.32) and (3.7), we obtain
() () 6 + 15 1 k(Bn )
(n) (M, N ) k(Rn ) 1 1 − + O . = = () () () 4 n n2 k(B )
(M, N ) k(R ) n+2
n+2
n+2
Now, by relations (3.5) and (3.12), one gets
6 + 5 1 1 () + E˜ n() (M, N ), +O n+2 = − 1 − 4 n n2 where |E˜ n() (M, N )|C
1 , n2+4
and we obtain the assertions of Theorem 3 for the sequence {(n) }(n ∈ Z+ ). From (2.7), (3.15) and (3.16) we deduce the following:
1 1 3 − 2 42 + 32 + 63 1 =− 1+ + E¯ n() (M, N ), +O + 2 3 () 4 2n 4 n n k(Bn+2 ) ()
l(Bn+2 )
where |E¯ n() (M, N )|C
1 n2+4
,
and the constant C > 0 is independent of n ∈ Z+ . Then, using (3.19), (3.20) and last two relations, one obtains () () () l(Bn+4 ) l(Bn+2 ) l(Bn ) − − 2 () () () k(Bn ) k(Bn+4 ) k(Bn+2 )
n 3 − 2 42 + 32 + 63 1 = + Fn() (M, N ), 1+ + + O 2 4 2n 2n n3 where |Fn() (M, N )|C
1 n2+4
,
and the constant C > 0 is independent of n ∈ Z+ . Combining this with (3.15) and (3.16), we get () () () Bn+4 l(Bn+2 ) l(Bn() ) l(Bn ) − − 2 − 2 1− () () () () k(Bn+2 ) k(Bn ) k(Bn+4 ) k(Bn ) 1 n 2 − 3 + F˜n() (M, N ), +O = − (3.21) 4 8 n2
Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
95
where |F˜n() (M, N )| C
1
(3.22)
n2+3
with the constant C > 0 independent of n ∈ Z+ . On the other hand, using (2.8), (3.17) and (3.18), we obtain
() () m(Bn ) m(Bn+4 ) n2 8 8 32 1 − = − + 2 − 3 +O + G(n) (M, N ), 4 () () 32 n n n n k(Bn ) k(Bn+4 ) where |Gn() (M, N )| C
1 n2+3
,
with the constant C > 0 independent of n ∈ Z+ . By (3.6), summing the last relations and (3.21), (3.22), we obtain 3 1 () ˜ (n) (M, N ), |G ˜ (n) (M, N )| C 1 , +G n = − + O 2 n n2+3 8 n with the constant C > 0 independent of n ∈ Z+ . () Thus we have the assertion of Theorem 3 for the sequence n . Theorem 3 is completely proved. Let us introduce the averaged Turan -determinant for the system of symmetric Gegenbauer– Sobolev orthonormal polynomials G(n) (x; ) =
n+4
()
()
()
(x)B (x)]2 − k B (x)]} k {k−4 [B k k−4 k+4
k=n+1 n+2
+
()
()
()
()
(x)B (x)B (x) − k B (x)]. k [k−2 B k k+2 k−2 k+4
k=n+1
Combining Theorem 2 with Lemma 2.6 and Theorem 3, we obtain the following statement. Theorem 4. At every point x ∈ (−1, 1) and uniformly on every closed subset of (−1, 1) the following statements are valid: 1. If for the sequence (1.8), conditions (1.12) and (1.15) are satisfied, then the following asymptotics lim G(n) (x; ) =
n→∞
3 22+5 ( + 1) 2 x (1 − 2x 2 )(1 − x 2 ) 2 − (2 + 2)
holds. 2. The generalized trace formula ∞ s() (x)]2 [4(2n+4 − 2n ) + 2(2n+2 − 2n )][B n=0
+
∞ n=0
() (x)B n() (x) [2n+2 (n+2 − n ) + 6(n+4 n+4 − n+2 n )]B n+2
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Boris P. Osilenker / Journal of Approximation Theory 141 (2006) 70 – 97
+4
∞
()
(x)B n() (x) n+4 (n+4 − n )B n+4
n=0
+2
∞
() (x)B n() (x) (n+4 n+6 − n+6 n+2 )B n+6
n=0 22+4 ( + 1)
=
(2 + 2)
x 2 (1 − x 2 ) 2 − . 5
holds. Acknowledgments This work was partially completed (the case of the Legendre polynomials) while the author visited the Departamento de Matematicas, Universidad Carlos III de Madrid for part of academic year 2002–2003. Author is grateful to Professors J. Arvesu, F. Marcellan and G. Lopez for several helpful comments. I am thankful to F. Marcellan for seeking and obtaining financial support for this period. Author is very grateful to the referee for his valuable suggestions and constructive comments, which have significantly improved the first draft of this paper. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1977. [2] J. Arvesu, R. Alvarez-Nodarse, F. Marcellan, K. Pan, Jacobi–Sobolev-type orthogonal polynomials: second-order differential equation and zeros, J. Comput. Appl. Math. 90 (1998) 135–156. [3] H. Bavinck, Differential operators having Sobolev-type Gegenbauer polynomials as eigenfunctions, J. Comput. Appl. Math. 118 (2000) 23–42. [4] H. Bavinck, H.G. Meijer, Orthogonal polynomials with respect to a symmetric inner product involving derivatives, Appl. Anal. 33 (1989) 103–117. [5] H. Bavinck, H.G. Meijer, On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations, Indag. Math. (N.S.) 1 (1990) 7–14. [6] E.T. Copson, Asymptotic Expansions, Cambridge University Press, Cambridge, 1965. [7] R. Courant, D. Hilbert, Methoden der mathematischen Physik, Berlin, 1931. [8] J. Dombrowski, P. Nevai, Orthogonal polynomials, measures and recurrence relation, SIAM J. Math. Anal. 17 (1986) 752–759. [9] W.D. Evans, L.L. Littlejohn, F. Marcellan, C. Markett, A. Ronveaux, On recurrence relations for Sobolev orthogonal polynomials, SIAM J. Math. Anal. 26 (1995) 446–467. [10] A. Foulquie Moreno, F. Marcellan, B.P. Osilenker, Estimates for polynomials orthogonal with respect to some Gegenbauer–Sobolev inner product, J. Ineq. Appl. 3 (1999) 401–419. [11] C.T. Fulton, S. Pruess, Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions, J. Math. Anal. Appl. 71 (1979) 431–462. [12] G. Gasper, On extension of Turan’s inequality to Jacobi polynomials, Duke Math. J. 38 (1971) 415–428. [13] J. Geronimo, W. Van Assche, Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991) 341–371. [14] A.A. Gonchar, On the convergence of Pade approximants for some classes of meromorphic functions, USSR Sb. 26 (1975) 555–575. [15] A. Iserles, P.E. Koch, S.P. Norsett, J.M. Sanz-Serna, Orthogonality and approximation in a Sobolev space, in: J.C. Masson, M.G. Cox (Eds.), Algorithms for Approximations, Chapman & Hall, New York, 1990, pp. 117–124. [16] S. Karlin, G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. d’Anal. Math. 8 (19601961) 1–157. [17] R. Koekoek, Differential equations for symmetric generalized ultraspherical polynomials, Trans. Amer. Math. Soc. 345 (1994) 47–72. [18] A. Krall, Hilbert space, Boundary value problems and orthogonal polynomials, Operator Theory: Advances and Applications, vol. 133, Birkhäuser, Basel, 2002.
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