GAUSSâTUR´AN QUADRATURE FORMULAS - Group for Dynamical ...
c 2009 Society for Industrial and Applied Mathematics
SIAM J. NUMER. ANAL. Vol. 47, No. 4, pp. 2638–2659
´ QUADRATURE FORMULAS: ASYMPTOTICS OF GAUSS–TURAN WEIGHTS∗ FRANZ PEHERSTORFER† Abstract. First we derive asymptotics for the derivatives of s-orthogonal polynomials and the derivatives of Cauchy principal value integrals associated with s-orthogonal polynomials. With the help of these results asymptotics for the quadrature weights of Gauss–Tur´ an quadrature formulas are obtained. So far asymptotics have been known with respect to the Chebyshev weight function only. Key words. asymptotics, s-orthogonal polynomials, Cauchy principal values, derivatives, Gauss–Tur´ an quadrature formulas, Szeg¨ o function AMS subject classifications. 65D32, 42C05 DOI. 10.1137/070695794
1. Introduction. In 1950 Tur´ an [29] introduced numerical quadrature formulas of the form n 2s (1.1) Aj,ν,n f (j) (xν,n ) + Rn,s (f ), f (t)dμ(t) = ν=1 j=0
−1 < xn,n < xn−1,n < · · · < x1,n < 1, exact for polynomials of degree less or equal 2(s + 1)n − 1, i.e., Rn,s (p) = 0 for p ∈ P2(s+1)n−1 ,
(1.2)
where Pm denotes the set of polynomials of degree ≤ m and μ(t) denotes a positive measure supported on [−1, 1]. Such quadrature formulas are called Gauss–Tur´ an quadrature formulas, nowadays. He has shown that (1.2) holds if and only if the nodes xν,n are chosen such that (1.3)
Tˆn (x; dμ) := Tˆn,s (x; dμ) :=
n
(x − xν,n )
ν=1
minimizes
xn + an−1 xn−1 + · · · + a0
2s+2
dμ(x)
or in other words, is a monic minimal polynomial with respect to the L2s+2 (dμ) norm which is equivalent to the conditions that 2s+1 (1.4) dμ(x) = 0, k = 0, . . . , n − 1. xk Tˆn,s (x; dμ) Orthogonality property (1.4) is called s-orthogonality. ∗ Received
by the editors June 29, 2007; accepted for publication (in revised form) March 17, 2009; published electronically July 22, 2009. This work was supported by the Austrian Science Fund FWF, project P20413-N18. http://www.siam.org/journals/sinum/47-4/69579.html † Abteilung f¨ ur Dynamische Systeme und Approximationstheorie, Institut f¨ ur Analysis, J.K. Universit¨ at Linz, Altenberger Str., 69, 4040 Linz, Austria ([email protected]). 2638
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2639
Naturally, and as pointed out by Tur´ an [30] himself, we would like to know the behavior of the quadrature weights Aj,ν,n , at least asymptotically. Note that the Aj,ν,n ’s can be represented in the form (1.5) Aj,ν,n = hj,ν,n (x)dμ(x), where the hj,ν,n (x) are the fundamental polynomials of Hermite interpolation. But the representation of the fundamental polynomials in terms of the fundamental Lagrange interpolation polynomials becomes rather involved at an early stage such that it is difficult to work with. So far with the help of (1.5) only the positivity of the even weights A2j,ν,n could be shown [12, 19]. By the way, this is the only information on the quadrature weights available for general weight functions. Other information is available only for the Chebyshev weight function, which will be discussed below. For s = 0, that is, when (1.1) becomes the Gauss-quadrature formula, the behavior is well known and given by the beautiful so-called circle theorem. More precisely, suppose that (1.6)
dμ(x) =
w(x) √ dx for x ∈ [−1, 1] π 1 − x2
and that w(x) is positive and from Lip γ, γ ∈ (0, 1], on [−1, 1]. Then the Gaussian weights AG ν,n := A0,ν,n associated with nodes xν,n ∈ [−1 + , 1 − ], > 0, behave like nAG ν,n = w(xν,n )(1 + o(1)).
(1.7)
This nice relation dates back to Szeg¨ o [27]. For a recent discussion of the circle theorem, see [5, 28]. Let us mention that now asymptotics for quadrature weights are available for positive interpolation quadrature formulas [21] even. Obviously, it would be of foremost interest to know whether there are similar asymptotic laws for the quadrature weights of Gauss–Tur´ an quadrature formulas. But so far, we have information on the weights for only the celebrated Chebyshev √ weight function dμ(x) = 1/ 1 − x2 dx [7, 9, 13, 14]. Recall that in this special case the s-orthogonal polynomial is for every s ∈ N0 := N ∪ {0} the Chebyshev polynomial, i.e., Tˆn,s (x; dμ) = Tˆn (x) = 2−n+1 cos(n arccos x), x ∈ [−1, 1]. Using the rich structure of the Chebyshev polynomials, Kis has shown in a lovely paper [9] that for any polynomial p ∈ P2(s+1)n−1
+1
(1.8) −1
s n p(x) 1 Sj d2j √ dx = (p(cos ϕν ))ϕν = (2ν−1)π , 2n n(s!)2 j=0 (2n)2j ν=1 dϕ2j π 1 − x2 ν
where Ss−j , j = 0, . . . , s, denote the elementary symmetric polynomials evaluated at 12 , 22 , . . . , s2 , i.e., Ss = 1, Ss−1 = 12 + 22 + · · · + s2 , . . . , S0 = 12 22 . . . s2 . For instance, transforming back we obtain for s = 1 that for any p ∈ P4n−1 n +1 n 1 p(x) xν,n √ dx = p(xν,n ) − p (xν,n ) 2 n 4n2 −1 π 1 − x ν=1 ν=1 n
1 − x2ν,n + p (xν,n ) , 4n2 ν=1
2640
FRANZ PEHERSTORFER
where xν,n = cos (2ν−1)π are the zeros of Tˆn (x). Denoting the weights of the Gauss– 2n Tur´ an quadrature formula with respect to the Chebyshev weight function by AC j,ν,n , one we may derive from (1.8), their asymptotic behavior, using (3.28) below in conjunction with induction arguments on the order of AC 2s−j,ν,n ; see pp. 545–546 and Theorem 4.1 in [14] (in Theorem 4.1 for s = 2r − 1, bs (x) should be replaced by −r(2r − 1)x(1 − x2 )r−1 ). Hence AC 2j,ν,n
(1.9)
j
Sj 1 − x2ν,n 1 = 1+O 2 2j 2j+1 n (s!) 2 n
and (1.10)
AC 2j−1,ν,n
j−1
(−Sj )j(2j − 1)xν,n 1 − x2ν,n 1 = 1+O . 2 2j 2j+1 n (s!) 2 n
For error estimates of Gauss–Tur´an quadrature formulas with Chebyshev nodes, see [4, 17, 22]. Concerning numerical constructions, see [6, 8, 18, 24, 26]. A survey on Gauss–Tur´ an quadrature formulas is given in [16]. Let us roughly outline how we obtain asymptotics for the quadrature weights of the Gauss–Tur´ an quadrature formulas. Put
2s+1 Tn (x; μ) (1.11) , Ων,2s+1 (x) := Ων (x) := x − xν where Tn (x; μ) = dn Tˆn (x; μ), dn ∈ R\{0}, is some suitable normalization of Tˆn (x; μ) (see (3.3)), and set (Tn (x; μ))2s+1 − (Tn (t; μ))2s+1 (1.12) dμ(t). I2s+1 (x) := I(x) := x−t Stancu [25] has shown that (1.13)
dj A˜2s−j,ν := j!(2s − j)!A2s−j,ν = j dx
I(x) Ων (x)
. x=xν
Next we observe that (1.14)
m−1 (m−j) I (m) (x) m Ων (x) ˜ ˜ A2s−m,ν (x) = A2s−j,ν (x), − j Ων (x) Ω (x) ν j=0
which can be proved by induction arguments (see Lemma 6.2 below) and that
+1 2s+1 Tn (t; μ) dm (m) dμ(t) (1.15) I (xν ) = m − dx t−x −1 x=xν if (1.16)
dm dxm
+1 dμ(t) − t−x −1
exists. As usual − denotes the Cauchy principal value. Thus, if we know asymptotics for the derivatives of the s-orthogonal polynomials and for the Cauchy principal values
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2641
from (1.15), then the quadrature weights can be calculated by (1.14) successively for m = 2s, 2s − 1, . . . , 1. In fact, this is how we obtain the general asymptotic form of the quadrature weights with respect to weight functions of the form (1.6), where w > 0 on [−1, 1] and is sufficiently smooth. Recently for weight functions of this type, asymptotics for s-orthogonal polynomials (in fact, for Lp -minimal polynomials, p ∈ (1, ∞)) have been obtained by Kro´o and Peherstorfer [10]. They build the basis for the derivation of asymptotics for the derivatives of the s-orthogonal polynomials, given in section 3. Concerning properties of zeros of s-orthogonal polynomials, see [20]. One of the main challenges is the determination of asymptotics of the Cauchyprincipal value integrals from (1.15). We first demonstrate in section 2, Theorem 2.1, that for the Bernstein–Szeg¨o weights they can be given explicitly even. With the help of this result, in section 4 we derive asymptotics for the Cauchy principal values with respect to general weight functions of the form (1.6), where for convenience w(x) = v 2s+2 (x) is considered. Finally in section 5 we present the asymptotics for the quadrature weights. 2. Bernstein–Szeg¨ o weight functions—Cauchy principal values. Let gk (z) be a real polynomial of degree k which has all its zeros in |z| > 1, and put ρk (cos ϕ) = gk (eiϕ |2 . (2.1) Note that ρk (cos ϕ) is a positive cosine polynomial of degree k. Bernstein [2, 3] (see also [1, pp. 278–280]) has shown that for p ∈ [1, ∞), n ≥ k, p
Tˆn,p cos ϕ; 1/ρk2 = cn Re e−inϕ gk eiϕ
= cn Re ei(n−k)ϕ gk∗ eiϕ , where cn ∈ R\{0} and gk∗ (z) = z k gk ( 1z ) denote the so-called reciprocal polynomial. In particular, by considering gk2 (z) instead of gk (z) and p = 2s + 2, s ∈ N0 , we obtain that
(2.2) = cn Re e−inϕ gk2 eiϕ . Tˆn,2s+2 cos ϕ; 1/ρ2s+2 k Notation 2.1. (a) Let gk (z) be a real polynomial of degree k which has all its zeros in |z| > 1. Then we put, x = cos ϕ,
Tn x; 1/ρ2k := Tn (x) := Re e−inϕ gk2 eiϕ . (2.3) Note that, x = cos ϕ, z = eiϕ , (2.4)
Tn (x) g ∗ (z) = Re z n−k k = cos Φk (ϕ), ρk (x) gk (z)
where Φk (ϕ) = arg z n−k
(2.5)
gk∗ (z) . gk (z)
(b) Let m ∈ N. We put (2.6)
sm (ϕ) =
m j=0
dj,m sin(m − j)ϕ = Im
⎧ m ⎨ ⎩
j=0
dj,m z m−j
⎫ ⎬ ⎭
,
2642
FRANZ PEHERSTORFER
where the dj,m ’s are the Fourier coefficients of (cos ϕ)m , i.e.,
(2.7)
2m−1 (cos ϕ)m =
m
dj,m cos(m − j)ϕ = Re
j=0
⎧ m ⎨ ⎩
dj,m z m−j
j=0
⎫ ⎬ ⎭
or explicitly that (2.8)
dm−2j,m
m = and dm−(2j+1),m = 0. j
Notation 2.2. For convenience we will use the differential operator D defined by Df (z) = iz
(2.9)
d f (z). dz
In particular, we have
dj iϕ = Dj f eiϕ . f e dϕj Theorem 2.1. Let m ∈ N, y ∈ C\[−1, 1], z = y − √ is chosen such that |z| < 1, and put for |z| ≤ 1 Hm (z) =
(2.10)
y 2 − 1, where that branch of
m−j g ∗ (z) dj,m z n−k k . gk (z) j=0
m
Then for |z| < 1 2 Hm (z) = y − 1
(2.11)
+1
−1
(Tn (t))m 1 dt √ m y − t ρk (t) π 1 − t2
and, ϕ ∈ [0, 2π], +1 dt (Tn (t))m 1 √ Im Hm (eiϕ ) = sin ϕ − . m (t) t − cos ϕ ρ π 1 − t2 −1 k
(2.12) Furthermore, (2.13)
dj dϕj
+1 dt (Tn (t))m 1 √ sin ϕ − = (sm ◦ Φk )(j) (ϕ). 2 t − cos ϕ ρm −1 k (t) π 1 − t
If m is odd, we have at the zeros xν = cos ψν , −1 < xn < xn−1 < · · · < x1 < 1, of Tn (x): (2.14)
s(j) m (Φk (ψν ))
(j) (−1)ν−1 sm ( π2 ) for j even, = 0 for j odd.
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2643
Proof. For the following let us point out that Hm (z) is analytic on the closed unit disk |z| ≤ 1. By Schwarz formula
(2.15)
2π
eiϕ + z ReHm eiϕ dϕ iϕ e −z 0
2 π ReHm eiϕ 1−z = dϕ 2 π 0 1 + z − 2z cos ϕ
m Tn (t) y 2 − 1 +1 1 dt √ = π y − t ρ (t) 1 − t2 k −1 =: y 2 − 1Q0 (y),
Hm (z) =
1 2π
where we used the facts that by (2.4) and (2.7)
m
Tn (cos ϕ) ReHm eiϕ = ; ρk (cos ϕ) moreover, ReHm (ei(π+ϕ) ) = ReHm (ei(π−ϕ) ) and that by the choice of the branch of square root
1 1 −z . y2 − 1 = 2 z This proves relation (2.11). Taking into consideration ± ± y 2 − 1 (x) = ±i 1 − x2 and y − y 2 − 1 (cos θ) = e∓iθ , we obtain by the definition of Q0 (y)
(2.16)
1 −ImHm eiϕ = sin ϕ lim (Q0 (x + i) + Q0 (x − i)) →0+ 2
m +1 Tn (t) 1 dt √ = sin ϕ − x − t ρk (t) π 1 − t2 −1 = −(sm ◦ Φk )(ϕ),
where the last equality follows by (2.10), (2.5), and (2.6). Thus relation (2.12) is proved. Since Hm (z) is analytic on |z| ≤ 1, we may moreover differentiate Hm and obtain, in conjunction with (2.16), relation (2.13):
dj Im Dj Hm (z) z=eiϕ = Im Hm eiϕ j dϕ = (sm ◦ Φk )(j) (ϕ). Concerning the last assertion (2.13), at the zeros xν = cos ψν of Tn (x), we have by (2.4) that π Φk (ψν ) = (2κ + 1) , 2
κ ∈ Z,
2644
FRANZ PEHERSTORFER
from which for j odd the vanishing property follows easily with the help of the first relation in (2.5) and (2.6), taking into account (2.8) and that m is odd. Now, if |zj | < 1,
z − zj z − zj d arg = Im D ln dϕ 1 − z¯j z 1 − z¯j z
z − zj d 1 − |zj |2 = Re z ln > 0. = iϕ dz 1 − z¯j z |e − zj |2 Thus Φk (ϕ) is strictly monotone increasing, with cos Φk (0) = Tn (1) > 0, and thus we may assume that Φk (xν ) = (2ν − 1) π2 , ν = 1, . . . , n. By the first relation from (2.6) and m odd it follows that sm (Φk (xν )) = (−1)ν−1 sm ( π2 ). Proceeding analogously for the even derivatives gives the assertion. For the special case ρk ≡ 1, we obtain
+1 dt Tnm (t) dj √ sin ϕ − dϕj t − cos ϕ π 1 − t2 ϕ= (2ν−1)π −1 2n (2.17) ν−1 j (j) π , = (−1) n sm 2 where Tn (t) = 2n−1 Tˆn (t), which can be deduced from the well-known relation (2.18)
+1 sin ϕ − −1
dt Tn (t) √ = sin nϕ t − cos ϕ π 1 − t2
and (2.6) and (2.7) also. In what follows, we need Lemma 2.1. Lemma 2.1. (a) Let m ∈ N. At the zeros xν = cos ψν of Tn (x) there holds
(2.19)
+1 dj (Tn (t))m 1 dt √ sin ϕ − dϕj t − cos ϕ ρm 1 − t2 ϕ=ψν −1 k (t)
+1 dt (Tn (t))m − (Tn (cos ϕ))m 1 dj √ sin ϕ = dϕj t − cos ϕ ρm 1 − t2 ϕ=ψν −1 k (t)
for j = 0, . . . , m − 1. (b) Let s ∈ N0 , n ≥ k. The following relation holds for x ∈ [−1, 1]: +1 1 (Tn (t))2s+1 dt √ − 2s+2 t−x ρk (t) 1 − t2 −1 +1 1 (Tn (t))2s+1 dt 1 √ − = . ρk (x) −1 t−x ρ2s+1 (t) 1 − t2 k Proof. (a) Obviously, +1 dt dt (Tn (t))m 1 (Tn (t))m − (Tn (cos ϕ))m 1 √ √ sin ϕ − = sin ϕ m t − cos ϕ ρk (t) 1 − t2 t − cos ϕ ρm (t) 1 − t2 −1 k +1 1 dt sin ϕ √ + (Tn (cos ϕ))m − . m t − cos ϕ ρk (t) 1 − t2 −1
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2645
Since Tn (cos ψν ) = 0, it follows by the product rule that the jth derivative of the second summand at ϕ = ψν is zero for j = 0, 1, . . . , m − 1, which proves part (a). Concerning part (b), since, by (2.2) √ and (2.3), Tn is minimal with respect to the L2s+2 (dμ)-norm, where dμ = 1/ρ2s+2 1 − t2 , it follows by its orthogonality property k (1.4) (note that (ρk (x) − ρk (t))/(x − t) ∈ Pk−1 ) that +1 ρk (x) − ρk (t) (Tn (t))2s+1 dt √ = 0, 2s+2 x − t ρ (t) 1 − t2 −1 k which is the stated relation. 3. Asymptotics for derivatives of s-orthogonal polynomials. First we need some notation. m+α Notation 3.1. By C2π , m ∈ N0 , α ∈ (0, 1), we denote the space of 2π-periodic functions which are m-times differentiable on [0, 2π] with the mth derivative in Lip α on [0, 2π]. Let v(cos ϕ) be positive and Lip α, 0 < α < 1, on [0, π], and let the so-called Szeg¨ o-function π(z; v(cos ϕ)) := π(z) of v(cos ϕ) be defined on the open unit disk by 2π iϕ e +z 1 log v(cos ϕ)dϕ (3.1) . π(z; v(cos ϕ)) := π(z) := exp − 4π 0 eiϕ − z Note that by the assumptions on v, the boundary values limr→1− π(reiϕ ) =: π(eiϕ ) exist; moreover, 2 v(cos ϕ) = 1/ π eiϕ . (3.2) Furthermore, for convenience, let us introduce the following normalized s-orthogonal polynomials, p = 2s + 2: 1/p p w(x) ˆ 1/p ˆ Tn,p (x) = λp Tn,p (x)/ (3.3) dx , Tn,p (x) √ 1 − x2 [−1,1] where (3.4)
Γ 12 Γ p+1 2
λp = Γ p2 + 1
(Tn,p is denoted in [10] by T˜n,p ). Thus for ρk (cos ϕ) given by ρk (cos ϕ) = |gk (eiϕ )|2 , we obtain by the normalization x = cos ϕ,
= Re e−inϕ gk2 e−iϕ = Tn (x). (3.5) Tn,2s+2 x; 1/ρ2s+2 k l+α , l ∈ N0 , α ∈ (0, 1), and let π(z) be the associated Lemma 3.1. Let v(cos ϕ) ∈ C2π Szeg¨ o-function. There exists a sequence of polynomials (gk (z)), gk (z) of degree k such that uniformly on |z| ≤ 1 for j = 0, 1, . . . , l,
1 (gk )(j) (z) − π (j) (z) = O (3.6) . k l+α−j
Furthermore, for each k ≥ k0 all zeros of gk (z) are in |z| > 1, and the zeros of gk∗ have no accumulation points on |z| = 1. Moreover, putting 2 ρk (cos ϕ) = gk eiϕ , (3.7)
2646
FRANZ PEHERSTORFER
we have for j = 0, . . . , l, n > k,
(3.8)
dj dj Tn cos ϕ; 1/ρk2s+2 = Re e−inϕ gk2 eiϕ j j dϕ dϕ
−inϕ 2 iϕ dj 1 = Re e π e . +O dϕj k l+α−j
m+α . Proof. By the assumptions on v(cos ϕ), it follows that log v(cos ϕ) ∈ C2π m (m) Hence, see [31, p. 121], the conjugate function log v(cos ϕ) ∈ C2π and (log v(cos ϕ)) m+α ∈ Lip α, and therefore π(eiϕ ; v(cos ϕ) ∈ C2π . Thus, see, e.g., [31, p. 67], there are polynomials gk (z) which satisfy (gk )(j) (z) −
1 (j) π (z) = O kl+α−j . Since π(z) is analytic on |z| < 1 and has, recall (3.1), no zero on |z| ≤ 1, it follows by the maximum principle that
(3.9)
0 < c1 ≤ |πp (z)| ≤ c2 for |z| ≤ 1
which implies, by (3.6), the assertion on the zeros. Relation (3.8) now follows with the help of (3.6) and (3.5).
1 , where l ∈ Lemma 3.2. Let tm ∈ Pm be such that ||tm ||∞,[−1,1] = O ml+α N0 , α ∈ [0, 1). Then the following statements hold:
1 dj (a) || dϕ ; j (tm (cos ϕ))||∞,[0,π] = O ml+α−j
dj tm (cos ϕ)−tm (cos ψ) 1 (b) || dϕj . )||∞,[0,π] = O ml+α−j−2 cos ϕ−cos ψ Proof. Part (a) follows immediately by Bernstein’s inequality. Concerning part (b), with the help of Markoff’s inequality, see, e.g., [11], we obtain tm (cos ϕ) − tm (cos ψ) cos ϕ − cos ψ
∞,[−π,π]
= tm (cos ξ) ∞,[−π,π] ≤ m2 ||tm (x)||∞,[−1,1] .
Hence, using Bernstein’s inequality it follows that j d tm (cos ϕ) − tm (cos ψ) j+2 dϕj ≤ m ||tm (x)||∞,[−1,1] , cos ϕ − cos ψ which is the assertion. l+α Proposition 3.1. Let v(cos ϕ) ∈ C2π , put μ = (l+α)/(2s+2), and let ρk (cos ϕ) be given as in Lemma 3.1. Then the following statement holds for 0 ≤ j < μ and n>k:
1 dj dj 2s+2 2s+2 = +O Tn cos ϕ; v Tn cos ϕ; 1/ρk (3.10) dϕj dϕj k μ−j ˙ and for 0 ≤ j < μ − 1 and n > k with k = O(n),
+1 m
Tn cos ϕ; v 2s+2 − Tnm t; v 2s+2 2s+2 dj dt v (t) √ sin ϕ dϕj cos ϕ − t 1 − t2 −1
+1 m − Tnm t; 1/ρk2s+2 Tn cos ϕ; 1/ρ2s+2 1 dt dj k √ (3.11) = sin ϕ dϕj cos ϕ − t ρk2s+2 (t) 1 − t2 −1
1 +O nμ−(j+2)
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
and
+1 m 2s+2 Tn t; v dt 2s+2 v (t) √ sin ϕ − t − cos ϕ 1 − t2 −1
+1 m
Tn t; 1/ρk2s+2 1 1 dt dj √ = sin ϕ − +O . dϕj t − cos ϕ nμ−(j+2) ρ2s+2 (t) 1 − t2 −1 k
dj dϕj (3.12)
2647
Proof. In [10] it has been shown by the Lipschitz continuity of the best approximation map that
1 Tn cos ϕ; v 2s+2 = Re e−inϕ π 2 eiϕ + O l+α 2s+2
n
1 2s+2 = Tn cos ϕ; 1/ρk +O , l+α k 2s+2 where the last equality follows by (3.8) and n > k. Applying Lemma 3.2 to
(3.13) rn (cos ϕ) = Tn cos ϕ; v 2s+2 − Tn cos ϕ; 1/ρk2s+2 , statement (3.10) follows. Concerning relation (3.11), first we claim that relation (3.11) holds when we (t) in the second integral by v 2s+2 (t). We put replace the weight 1/ρ2s+2 k (3.14)
Qn,m (x) =
m−1
Tnm−1−κ cos ϕ; v 2s+2 Tnκ cos ϕ; 1/ρk2s+2
κ=0
and write the difference of the two integrands in the form
Qn,m (cos ϕ) − Qn,m (t) rn (cos ϕ) − rn (t) (3.15) Qn,m (cos ϕ) + rn (t). cos ϕ − t cos ϕ − t Since Tn (x; v 2s+2 ) and Tn (x; 1/ρ2s+2 ) are uniformly bounded on [−1, 1], it follows by k di l i Bernstein’s inequality that l ∈ {0, . . . , m − 1} and that dϕ i Tn (cos ϕ; .) = O(n ), which i
d i gives with the help of Leibniz’s rule that dϕ i Qn,m (cos ϕ) = O(n ) and, by Lemma 3.2,
di Qn,m (cos ϕ) − Qn,m (t) (3.16) = O ni+2 . i dϕ cos ϕ − t
Thus by (3.15), (3.10), and Leibniz’s rule the claim follows. Finally, relation (3.11) follows by recalling the fact that by (3.6) and (3.7) uniformly for ϕ ∈ [0, 2π],
1 1 2s+2 (3.17) . (cos ϕ) + O =v k l+α ρ2s+2 (cos ϕ) k Concerning relation (3.12), the difference of the jth derivative of the integrals in (3.12) is by (3.11) of the form
m
+1 v 2s+2 (t) dj dt 2s+2 2s+2 m √ T cos ϕ; v − T cos ϕ; 1/ρ − n n k dϕj cos ϕ − t 1 − t2 −1
1 +O , nμ−(j+2) which gives with the help of the assertion (3.10).
2648
FRANZ PEHERSTORFER
l+α Theorem 3.1. Let v(cos ϕ) ∈ C2π , put μ = (l + α)/(2s + 2), and let π(z) be the Szeg¨ o function associated with v(cos ϕ). Then the following statements hold: (a) For j < μ, z = eiϕ ,
dj
Tn cos ϕ; v 2s+2 = Re (−i)j nj z −n π 2 (z) + i(−in)j−1 2jz −n+1 π(z)π (z) dϕj
1 +O . nμ−j (b) At the zeros xν = cos ψν of Tn (x; v 2s+2 ), there holds, 2κ + 1 < μ, (3.18)
(−1)ν+κ n2κ+1 d2κ+1
2s+2 T cos ϕ; v = n ϕ=cos ψν dϕ2κ+1 v(xν )
1 1+O n
and, 2κ < μ, v (xν ) d2κ
2s+2 ν−1+κ 2κ−1 2 T cos ϕ; v = (−1) n 2κ 1 − x n ν ϕ=cos ψν dϕ2κ v 2 (xν ) (3.19)
1 1+O . n
Proof. For abbreviation put G(z) = π 2 (z). By (3.8) and (3.10), for instance, choose k = [n/2], j
d −inϕ iϕ 1 dj
2s+2 Tn cos ϕ; v = Re e G e +O dϕj dϕj nμ−j = Re (−in)j z −n G(z) + (−in)j−1 ijz −n+1 G (z) (3.20)
1 + O nj−κ +O , nμ−j where the second equality follows by straightforward calculation. Thus part (a) is proved. Concerning (b), for j = 2κ + 1 we obtain by (a) considering the first summand at the right-hand side only that (3.21)
−n 1 d2κ+1
2s+2 κ 2κ+1 T cos ϕ; v = (−1) n G(z) + O Im z . n dϕ2κ+1 n
Now, by (3.10) and (3.8) for j = 0, at the zeros xν = cos ψν of Tn (x; v 2s+2 ), (3.22)
Re {z
−n
G(z)}z=eiψν = O
1 nμ
,
and thus by (3.23)
(Re {z −n G(z)})2 + (Im {z −n G(z)})2 = |G(z)|2
and sgn Im{z −n G(z)}z=eiψν = (−1)ν , note that Re{e−inϕ gk2 (eiϕ )} and Im{e−inϕ gk2 (eiϕ )} have strictly interlacing zeros; relation (3.18) follows.
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2649
For j = 2κ we obtain by (a) and (3.22) that
d2κ Tn cos ϕ; v 2s+2 = (−1)κ Re n2κ z −n G(z) − 2κn2κ−1 z −n+1 G (z) 2κ dϕ
2κ−2 1 +O +O n (3.24) nμ
−n+1 1 κ+1 2κ−1 2κn Re z G (z) 1 + O = (−1) . n Observing that, with the help of (3.22), at the zeros of Tn (x; v 2s+2 ), z = eiϕ , 2Re {z −n G(z)}
d Re {z −n G(z)} = O dϕ
1 nμ−1
,
and, again by (3.22),
(3.25)
d
Im e−inϕ G eiϕ ϕ=ψν Im e−inψν G eiψν dϕ
−inψν iψν −n+1 1 = Im e G e G (z) z=eiψν , + Re z O nμ−1
it follows, by differentiating (3.23) with respect to ϕ, that d
Im e−inϕ G eiϕ ϕ=ψν Im e−inψν G eiψν dϕ
1 d iϕ 2 1 G e = +O . 2 dϕ nμ−1 ϕ=ψν
(3.26)
Since
Im e−inψν G eiψν = (−1)ν G eiψν + O
(3.27)
1 nμ
where we have used (3.23) in conjunction with (3.22), the assertion follows by (3.24) taking into consideration (3.25)–(3.27) and the fact that |G(eiϕ )| = 1/v(cos ϕ). Obviously, taking a look at (3.20) and the proof, one could state (3.18) and (3.19) up to higher order, also. Remark 3.1. (−1)j (3.28)
dj (sin ϕ f (cos ϕ)) dϕj = (sin ϕ)j+1 f (j) (cos ϕ) − +
j−2
j(j + 1) (sin ϕ)j−1 cos ϕ f (j−1) (cos ϕ) 2
τν (cos ϕ, sin ϕ)f (ν) (cos ϕ),
ν=0
where τν (cos ϕ, sin ϕ) is a trigonometric polynomial of degree ≤ j − 1.
2650
FRANZ PEHERSTORFER
Corollary 3.1. (3.29)
Tn(2κ+1)
(−1)ν−1+κ n2κ+1 xν ; v 2s+2 = (sin ψν )2κ+1 v(xν )
1 1+O n
if 2κ + 1 < μ, and
(−1)ν−1+κ n2κ−1 2κ Tn(2κ) xν ; v 2s+2 = (sin ψν )2κ−1 v(xν ) (3.30)
v (xν ) 2κ − 1 xν − v(xν ) 2 1 − x2ν
1 1+O n
if 2κ < μ. Proof. First let us observe that (−1)
dj+1 dj (f (cos ϕ)) = (sin ϕ f (cos ϕ)) . dϕj+1 dϕj
By Theorem 3.1 and Remark 3.1, it follows by induction arguments that
j+1 1 Tn(j) (xν ) = O n2[ 2 ]−1 1+O (3.31) n which yields, by Remark 3.1 again, relation (3.29). Relation (3.30) follows by observing that by Remark 3.1 and (3.31), d2κ (Tn (cos ϕ))ϕ=ψν = (sin ψν )2κ Tn(2κ) (xν ) dϕ2κ (3.32)
− κ(2κ − 1)(sin ψν )2κ−2 cos ψν Tn(2κ−1) (xν )
1 + O n2κ−3 1 + O n
which gives, by (3.29) and Theorem 3.1, relation (3.30). 4. Cauchy principal values. The following result is crucial in what follows. l+α Theorem 4.1. Let s ∈ N0 , v(cos ϕ) ∈ C2π , put μ = (l + α)/(2s + 2), and let π(z) be the Szeg¨ o function associated with v(cos ϕ). Then uniformly on [0, π] for 0 ≤ j < μ − 2, +1 2s+1 2s+2 t, v Tn dt dj 2s+2 v (t) √ sin ϕ − dϕj t − cos ϕ 1 − t2 −1 (4.1)
1 dj = (v(cos ϕ)s (Φ(ϕ))) + O , 2s+1 dϕj nμ−j−2 where s2s+1 is given by (2.6) and Φ and its derivatives are given by, j = 0, . . . , 2s,
dj zn j (4.2) Φ(ϕ) = Im D ln 2 . dϕj π (z) z=eiϕ Furthermore, at the zeros xν = cos ψν of Tn (x; v 2s+2 ), we have
π 1 (j) ν−1 (j) s2s+1 (Φ(ψν )) = (−1) +O s2s+1 (4.3) . 2 nμ−j
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2651
Proof. By Proposition 3.1 we know that the left-hand side of (4.1) is for n > k, 1 ˙ with k = O(n), up to order O( nμ−j−2 ), equal to 2s+1
Tn t; 1/ρk2s+2 dt 1 √ sin ϕ − t − x 1 − t2 ρ2s+2 (t) k 2s+1 +1 Tn (t) dt 1 dj 1 √ sin ϕ − = dϕj ρk (cos ϕ) ρk (t) x − t 1 − t2 −1
2s+1 j 1 dκ j dj−κ = dμ,2s+1 κ (sin Φk (ϕ)) , j−κ κ dϕ ρk (cos ϕ) μ=0 dϕ κ=0
dj dϕj (4.4)
where the second equality follows by Lemma 2.1 and the third equality by Theo˙ ) denotes the precise order. Recall that by (2.5) rem 2.1. As usual O( Φk (ϕ) = arg z n − 2 arg gk (z) = arg z n − 2Im {ln gk (z)}. Since by (3.6) uniformly on |z| ≤ 1
D ln gk (z) = D ln π(z) + O j
1
j
k l+α−j
,
it follows moreover that uniformly for ϕ ∈ [0, π], j = 0, . . . , 2s,
1 dj Φ dj Φk (ϕ) = (ϕ) + O (4.5) , dϕj dϕj k l+α−j where Φ(ϕ) is given by (4.2), i.e., Φ(ϕ) = arg z n − 2 arg π(z). Observing that
d
1 d 2 Re =− ln gk eiϕ dϕ ρk (cos ϕ) ρk (cos ϕ) dϕ
2 =− Re D ln gk eiϕ , iϕ 2 |ρk (e )| j
d 1 it follows that the higher derivatives dϕ j ( ρ (cos ϕ) ) can be expressed with the help k of Re Dκ ln gk (z), κ ∈ {1, . . . , j}, and 1/|gk (eiϕ )|2 which implies by (3.6) again that uniformly on [0, π],
1 1 dj dj (v(cos ϕ)) + O . = dϕj ρk (cos ϕ) dϕj k l+α−j
Thus we may replace in the last line from (4.4) the argument Φk (ϕ) by Φ(ϕ) and 1/|gk (cos ϕ)|2 by v(cos ϕ), which yields relation (4.1). Concerning (4.3), we first observe that by (3.10) and (3.8), put j = 0,
Tn xν ; v 2s+2 1 0= = cos Φ(ψ ) + O , ν iϕ 2 |π(e )| nμ and thus (2l + 1)π +O Φ(ψν ) = 2
1 nμ
,
2652
FRANZ PEHERSTORFER
where l ∈ Z. Now for sufficiently large k the argument Φk is by (4.5) close to Φ; hence (see the proof of Theorem 2.1, last part), we may put l = ν. Thus
1 (2ν + 1)π (2ν + 1)π 1 (j) (j) +O s2s+1 = s2s+1 +O 2 nμ 2 nμ−j
π 1 ν−1 (j) +O s2s+1 , = (−1) 2 nμ−j where the last equality has been demonstrated at the end of the proof of Theorem 2.1. (j) Remark 4.1. Recall that s2s+1 ( π2 ) can be calculated by (2.6), in particular (2j+1) π s2s+1 ( 2 ) = 0 for j ∈ N0 . To get closed expressions it is sometimes of advantage +1 t2s+1 (j) dj √ dt π to use the relation; see (2.17), s2s+1 ( π2 ) = dϕ j (sin ϕ −−1 t−cos ϕ 1−t2 )ϕ= 2 in conjunction with Remark 3.1 and the relation ⎧
+1 2s+1 ⎨ π(2μ)! 2(s − μ) dj dt t if j = 2μ, √ s−μ − = 22(s−μ) dxj −1 t − x 1 − t2 x=0 ⎩ 0 otherwise. Proof. Only the last relation has to be shown. Since t2s−ν =
2s−ν
(2s−ν)
B2s−ν−2j T2s−ν−2j (x),
j=0
where, by [23, equation (1.148)], ⎧
⎨ 1−2(s−μ) 2(s − μ) if ν = 2μ, 2 (2s−ν) s−μ = B0 ⎩ 0 otherwise, it follows that
+1 ν ν+1
+1 2s+1 t dν dt t d − xν+1 2s−ν dt √ √ = − − t dxν −1 t − x 1 − t2 x=0 dxν t−x 1 − t2 x=0 −1
+1 2s−ν dt t dν √ (4.6) + ν xν+1 − dx t − x 1 − t2 x=0 −1 (2s−ν)
= (ν)!B0
.
Corollary 4.1. Put
+1 2s+1 2s+2 t; v Tn dt v 2s+2 (t) √ I(x) = − . t−x 1 − t2 −1
Then the following asymptotics hold if 2κ + 2 < μ: 2κ+1 (2κ)
(sin ψν )
(4.7) (2κ)
I
(xν ) = n
2κ
(2κ) v(xν )s2s+1 (Φ(ψν ))
1 1+O , n
where s2s+1 (Φ(ψν )) is given by (4.3), and, if 2κ + 3 < μ,
xν v (xν ) 1 (2κ+1) (2κ) + (κ + 1) (xν ) = (2κ + 1)I (xν ) 1+O (4.8) I . 2 v(xν ) 1 − xν n
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2653
Proof. First let us note that dj+1 j+1 (sm (Φ(ϕ))) = s(j+1) (Φ(ϕ)) (Φ (ϕ)) m dϕj+1 j j−1 + i s(j) Φ (ϕ) m (Φ(ϕ)) (Φ (ϕ)) i=0
+ O (Φ (ϕ))
j−3
.
˙ Since Φ (ϕ) = O(n), it follows that (4.9)
1 d2κ (2κ) 2κ (v(cos ϕ)sm (Φ(ϕ))) = sm (Φ(ϕ))n v(cos ϕ) 1 + O , dϕ2κ n (2κ+1)
and since sm
(4.10)
(Φ(ψν )) = 0, that
d2κ+1 (v(cos ϕ)sm (Φ(ϕ)))ϕ=ψν dϕ2κ+1
d d2κ (v(cos ϕ)) = (2κ + 1) (sm (Φ(ϕ)))ϕ=ψν 2κ dϕ ϕ=ψν dϕ 2κ−1 + O (Φ (ψν ))
1 = −(2κ + 1)v (xν ) sin ψν n2κ s(2κ) (Φ(ψ )) 1 + O . ν m n
Now by Theorem 4.1 dj dj (sin ϕ I(cos ϕ)) = (v(cos ϕ)sm (Φ(ϕ)))ϕ=ψν + O (4.11) ϕ=ψ ν dϕj dϕj
1 nμ−j−1
.
With the help of Remark 3.1, (4.9), (4.10), and induction arguments, it follows that j (4.12) I (j) (x)x=xν = O n2[ 2 ] which yields, by Remark 3.1, (4.11) in conjunction with (4.9), relation (4.7). Concerning relation (4.8), we observe that by Remark 3.1 and (4.12), (−1)
d2κ+1 (sin ϕ I(cos ϕ))ϕ=ψν = (sin ψν )2κ+2 I (2κ+1) (xν ) dϕ2κ+1 − (2κ + 1)(κ + 1)(sin ψν )2κ (cos ψν )I (2κ) (xν )
+ O n2κ−2 ,
which gives by (4.11), (4.10), and (4.7) the assertion. Again the asymptotics could be given up to higher order. 5. Asymptotics of Gauss–Tur´ an quadrature weights. The main result of the paper is now as follows. Theorem 5.1. Let s ∈ N0 , and suppose that w is positive on [−1, 1] and that m+α , with m + α > (2s + 2)2 , where m ∈ N, α ∈ [0, 1). Put w(cos ϕ) ∈ C2π 2−2s
s s
x + (2j)2 = aj,s xs−j , j=1
j=0
2654
FRANZ PEHERSTORFER
and let
n 2s w(x) f (x) √ dx = Aj,ν,n,s f (j) (xν,n ) + Rn,s (f ) π 1 − x2 −1 ν=1 j=0 +
be the Gauss–Tur´ an quadrature formula, i.e., Rn,s (p) = 0 for p ∈ P2(s+1)n−1 . Then the quadrature weights associated with nodes xν,n ∈ [−1+, 1−], where > 0 arbitrary but fixed, have the following asymptotic representation, j = 0, . . . , s:
s−j
aj,s 1 − x2ν,n w(xν,n ) 1 A2s−2j,ν,n,s = 1 + O (5.1) (s!)2 n2s+1−2j n and
A2s−2j−1,ν,n,s =
−(2s − 2j − 1) xν,n w (xν,n ) + b j,s 2 1 − x2ν,n w(xν,n )
(2s − 2j)A2s−2j,ν,n,s ,
(5.2) where bj,s = 1 for j = 0, 1 and all s ∈ N0 , and otherwise bj,s is an unknown constant which depends on j and s only, in particular not on the weight function w. Conjecture: We conjecture that for all s ∈ N0 (5.3)
bj,s = 1 for j = 0, . . . , s.
Corollary 5.1. For fixed s ∈ N0 let +1 n w(x) 2s G G (5.4) g(x)Tn,s (x; w) √ dx = Bν,n,s g(xν,n ) + Rn,s (g) π 1 − x2 −1 ν=1 G be the Gaussian quadrature formula, i.e., Rn,s (g) = 0 for g ∈ P2n−1 . Then, under the assumptions of Theorem 5.1 on the weight function w, the Gaussian weights associated with nodes xν,n ∈ [−1 + , 1 − ], > 0, satisfy
1 1 1 2s G nBν,n,s = 2s (w(xν,n )) s+1 1 + O . s 2 n
The connection between Gauss–Tur´ an quadrature formulas and (5.4) has been observed in [15]. 6. Proof of Theorem 5.1. First we need some auxiliaries concerning the representation of the quadrature weights, having in mind (1.11) and (1.14). Lemma 6.1. Let pn−1 (x) be a polynomial of degree n−1 such that for j = 1, . . . , s, (6.1)
(j) pn−1 (x) 1 2[ 2j ] = cj (x)n 1+O pn−1 (x) n
for x ∈ En ⊂ [−1, 1], where cj (x) is bounded on ∪n∈N En . Put ˜ r (x) = (pn−1 (x))r for r ∈ N. Ω Then for any r ∈ N, j = 1, . . . , s, (6.2)
j ˜ (j) Ω r (x) = O n2[ 2 ] ˜ r (x) Ω
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2655
on En , where the constant in the O() term does not depend on j and n. Now suppose that for every j = 2l, 1 ≤ j ≤ s, the function c2l (x) in (6.1) is of the form c2l (x) = cl /(c(x))2l ,
(6.3)
cl ∈ R\{0}. Then for any r ∈ N, 1 ≤ 2l ≤ s,
2l ˜ (2l) Ω n 1 r (x) = c˜l,r (6.4) 1+O for x ∈ En , ˜ r (x) c(x) n Ω where c˜l,r ∈ R. If in addition for every j = 2l + 1, 1 ≤ 2l + 1 ≤ s, c2l+1 (x) =
(6.5)
dl f (x) + el g(x) , (c(x))2l
then for any r ∈ N, 1 ≤ 2l + 1 ≤ s,
2l ˜ (2l+1) Ω 1 n (x) r ˜ (6.6) (dl,r f (x) + e˜l,r g(x)) 1 + O = for x ∈ En . ˜ c(x) n Ωr (x) Proof. Relation (6.2) follows immediately by
(6.7)
˜ (m) Ω 1 dm−1 r+1 = r+1 m−1 ((r + 1)prn−1 pn−1 ) ˜ r+1 p dx Ω m−1 m − 1 (prn−1 )(m−1−κ) p(κ+1) n−1 = (r + 1) r κ p p n−1 n−1 κ=0 =
m−1
m−(κ+1) (κ+1) O n2[ 2 ] O n2[ 2 ] ,
κ=0
where in the last equality we used the induction hypothesis with respect to m on (6.2) and (6.1). Concerning (6.4), since ˜ (2l)
2 Ω 1 d2l−2 r+1 r−1 r (r + 1)rp p = + rp p n−1 n−1 n−1 n−1 ˜ r+1 ˜ r+1 dx2l−2 Ω Ω 2l−2 κ (κ+1−ν) (ν+1) ˜ (2l−2−κ) Ω pn−1 κ pn−1 r−1 = r(r − 1) ˜ r−1 ν pn−1 pn−1 Ω κ=0 ν=0 (2l−2−κ) ˜r 2l−2 (κ+2) Ω pn−1 +r , ˜r pn−1 Ω κ=0 it follows by (6.2) and (6.1) that the summands in the first sum may reach order ˙ 2l ) only if all three indices κ, κ + 1 − ν, and ν + 1 are even. Recall that O( ˙ ) O(n denotes the precise order. Similarly the summands in the second sum may reach ˙ 2l ) only if κ is even. By induction hypothesis on (6.4) and the form of p(2j) /pn−1 O(n n−1 given by (6.1) and (6.2), the assertion follows. Equation (6.6) follows with the help of (6.7), (6.4), and induction arguments, observing that in (6.7) m − 1 − κ is odd and κ + 1 is even or conversely, since m is supposed to be odd.
2656
FRANZ PEHERSTORFER
Let
(6.8)
Ων (x) =
Tn (x; w) x − xν
2s+1 ,
where xν is a zero of Tn (x; w) and +1 dt Tn (x; w)2s+1 − Tn (t; w)2s+1 w(t) √ (6.9) . I(x) = x − t 1 − t2 −1 As already mentioned it has been shown by Stancu [25] that the weights Ai,ν are given by
I d2s−i ˜ Ai,ν := i!(2s − i)!Ai,ν = 2s−i (6.10) (xν ), dx Ων (j)
and thus if we know Ων (xν ) and I (j) (xν ) (or asymptotics of them) for j = 0, . . . , 2s, then (asymptotics of) the weights A˜2s−m,ν , m = 0, . . . , 2s, can be determined successively for 2s, 2s − 1, . . . with the help of the following relation (6.13) not observed so far. (j) Asymptotics for Ων (xν ) can be derived by Corollary 3.1 using the fact that by Taylor expansion of Tn (x; w) at the zero x = xν ,
(6.11)
Tn (x; w) x − xν
(j)
(j+1)
(xν ) =
Tn
(xν ; w) . j+1
Corollary 4.1 provides us with asymptotics for I (j) (xν ) since at the zeros xν of Tn (x; w), I (j) (xν ) = I (j) (xν ).
(6.12)
+1 1 w(t) dj √ Note that dx j (−−1 x−t 1−t2 dt) exists for j = 1, . . . , 2s, since w is supposed to be 2s-times continuously differentiable at least. Lemma 6.2. Put w(x) = v 2s+2 (x), and let m ∈ {0, . . . , 2s}, ν ∈ {1, . . . , n}, and set
I(x) dj ˜ A2s−j,ν (x) = j . dx Ων (x) Then m−1 (m−j) (x) ˜ I (m) (x) m Ων − A2s−j,ν (x). A˜2s−m,ν (x) = j Ων (x) Ω (x) ν j=0
(6.13) Furthermore,
A˜2s−m,ν (xν,n ) = O
(6.14)
1 m
n2s+1−2[ 2 ]
,
and, in particular, (6.15)
A˜2s−2l,ν (xν,n ) =
s−l
cl,s v 2s+2 (xν,n ) 1 − x2ν,n 1 1+O , 2s+1−2l n n
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2657
where the constant cs,l depends on s and l only. Furthermore,
v (xν,n ) xν,n A˜2s−2l−1,ν,n (xν ) = A˜2s−2l,ν,n (xν,n ) dl,s − el,s (6.16) , v(xν,n ) 1 − x2ν,n where the constants dl,s and el,s depend on s and l only. Proof. The relation (6.13) follows by induction arguments using the fact that −A˜2s−(m+1),ν = − A˜2s−m,ν m−1 (m−j) m Ων(m+1−j) Ω ν A˜2s−j,ν + A˜2s−(j+1),ν = j Ω Ω ν ν j=0 ⎛ ⎞
(m−j) m−1 m Ω I (x) ⎠ I (m+1) (x) m Ων − ν⎝ A˜2s−j,ν − − , Ων j=0 j Ων Ων Ων where the second expression is by induction hypothesis equal to (Ων /Ων )A˜2s−m,ν . Concerning the other statements, in Lemma 6.1 put pn−1 (x) = ˜ Ω2s+1 = Ων , and by (6.11) (j+1)
xν ; v 2s+2 Tn (j) pn−1 (xν ) = (6.17) . j+1
Tn (x;v 2s+2 ) ; x−xν
hence
Thus by Corollary√3.1 the assumptions of Lemma 6.1 are satisfied on [−1+, 1−], > 0, where c(x) = 1 − x2 . Hence, relation (6.2) from Lemma 6.1 holds. Now (6.14) follows by (6.13), using Corollary 4.1, Corollary 3.1, and induction arguments with respect to m. Concerning (6.15), let us first observe that by (6.14) and (6.2), the even terms in the sum (6.13) can be of maximal order 1/n2s+1−2l only. Now the assertion follows by relation (6.4) from Lemma 6.1, Corollary 3.1, and Corollary 4.1. The last statement follows with the help of (6.13), Corollary 4.1, (6.15), Lemma 6.1, and induction arguments using the facts that each summand in (6.13) is the product ˜ or, conversely, since m is odd now, and that, of an odd derivative and an even A, putting f (x) = v (x)/v(x) and g(x) = x/(1 − x2 ), (2l+1−2j)
Ων
Ων
A˜2s−2j,ν s−l
v 2s+2 (xν ) 1 − x2ν 1 ˜s,j f (xν ) + e˜s,j g(xν ) d = . 1 + O n2s+1−2l n
6.1. Proof of Theorem 5.1. By Lemma 6.2 only the constants in (6.15) and (6.16) remain to be determined. Recall that they do not depend on the weight function. Therefore, considering the Chebyshev weight function, i.e., v(x) ≡ 1, we obtain by (1.9) and (1.10) the constants we are looking for, up to the dl,s in front of v (xν,n )/v(xν,n ) in (6.16). To demonstrate that b0,s and b1,s in (5.2) are equal to one for arbitrary s ∈ N0 , we calculate with the help of (6.13) A˜2s,ν , . . . , A˜2s−3,ν explicitly. In fact, by what we know already, it would not be necessary to calculate the even A˜2s−2j,ν . First we note that
v(x ) (−1)ν−1 2s 1 I(xν ) ˜ ν A2s,ν = (6.18) , (x ))2s+1 2 s Ων (xν ) 22s (T 1 − xν n ν
2658
FRANZ PEHERSTORFER
where in the last equality we have used Corollary 4.1, (4.3), (6.12) and Remark 4.1. Concerning A˜2s−1,ν , we observe that cos ψν I(cos ψν ) − sin2 ψν I (cos ψν ) =
d v (xν ) (sin ϕI(cos ϕ))ϕ=ψν − sin2 ψν I(xν ), dϕ v(xν )
where in the last equality we used Theorem 4.1 in conjunction with (4.3). Hence
I (xν ) I(xν ) v (xν ) xν + , Ων (xν ) Ων (xν ) 1 − x2ν v(xν )
Ω (x ) v (xν ) xν ν) (− Ωνν (xνν ) + 1−x which yields by (6.13) A˜2s−1,ν ΩI(x 2 + v(x ) ). With the help of ν (xν ) ν ν (6.11) and Corollary 3.1, we get
2s + 1 xν v (xν ) Ων (xν ) − 2 , Ων (xν ) 2 1 − x2ν v(xν )
and by the first relation in (6.18) b0,s = 1 follows. Recall that w = v 2s+2 . Concerning A˜2s−2,ν , we note that with respect to n by the lower order of Ων (xν )/ Ων (xν ) and A˜2s−1,ν /Ων (xν ), we have by (6.13)
A˜2s−2,ν
(6.19)
I (2) (xν ) Ων (xν ) ˜ A2s,ν + Ων (xν ) Ων (xν )
,
which gives by Corollaries 3.1 and 4.1, (4.3), and Remark 4.1 the asymptotic value for A˜2s−2,ν . Finally, to calculate the asymptotic value of A˜2s−3,ν in a short way, we recall that by Corollary 4.1 I (3) (xν ) Ων (xν )
cos ψν v (xν ) I (2) (xν ) 6 2 +3 v(xν ) Ων (xν ) sin ψν
and insert for I (2) (xν )/Ων (xν ) the value from (6.19). Then using (6.13) and %
& (3) (2s + 1) −n2 Ων (xν ) v (xν ) xν = − (4s + 2) (2s + 3) , Ων (xν ) 2 1 − x2ν 1 − x2ν v(xν ) it follows, after surprising cancellation, that (3) xν Ων (xν ) v (xν ) ˜ ˜ A2s−2,ν , −6 −3 −A2s−3,ν 3 Ων (xν ) 1 − x2ν v(xν ) which yields b1,s = 1. 6.2. Proof of Corollary 5.1. It is well known that G Bν,n,s
1 = Tn (xν ; w)
+1
−1
Tn (t; w) 2s w(t) I(xν ) , T (t; w) √ dt = 2 t − xν n πT π 1−t n (xν ; w)
which gives, by (6.12), Corollaries 4.1 and 3.1, and Remark 4.1, the assertion.
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2659
REFERENCES [1] N.I. Achieser, Vorlesungen u ¨ber Approximationstheorie, Akademie Verlag, Berlin, 1967. [2] S.N. Bernstein, Sur les polynomes orthogonaux relatifs a un segment fini, J. Math., 9 (1930), pp. 127–177, 10 (1931), pp. 219–286. [3] S.N. Bernstein, Sur une classe de polynomes orthogonaux, Commun. Soc. Math. de Kharkov, IV (1930), pp. 79–93. [4] B. Bojanov, On a quadrature formula of Micchelli and Rivlin, J. Comput. Appl. Math., 70 (1996), pp. 349–356. [5] W. Gautschi, The circle theorem and related theorem for Gauss-type quadrature rules, Electron. Trans. Numer. Anal., 25 (2006), pp. 129–137. ´, S-orthogonality and construction of Gauss-Tur´ [6] W. Gautschi and G.V. Milovanovic an-type quadrature formulae, J. Comput. Appl. Math., 86 (1997), pp. 205–218. [7] A. Ghizzetti and A. Ossicini, Quadrature Formulae, Birkh¨ auser, Basel, 1970. [8] L. Gori and C.A. Micchelli, On weight functions which admit explicit Gauss-Tur´ an quadrature formulas, Math. Comp., 65 (1996), pp. 1567–1581. [9] O. Kis, Remark on mechanical quadrature, Acta Math. Acad. Sci. Hungar., 8 (1957), pp. 473– 476 (in Russian). ´ and F. Peherstorfer, Asymptotic representation of Lp -minimal polynomials 1 < [10] A. Kroo p < ∞, Constr. Approx., 25 (2007), pp. 29–39. [11] G.G. Lorenz, Approximation of Functions, Chelsea, New York, 1986. [12] C.A. Micchelli, The Fundamental Theorem of Algebra for Monosplines with Multiplicities, ISNM 20, Birkh¨ auser, Basel, 1972, pp. 419–430. [13] C.A. Micchelli and T.J. Rivlin, Tur´ an formulae and highest precision quadrature rules for Chebyshev coefficients, IBM J. Res. Develop., 16 (1972), pp. 372–379. [14] C.A. Micchelli and A. Sharma, On a problem of Tur´ an: Multiple node Gaussian quadrature, Rend. Mat. Ser. VII, 3 (1983), pp. 529–552. ´, Construction of s-orthogonal polynomials and Tur´ [15] G.V. Milovanovic an quadrature formulae, in Proceedings of the Numerical Methods and Approximation Theory III Niˇs, 1987, pp. 311– 328. ´, Quadrature with multiple nodes, power orthogonality and moment pre[16] G.V. Milovanovic serving spline approximation, J. Comput. Appl. Math., 127 (2001), pp. 267–286. ´ and M.M. Spalevic ´, Error bounds for Gauss-Tur´ [17] G.V. Milovanovic an quadrature formulae of analytic functions, Math. Comp., 72 (2003), pp. 1855–1872. ´, M.M. Spalevic ´, and A.S. Cvetkovic ´, Calculation of Gaussian-type [18] G.V. Milovanovic quadratures with multiple nodes, Math. Comput. Modelling, 39 (2004), pp. 325–347. [19] A. Ossicini and F. Rosati, Sulla convergenza dei funzionali ipergaussiani, Rend. Mat., 11 (1978), pp. 97–108. [20] F. Peherstorfer, Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and Lq -minimal polynomials, q ∈ [1, ∞], J. Approx. Theory, to appear. [21] F. Peherstorfer, Positive quadrature formulas III: Asymptotics of weights, Math. Comp., 77 (2008), pp. 2241–2259. [22] R.D. Riess, Gauss-Tur´ an quadratures of Chebyshev type and error formulae, Computing, 15 (1975), pp. 173–179. [23] T.J. Rivlin, Chebyshev polynomials. From Approximation Theory to Algebra and Number Theory, 2nd ed., Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1990. [24] Y.G. Shi and G. Xu, Construction of sigma-orthogonal polynomials and Gaussian quadrature formulas, Adv. Comput. Math., 27 (2007), pp. 79–94. [25] D.D. Stancu, Asupra unor formule generale de integrare numeric˘ a, Acad. R.P. Romˆ ane Stud. Cerc. Mat., 9 (1958), pp. 209–216. [26] A.H. Stroud and D.D. Stancu, Quadrature formulas with multiple Gaussian nodes, J. SIAM Ser. B: Numer. Anal., 2 (1965), pp. 129–143. ¨ , Orthogonal Polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. [27] G. Szego Soc., Providence, RI, 1967. [28] V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math., 81 (2000), pp. 283–303. ´ n, On the theory of the mechanical quadrature, Acta Sci. Math. (Szeged), 12 (1950), [29] P. Tura pp. 30–37. ´ n, On some open problems of approximation theory, J. Approx. Theory, 29 (1980), [30] P. Tura pp. 23–85. [31] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge, University Press, London, 1959.
SIAM J. NUMER. ANAL. Vol. 47, No. 4, pp. 2638–2659
´ QUADRATURE FORMULAS: ASYMPTOTICS OF GAUSS–TURAN WEIGHTS∗ FRANZ PEHERSTORFER† Abstract. First we derive asymptotics for the derivatives of s-orthogonal polynomials and the derivatives of Cauchy principal value integrals associated with s-orthogonal polynomials. With the help of these results asymptotics for the quadrature weights of Gauss–Tur´ an quadrature formulas are obtained. So far asymptotics have been known with respect to the Chebyshev weight function only. Key words. asymptotics, s-orthogonal polynomials, Cauchy principal values, derivatives, Gauss–Tur´ an quadrature formulas, Szeg¨ o function AMS subject classifications. 65D32, 42C05 DOI. 10.1137/070695794
1. Introduction. In 1950 Tur´ an [29] introduced numerical quadrature formulas of the form n 2s (1.1) Aj,ν,n f (j) (xν,n ) + Rn,s (f ), f (t)dμ(t) = ν=1 j=0
−1 < xn,n < xn−1,n < · · · < x1,n < 1, exact for polynomials of degree less or equal 2(s + 1)n − 1, i.e., Rn,s (p) = 0 for p ∈ P2(s+1)n−1 ,
(1.2)
where Pm denotes the set of polynomials of degree ≤ m and μ(t) denotes a positive measure supported on [−1, 1]. Such quadrature formulas are called Gauss–Tur´ an quadrature formulas, nowadays. He has shown that (1.2) holds if and only if the nodes xν,n are chosen such that (1.3)
Tˆn (x; dμ) := Tˆn,s (x; dμ) :=
n
(x − xν,n )
ν=1
minimizes
xn + an−1 xn−1 + · · · + a0
2s+2
dμ(x)
or in other words, is a monic minimal polynomial with respect to the L2s+2 (dμ) norm which is equivalent to the conditions that 2s+1 (1.4) dμ(x) = 0, k = 0, . . . , n − 1. xk Tˆn,s (x; dμ) Orthogonality property (1.4) is called s-orthogonality. ∗ Received
by the editors June 29, 2007; accepted for publication (in revised form) March 17, 2009; published electronically July 22, 2009. This work was supported by the Austrian Science Fund FWF, project P20413-N18. http://www.siam.org/journals/sinum/47-4/69579.html † Abteilung f¨ ur Dynamische Systeme und Approximationstheorie, Institut f¨ ur Analysis, J.K. Universit¨ at Linz, Altenberger Str., 69, 4040 Linz, Austria ([email protected]). 2638
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2639
Naturally, and as pointed out by Tur´ an [30] himself, we would like to know the behavior of the quadrature weights Aj,ν,n , at least asymptotically. Note that the Aj,ν,n ’s can be represented in the form (1.5) Aj,ν,n = hj,ν,n (x)dμ(x), where the hj,ν,n (x) are the fundamental polynomials of Hermite interpolation. But the representation of the fundamental polynomials in terms of the fundamental Lagrange interpolation polynomials becomes rather involved at an early stage such that it is difficult to work with. So far with the help of (1.5) only the positivity of the even weights A2j,ν,n could be shown [12, 19]. By the way, this is the only information on the quadrature weights available for general weight functions. Other information is available only for the Chebyshev weight function, which will be discussed below. For s = 0, that is, when (1.1) becomes the Gauss-quadrature formula, the behavior is well known and given by the beautiful so-called circle theorem. More precisely, suppose that (1.6)
dμ(x) =
w(x) √ dx for x ∈ [−1, 1] π 1 − x2
and that w(x) is positive and from Lip γ, γ ∈ (0, 1], on [−1, 1]. Then the Gaussian weights AG ν,n := A0,ν,n associated with nodes xν,n ∈ [−1 + , 1 − ], > 0, behave like nAG ν,n = w(xν,n )(1 + o(1)).
(1.7)
This nice relation dates back to Szeg¨ o [27]. For a recent discussion of the circle theorem, see [5, 28]. Let us mention that now asymptotics for quadrature weights are available for positive interpolation quadrature formulas [21] even. Obviously, it would be of foremost interest to know whether there are similar asymptotic laws for the quadrature weights of Gauss–Tur´ an quadrature formulas. But so far, we have information on the weights for only the celebrated Chebyshev √ weight function dμ(x) = 1/ 1 − x2 dx [7, 9, 13, 14]. Recall that in this special case the s-orthogonal polynomial is for every s ∈ N0 := N ∪ {0} the Chebyshev polynomial, i.e., Tˆn,s (x; dμ) = Tˆn (x) = 2−n+1 cos(n arccos x), x ∈ [−1, 1]. Using the rich structure of the Chebyshev polynomials, Kis has shown in a lovely paper [9] that for any polynomial p ∈ P2(s+1)n−1
+1
(1.8) −1
s n p(x) 1 Sj d2j √ dx = (p(cos ϕν ))ϕν = (2ν−1)π , 2n n(s!)2 j=0 (2n)2j ν=1 dϕ2j π 1 − x2 ν
where Ss−j , j = 0, . . . , s, denote the elementary symmetric polynomials evaluated at 12 , 22 , . . . , s2 , i.e., Ss = 1, Ss−1 = 12 + 22 + · · · + s2 , . . . , S0 = 12 22 . . . s2 . For instance, transforming back we obtain for s = 1 that for any p ∈ P4n−1 n +1 n 1 p(x) xν,n √ dx = p(xν,n ) − p (xν,n ) 2 n 4n2 −1 π 1 − x ν=1 ν=1 n
1 − x2ν,n + p (xν,n ) , 4n2 ν=1
2640
FRANZ PEHERSTORFER
where xν,n = cos (2ν−1)π are the zeros of Tˆn (x). Denoting the weights of the Gauss– 2n Tur´ an quadrature formula with respect to the Chebyshev weight function by AC j,ν,n , one we may derive from (1.8), their asymptotic behavior, using (3.28) below in conjunction with induction arguments on the order of AC 2s−j,ν,n ; see pp. 545–546 and Theorem 4.1 in [14] (in Theorem 4.1 for s = 2r − 1, bs (x) should be replaced by −r(2r − 1)x(1 − x2 )r−1 ). Hence AC 2j,ν,n
(1.9)
j
Sj 1 − x2ν,n 1 = 1+O 2 2j 2j+1 n (s!) 2 n
and (1.10)
AC 2j−1,ν,n
j−1
(−Sj )j(2j − 1)xν,n 1 − x2ν,n 1 = 1+O . 2 2j 2j+1 n (s!) 2 n
For error estimates of Gauss–Tur´an quadrature formulas with Chebyshev nodes, see [4, 17, 22]. Concerning numerical constructions, see [6, 8, 18, 24, 26]. A survey on Gauss–Tur´ an quadrature formulas is given in [16]. Let us roughly outline how we obtain asymptotics for the quadrature weights of the Gauss–Tur´ an quadrature formulas. Put
2s+1 Tn (x; μ) (1.11) , Ων,2s+1 (x) := Ων (x) := x − xν where Tn (x; μ) = dn Tˆn (x; μ), dn ∈ R\{0}, is some suitable normalization of Tˆn (x; μ) (see (3.3)), and set (Tn (x; μ))2s+1 − (Tn (t; μ))2s+1 (1.12) dμ(t). I2s+1 (x) := I(x) := x−t Stancu [25] has shown that (1.13)
dj A˜2s−j,ν := j!(2s − j)!A2s−j,ν = j dx
I(x) Ων (x)
. x=xν
Next we observe that (1.14)
m−1 (m−j) I (m) (x) m Ων (x) ˜ ˜ A2s−m,ν (x) = A2s−j,ν (x), − j Ων (x) Ω (x) ν j=0
which can be proved by induction arguments (see Lemma 6.2 below) and that
+1 2s+1 Tn (t; μ) dm (m) dμ(t) (1.15) I (xν ) = m − dx t−x −1 x=xν if (1.16)
dm dxm
+1 dμ(t) − t−x −1
exists. As usual − denotes the Cauchy principal value. Thus, if we know asymptotics for the derivatives of the s-orthogonal polynomials and for the Cauchy principal values
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2641
from (1.15), then the quadrature weights can be calculated by (1.14) successively for m = 2s, 2s − 1, . . . , 1. In fact, this is how we obtain the general asymptotic form of the quadrature weights with respect to weight functions of the form (1.6), where w > 0 on [−1, 1] and is sufficiently smooth. Recently for weight functions of this type, asymptotics for s-orthogonal polynomials (in fact, for Lp -minimal polynomials, p ∈ (1, ∞)) have been obtained by Kro´o and Peherstorfer [10]. They build the basis for the derivation of asymptotics for the derivatives of the s-orthogonal polynomials, given in section 3. Concerning properties of zeros of s-orthogonal polynomials, see [20]. One of the main challenges is the determination of asymptotics of the Cauchyprincipal value integrals from (1.15). We first demonstrate in section 2, Theorem 2.1, that for the Bernstein–Szeg¨o weights they can be given explicitly even. With the help of this result, in section 4 we derive asymptotics for the Cauchy principal values with respect to general weight functions of the form (1.6), where for convenience w(x) = v 2s+2 (x) is considered. Finally in section 5 we present the asymptotics for the quadrature weights. 2. Bernstein–Szeg¨ o weight functions—Cauchy principal values. Let gk (z) be a real polynomial of degree k which has all its zeros in |z| > 1, and put ρk (cos ϕ) = gk (eiϕ |2 . (2.1) Note that ρk (cos ϕ) is a positive cosine polynomial of degree k. Bernstein [2, 3] (see also [1, pp. 278–280]) has shown that for p ∈ [1, ∞), n ≥ k, p
Tˆn,p cos ϕ; 1/ρk2 = cn Re e−inϕ gk eiϕ
= cn Re ei(n−k)ϕ gk∗ eiϕ , where cn ∈ R\{0} and gk∗ (z) = z k gk ( 1z ) denote the so-called reciprocal polynomial. In particular, by considering gk2 (z) instead of gk (z) and p = 2s + 2, s ∈ N0 , we obtain that
(2.2) = cn Re e−inϕ gk2 eiϕ . Tˆn,2s+2 cos ϕ; 1/ρ2s+2 k Notation 2.1. (a) Let gk (z) be a real polynomial of degree k which has all its zeros in |z| > 1. Then we put, x = cos ϕ,
Tn x; 1/ρ2k := Tn (x) := Re e−inϕ gk2 eiϕ . (2.3) Note that, x = cos ϕ, z = eiϕ , (2.4)
Tn (x) g ∗ (z) = Re z n−k k = cos Φk (ϕ), ρk (x) gk (z)
where Φk (ϕ) = arg z n−k
(2.5)
gk∗ (z) . gk (z)
(b) Let m ∈ N. We put (2.6)
sm (ϕ) =
m j=0
dj,m sin(m − j)ϕ = Im
⎧ m ⎨ ⎩
j=0
dj,m z m−j
⎫ ⎬ ⎭
,
2642
FRANZ PEHERSTORFER
where the dj,m ’s are the Fourier coefficients of (cos ϕ)m , i.e.,
(2.7)
2m−1 (cos ϕ)m =
m
dj,m cos(m − j)ϕ = Re
j=0
⎧ m ⎨ ⎩
dj,m z m−j
j=0
⎫ ⎬ ⎭
or explicitly that (2.8)
dm−2j,m
m = and dm−(2j+1),m = 0. j
Notation 2.2. For convenience we will use the differential operator D defined by Df (z) = iz
(2.9)
d f (z). dz
In particular, we have
dj iϕ = Dj f eiϕ . f e dϕj Theorem 2.1. Let m ∈ N, y ∈ C\[−1, 1], z = y − √ is chosen such that |z| < 1, and put for |z| ≤ 1 Hm (z) =
(2.10)
y 2 − 1, where that branch of
m−j g ∗ (z) dj,m z n−k k . gk (z) j=0
m
Then for |z| < 1 2 Hm (z) = y − 1
(2.11)
+1
−1
(Tn (t))m 1 dt √ m y − t ρk (t) π 1 − t2
and, ϕ ∈ [0, 2π], +1 dt (Tn (t))m 1 √ Im Hm (eiϕ ) = sin ϕ − . m (t) t − cos ϕ ρ π 1 − t2 −1 k
(2.12) Furthermore, (2.13)
dj dϕj
+1 dt (Tn (t))m 1 √ sin ϕ − = (sm ◦ Φk )(j) (ϕ). 2 t − cos ϕ ρm −1 k (t) π 1 − t
If m is odd, we have at the zeros xν = cos ψν , −1 < xn < xn−1 < · · · < x1 < 1, of Tn (x): (2.14)
s(j) m (Φk (ψν ))
(j) (−1)ν−1 sm ( π2 ) for j even, = 0 for j odd.
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2643
Proof. For the following let us point out that Hm (z) is analytic on the closed unit disk |z| ≤ 1. By Schwarz formula
(2.15)
2π
eiϕ + z ReHm eiϕ dϕ iϕ e −z 0
2 π ReHm eiϕ 1−z = dϕ 2 π 0 1 + z − 2z cos ϕ
m Tn (t) y 2 − 1 +1 1 dt √ = π y − t ρ (t) 1 − t2 k −1 =: y 2 − 1Q0 (y),
Hm (z) =
1 2π
where we used the facts that by (2.4) and (2.7)
m
Tn (cos ϕ) ReHm eiϕ = ; ρk (cos ϕ) moreover, ReHm (ei(π+ϕ) ) = ReHm (ei(π−ϕ) ) and that by the choice of the branch of square root
1 1 −z . y2 − 1 = 2 z This proves relation (2.11). Taking into consideration ± ± y 2 − 1 (x) = ±i 1 − x2 and y − y 2 − 1 (cos θ) = e∓iθ , we obtain by the definition of Q0 (y)
(2.16)
1 −ImHm eiϕ = sin ϕ lim (Q0 (x + i) + Q0 (x − i)) →0+ 2
m +1 Tn (t) 1 dt √ = sin ϕ − x − t ρk (t) π 1 − t2 −1 = −(sm ◦ Φk )(ϕ),
where the last equality follows by (2.10), (2.5), and (2.6). Thus relation (2.12) is proved. Since Hm (z) is analytic on |z| ≤ 1, we may moreover differentiate Hm and obtain, in conjunction with (2.16), relation (2.13):
dj Im Dj Hm (z) z=eiϕ = Im Hm eiϕ j dϕ = (sm ◦ Φk )(j) (ϕ). Concerning the last assertion (2.13), at the zeros xν = cos ψν of Tn (x), we have by (2.4) that π Φk (ψν ) = (2κ + 1) , 2
κ ∈ Z,
2644
FRANZ PEHERSTORFER
from which for j odd the vanishing property follows easily with the help of the first relation in (2.5) and (2.6), taking into account (2.8) and that m is odd. Now, if |zj | < 1,
z − zj z − zj d arg = Im D ln dϕ 1 − z¯j z 1 − z¯j z
z − zj d 1 − |zj |2 = Re z ln > 0. = iϕ dz 1 − z¯j z |e − zj |2 Thus Φk (ϕ) is strictly monotone increasing, with cos Φk (0) = Tn (1) > 0, and thus we may assume that Φk (xν ) = (2ν − 1) π2 , ν = 1, . . . , n. By the first relation from (2.6) and m odd it follows that sm (Φk (xν )) = (−1)ν−1 sm ( π2 ). Proceeding analogously for the even derivatives gives the assertion. For the special case ρk ≡ 1, we obtain
+1 dt Tnm (t) dj √ sin ϕ − dϕj t − cos ϕ π 1 − t2 ϕ= (2ν−1)π −1 2n (2.17) ν−1 j (j) π , = (−1) n sm 2 where Tn (t) = 2n−1 Tˆn (t), which can be deduced from the well-known relation (2.18)
+1 sin ϕ − −1
dt Tn (t) √ = sin nϕ t − cos ϕ π 1 − t2
and (2.6) and (2.7) also. In what follows, we need Lemma 2.1. Lemma 2.1. (a) Let m ∈ N. At the zeros xν = cos ψν of Tn (x) there holds
(2.19)
+1 dj (Tn (t))m 1 dt √ sin ϕ − dϕj t − cos ϕ ρm 1 − t2 ϕ=ψν −1 k (t)
+1 dt (Tn (t))m − (Tn (cos ϕ))m 1 dj √ sin ϕ = dϕj t − cos ϕ ρm 1 − t2 ϕ=ψν −1 k (t)
for j = 0, . . . , m − 1. (b) Let s ∈ N0 , n ≥ k. The following relation holds for x ∈ [−1, 1]: +1 1 (Tn (t))2s+1 dt √ − 2s+2 t−x ρk (t) 1 − t2 −1 +1 1 (Tn (t))2s+1 dt 1 √ − = . ρk (x) −1 t−x ρ2s+1 (t) 1 − t2 k Proof. (a) Obviously, +1 dt dt (Tn (t))m 1 (Tn (t))m − (Tn (cos ϕ))m 1 √ √ sin ϕ − = sin ϕ m t − cos ϕ ρk (t) 1 − t2 t − cos ϕ ρm (t) 1 − t2 −1 k +1 1 dt sin ϕ √ + (Tn (cos ϕ))m − . m t − cos ϕ ρk (t) 1 − t2 −1
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2645
Since Tn (cos ψν ) = 0, it follows by the product rule that the jth derivative of the second summand at ϕ = ψν is zero for j = 0, 1, . . . , m − 1, which proves part (a). Concerning part (b), since, by (2.2) √ and (2.3), Tn is minimal with respect to the L2s+2 (dμ)-norm, where dμ = 1/ρ2s+2 1 − t2 , it follows by its orthogonality property k (1.4) (note that (ρk (x) − ρk (t))/(x − t) ∈ Pk−1 ) that +1 ρk (x) − ρk (t) (Tn (t))2s+1 dt √ = 0, 2s+2 x − t ρ (t) 1 − t2 −1 k which is the stated relation. 3. Asymptotics for derivatives of s-orthogonal polynomials. First we need some notation. m+α Notation 3.1. By C2π , m ∈ N0 , α ∈ (0, 1), we denote the space of 2π-periodic functions which are m-times differentiable on [0, 2π] with the mth derivative in Lip α on [0, 2π]. Let v(cos ϕ) be positive and Lip α, 0 < α < 1, on [0, π], and let the so-called Szeg¨ o-function π(z; v(cos ϕ)) := π(z) of v(cos ϕ) be defined on the open unit disk by 2π iϕ e +z 1 log v(cos ϕ)dϕ (3.1) . π(z; v(cos ϕ)) := π(z) := exp − 4π 0 eiϕ − z Note that by the assumptions on v, the boundary values limr→1− π(reiϕ ) =: π(eiϕ ) exist; moreover, 2 v(cos ϕ) = 1/ π eiϕ . (3.2) Furthermore, for convenience, let us introduce the following normalized s-orthogonal polynomials, p = 2s + 2: 1/p p w(x) ˆ 1/p ˆ Tn,p (x) = λp Tn,p (x)/ (3.3) dx , Tn,p (x) √ 1 − x2 [−1,1] where (3.4)
Γ 12 Γ p+1 2
λp = Γ p2 + 1
(Tn,p is denoted in [10] by T˜n,p ). Thus for ρk (cos ϕ) given by ρk (cos ϕ) = |gk (eiϕ )|2 , we obtain by the normalization x = cos ϕ,
= Re e−inϕ gk2 e−iϕ = Tn (x). (3.5) Tn,2s+2 x; 1/ρ2s+2 k l+α , l ∈ N0 , α ∈ (0, 1), and let π(z) be the associated Lemma 3.1. Let v(cos ϕ) ∈ C2π Szeg¨ o-function. There exists a sequence of polynomials (gk (z)), gk (z) of degree k such that uniformly on |z| ≤ 1 for j = 0, 1, . . . , l,
1 (gk )(j) (z) − π (j) (z) = O (3.6) . k l+α−j
Furthermore, for each k ≥ k0 all zeros of gk (z) are in |z| > 1, and the zeros of gk∗ have no accumulation points on |z| = 1. Moreover, putting 2 ρk (cos ϕ) = gk eiϕ , (3.7)
2646
FRANZ PEHERSTORFER
we have for j = 0, . . . , l, n > k,
(3.8)
dj dj Tn cos ϕ; 1/ρk2s+2 = Re e−inϕ gk2 eiϕ j j dϕ dϕ
−inϕ 2 iϕ dj 1 = Re e π e . +O dϕj k l+α−j
m+α . Proof. By the assumptions on v(cos ϕ), it follows that log v(cos ϕ) ∈ C2π m (m) Hence, see [31, p. 121], the conjugate function log v(cos ϕ) ∈ C2π and (log v(cos ϕ)) m+α ∈ Lip α, and therefore π(eiϕ ; v(cos ϕ) ∈ C2π . Thus, see, e.g., [31, p. 67], there are polynomials gk (z) which satisfy (gk )(j) (z) −
1 (j) π (z) = O kl+α−j . Since π(z) is analytic on |z| < 1 and has, recall (3.1), no zero on |z| ≤ 1, it follows by the maximum principle that
(3.9)
0 < c1 ≤ |πp (z)| ≤ c2 for |z| ≤ 1
which implies, by (3.6), the assertion on the zeros. Relation (3.8) now follows with the help of (3.6) and (3.5).
1 , where l ∈ Lemma 3.2. Let tm ∈ Pm be such that ||tm ||∞,[−1,1] = O ml+α N0 , α ∈ [0, 1). Then the following statements hold:
1 dj (a) || dϕ ; j (tm (cos ϕ))||∞,[0,π] = O ml+α−j
dj tm (cos ϕ)−tm (cos ψ) 1 (b) || dϕj . )||∞,[0,π] = O ml+α−j−2 cos ϕ−cos ψ Proof. Part (a) follows immediately by Bernstein’s inequality. Concerning part (b), with the help of Markoff’s inequality, see, e.g., [11], we obtain tm (cos ϕ) − tm (cos ψ) cos ϕ − cos ψ
∞,[−π,π]
= tm (cos ξ) ∞,[−π,π] ≤ m2 ||tm (x)||∞,[−1,1] .
Hence, using Bernstein’s inequality it follows that j d tm (cos ϕ) − tm (cos ψ) j+2 dϕj ≤ m ||tm (x)||∞,[−1,1] , cos ϕ − cos ψ which is the assertion. l+α Proposition 3.1. Let v(cos ϕ) ∈ C2π , put μ = (l+α)/(2s+2), and let ρk (cos ϕ) be given as in Lemma 3.1. Then the following statement holds for 0 ≤ j < μ and n>k:
1 dj dj 2s+2 2s+2 = +O Tn cos ϕ; v Tn cos ϕ; 1/ρk (3.10) dϕj dϕj k μ−j ˙ and for 0 ≤ j < μ − 1 and n > k with k = O(n),
+1 m
Tn cos ϕ; v 2s+2 − Tnm t; v 2s+2 2s+2 dj dt v (t) √ sin ϕ dϕj cos ϕ − t 1 − t2 −1
+1 m − Tnm t; 1/ρk2s+2 Tn cos ϕ; 1/ρ2s+2 1 dt dj k √ (3.11) = sin ϕ dϕj cos ϕ − t ρk2s+2 (t) 1 − t2 −1
1 +O nμ−(j+2)
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
and
+1 m 2s+2 Tn t; v dt 2s+2 v (t) √ sin ϕ − t − cos ϕ 1 − t2 −1
+1 m
Tn t; 1/ρk2s+2 1 1 dt dj √ = sin ϕ − +O . dϕj t − cos ϕ nμ−(j+2) ρ2s+2 (t) 1 − t2 −1 k
dj dϕj (3.12)
2647
Proof. In [10] it has been shown by the Lipschitz continuity of the best approximation map that
1 Tn cos ϕ; v 2s+2 = Re e−inϕ π 2 eiϕ + O l+α 2s+2
n
1 2s+2 = Tn cos ϕ; 1/ρk +O , l+α k 2s+2 where the last equality follows by (3.8) and n > k. Applying Lemma 3.2 to
(3.13) rn (cos ϕ) = Tn cos ϕ; v 2s+2 − Tn cos ϕ; 1/ρk2s+2 , statement (3.10) follows. Concerning relation (3.11), first we claim that relation (3.11) holds when we (t) in the second integral by v 2s+2 (t). We put replace the weight 1/ρ2s+2 k (3.14)
Qn,m (x) =
m−1
Tnm−1−κ cos ϕ; v 2s+2 Tnκ cos ϕ; 1/ρk2s+2
κ=0
and write the difference of the two integrands in the form
Qn,m (cos ϕ) − Qn,m (t) rn (cos ϕ) − rn (t) (3.15) Qn,m (cos ϕ) + rn (t). cos ϕ − t cos ϕ − t Since Tn (x; v 2s+2 ) and Tn (x; 1/ρ2s+2 ) are uniformly bounded on [−1, 1], it follows by k di l i Bernstein’s inequality that l ∈ {0, . . . , m − 1} and that dϕ i Tn (cos ϕ; .) = O(n ), which i
d i gives with the help of Leibniz’s rule that dϕ i Qn,m (cos ϕ) = O(n ) and, by Lemma 3.2,
di Qn,m (cos ϕ) − Qn,m (t) (3.16) = O ni+2 . i dϕ cos ϕ − t
Thus by (3.15), (3.10), and Leibniz’s rule the claim follows. Finally, relation (3.11) follows by recalling the fact that by (3.6) and (3.7) uniformly for ϕ ∈ [0, 2π],
1 1 2s+2 (3.17) . (cos ϕ) + O =v k l+α ρ2s+2 (cos ϕ) k Concerning relation (3.12), the difference of the jth derivative of the integrals in (3.12) is by (3.11) of the form
m
+1 v 2s+2 (t) dj dt 2s+2 2s+2 m √ T cos ϕ; v − T cos ϕ; 1/ρ − n n k dϕj cos ϕ − t 1 − t2 −1
1 +O , nμ−(j+2) which gives with the help of the assertion (3.10).
2648
FRANZ PEHERSTORFER
l+α Theorem 3.1. Let v(cos ϕ) ∈ C2π , put μ = (l + α)/(2s + 2), and let π(z) be the Szeg¨ o function associated with v(cos ϕ). Then the following statements hold: (a) For j < μ, z = eiϕ ,
dj
Tn cos ϕ; v 2s+2 = Re (−i)j nj z −n π 2 (z) + i(−in)j−1 2jz −n+1 π(z)π (z) dϕj
1 +O . nμ−j (b) At the zeros xν = cos ψν of Tn (x; v 2s+2 ), there holds, 2κ + 1 < μ, (3.18)
(−1)ν+κ n2κ+1 d2κ+1
2s+2 T cos ϕ; v = n ϕ=cos ψν dϕ2κ+1 v(xν )
1 1+O n
and, 2κ < μ, v (xν ) d2κ
2s+2 ν−1+κ 2κ−1 2 T cos ϕ; v = (−1) n 2κ 1 − x n ν ϕ=cos ψν dϕ2κ v 2 (xν ) (3.19)
1 1+O . n
Proof. For abbreviation put G(z) = π 2 (z). By (3.8) and (3.10), for instance, choose k = [n/2], j
d −inϕ iϕ 1 dj
2s+2 Tn cos ϕ; v = Re e G e +O dϕj dϕj nμ−j = Re (−in)j z −n G(z) + (−in)j−1 ijz −n+1 G (z) (3.20)
1 + O nj−κ +O , nμ−j where the second equality follows by straightforward calculation. Thus part (a) is proved. Concerning (b), for j = 2κ + 1 we obtain by (a) considering the first summand at the right-hand side only that (3.21)
−n 1 d2κ+1
2s+2 κ 2κ+1 T cos ϕ; v = (−1) n G(z) + O Im z . n dϕ2κ+1 n
Now, by (3.10) and (3.8) for j = 0, at the zeros xν = cos ψν of Tn (x; v 2s+2 ), (3.22)
Re {z
−n
G(z)}z=eiψν = O
1 nμ
,
and thus by (3.23)
(Re {z −n G(z)})2 + (Im {z −n G(z)})2 = |G(z)|2
and sgn Im{z −n G(z)}z=eiψν = (−1)ν , note that Re{e−inϕ gk2 (eiϕ )} and Im{e−inϕ gk2 (eiϕ )} have strictly interlacing zeros; relation (3.18) follows.
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2649
For j = 2κ we obtain by (a) and (3.22) that
d2κ Tn cos ϕ; v 2s+2 = (−1)κ Re n2κ z −n G(z) − 2κn2κ−1 z −n+1 G (z) 2κ dϕ
2κ−2 1 +O +O n (3.24) nμ
−n+1 1 κ+1 2κ−1 2κn Re z G (z) 1 + O = (−1) . n Observing that, with the help of (3.22), at the zeros of Tn (x; v 2s+2 ), z = eiϕ , 2Re {z −n G(z)}
d Re {z −n G(z)} = O dϕ
1 nμ−1
,
and, again by (3.22),
(3.25)
d
Im e−inϕ G eiϕ ϕ=ψν Im e−inψν G eiψν dϕ
−inψν iψν −n+1 1 = Im e G e G (z) z=eiψν , + Re z O nμ−1
it follows, by differentiating (3.23) with respect to ϕ, that d
Im e−inϕ G eiϕ ϕ=ψν Im e−inψν G eiψν dϕ
1 d iϕ 2 1 G e = +O . 2 dϕ nμ−1 ϕ=ψν
(3.26)
Since
Im e−inψν G eiψν = (−1)ν G eiψν + O
(3.27)
1 nμ
where we have used (3.23) in conjunction with (3.22), the assertion follows by (3.24) taking into consideration (3.25)–(3.27) and the fact that |G(eiϕ )| = 1/v(cos ϕ). Obviously, taking a look at (3.20) and the proof, one could state (3.18) and (3.19) up to higher order, also. Remark 3.1. (−1)j (3.28)
dj (sin ϕ f (cos ϕ)) dϕj = (sin ϕ)j+1 f (j) (cos ϕ) − +
j−2
j(j + 1) (sin ϕ)j−1 cos ϕ f (j−1) (cos ϕ) 2
τν (cos ϕ, sin ϕ)f (ν) (cos ϕ),
ν=0
where τν (cos ϕ, sin ϕ) is a trigonometric polynomial of degree ≤ j − 1.
2650
FRANZ PEHERSTORFER
Corollary 3.1. (3.29)
Tn(2κ+1)
(−1)ν−1+κ n2κ+1 xν ; v 2s+2 = (sin ψν )2κ+1 v(xν )
1 1+O n
if 2κ + 1 < μ, and
(−1)ν−1+κ n2κ−1 2κ Tn(2κ) xν ; v 2s+2 = (sin ψν )2κ−1 v(xν ) (3.30)
v (xν ) 2κ − 1 xν − v(xν ) 2 1 − x2ν
1 1+O n
if 2κ < μ. Proof. First let us observe that (−1)
dj+1 dj (f (cos ϕ)) = (sin ϕ f (cos ϕ)) . dϕj+1 dϕj
By Theorem 3.1 and Remark 3.1, it follows by induction arguments that
j+1 1 Tn(j) (xν ) = O n2[ 2 ]−1 1+O (3.31) n which yields, by Remark 3.1 again, relation (3.29). Relation (3.30) follows by observing that by Remark 3.1 and (3.31), d2κ (Tn (cos ϕ))ϕ=ψν = (sin ψν )2κ Tn(2κ) (xν ) dϕ2κ (3.32)
− κ(2κ − 1)(sin ψν )2κ−2 cos ψν Tn(2κ−1) (xν )
1 + O n2κ−3 1 + O n
which gives, by (3.29) and Theorem 3.1, relation (3.30). 4. Cauchy principal values. The following result is crucial in what follows. l+α Theorem 4.1. Let s ∈ N0 , v(cos ϕ) ∈ C2π , put μ = (l + α)/(2s + 2), and let π(z) be the Szeg¨ o function associated with v(cos ϕ). Then uniformly on [0, π] for 0 ≤ j < μ − 2, +1 2s+1 2s+2 t, v Tn dt dj 2s+2 v (t) √ sin ϕ − dϕj t − cos ϕ 1 − t2 −1 (4.1)
1 dj = (v(cos ϕ)s (Φ(ϕ))) + O , 2s+1 dϕj nμ−j−2 where s2s+1 is given by (2.6) and Φ and its derivatives are given by, j = 0, . . . , 2s,
dj zn j (4.2) Φ(ϕ) = Im D ln 2 . dϕj π (z) z=eiϕ Furthermore, at the zeros xν = cos ψν of Tn (x; v 2s+2 ), we have
π 1 (j) ν−1 (j) s2s+1 (Φ(ψν )) = (−1) +O s2s+1 (4.3) . 2 nμ−j
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2651
Proof. By Proposition 3.1 we know that the left-hand side of (4.1) is for n > k, 1 ˙ with k = O(n), up to order O( nμ−j−2 ), equal to 2s+1
Tn t; 1/ρk2s+2 dt 1 √ sin ϕ − t − x 1 − t2 ρ2s+2 (t) k 2s+1 +1 Tn (t) dt 1 dj 1 √ sin ϕ − = dϕj ρk (cos ϕ) ρk (t) x − t 1 − t2 −1
2s+1 j 1 dκ j dj−κ = dμ,2s+1 κ (sin Φk (ϕ)) , j−κ κ dϕ ρk (cos ϕ) μ=0 dϕ κ=0
dj dϕj (4.4)
where the second equality follows by Lemma 2.1 and the third equality by Theo˙ ) denotes the precise order. Recall that by (2.5) rem 2.1. As usual O( Φk (ϕ) = arg z n − 2 arg gk (z) = arg z n − 2Im {ln gk (z)}. Since by (3.6) uniformly on |z| ≤ 1
D ln gk (z) = D ln π(z) + O j
1
j
k l+α−j
,
it follows moreover that uniformly for ϕ ∈ [0, π], j = 0, . . . , 2s,
1 dj Φ dj Φk (ϕ) = (ϕ) + O (4.5) , dϕj dϕj k l+α−j where Φ(ϕ) is given by (4.2), i.e., Φ(ϕ) = arg z n − 2 arg π(z). Observing that
d
1 d 2 Re =− ln gk eiϕ dϕ ρk (cos ϕ) ρk (cos ϕ) dϕ
2 =− Re D ln gk eiϕ , iϕ 2 |ρk (e )| j
d 1 it follows that the higher derivatives dϕ j ( ρ (cos ϕ) ) can be expressed with the help k of Re Dκ ln gk (z), κ ∈ {1, . . . , j}, and 1/|gk (eiϕ )|2 which implies by (3.6) again that uniformly on [0, π],
1 1 dj dj (v(cos ϕ)) + O . = dϕj ρk (cos ϕ) dϕj k l+α−j
Thus we may replace in the last line from (4.4) the argument Φk (ϕ) by Φ(ϕ) and 1/|gk (cos ϕ)|2 by v(cos ϕ), which yields relation (4.1). Concerning (4.3), we first observe that by (3.10) and (3.8), put j = 0,
Tn xν ; v 2s+2 1 0= = cos Φ(ψ ) + O , ν iϕ 2 |π(e )| nμ and thus (2l + 1)π +O Φ(ψν ) = 2
1 nμ
,
2652
FRANZ PEHERSTORFER
where l ∈ Z. Now for sufficiently large k the argument Φk is by (4.5) close to Φ; hence (see the proof of Theorem 2.1, last part), we may put l = ν. Thus
1 (2ν + 1)π (2ν + 1)π 1 (j) (j) +O s2s+1 = s2s+1 +O 2 nμ 2 nμ−j
π 1 ν−1 (j) +O s2s+1 , = (−1) 2 nμ−j where the last equality has been demonstrated at the end of the proof of Theorem 2.1. (j) Remark 4.1. Recall that s2s+1 ( π2 ) can be calculated by (2.6), in particular (2j+1) π s2s+1 ( 2 ) = 0 for j ∈ N0 . To get closed expressions it is sometimes of advantage +1 t2s+1 (j) dj √ dt π to use the relation; see (2.17), s2s+1 ( π2 ) = dϕ j (sin ϕ −−1 t−cos ϕ 1−t2 )ϕ= 2 in conjunction with Remark 3.1 and the relation ⎧
+1 2s+1 ⎨ π(2μ)! 2(s − μ) dj dt t if j = 2μ, √ s−μ − = 22(s−μ) dxj −1 t − x 1 − t2 x=0 ⎩ 0 otherwise. Proof. Only the last relation has to be shown. Since t2s−ν =
2s−ν
(2s−ν)
B2s−ν−2j T2s−ν−2j (x),
j=0
where, by [23, equation (1.148)], ⎧
⎨ 1−2(s−μ) 2(s − μ) if ν = 2μ, 2 (2s−ν) s−μ = B0 ⎩ 0 otherwise, it follows that
+1 ν ν+1
+1 2s+1 t dν dt t d − xν+1 2s−ν dt √ √ = − − t dxν −1 t − x 1 − t2 x=0 dxν t−x 1 − t2 x=0 −1
+1 2s−ν dt t dν √ (4.6) + ν xν+1 − dx t − x 1 − t2 x=0 −1 (2s−ν)
= (ν)!B0
.
Corollary 4.1. Put
+1 2s+1 2s+2 t; v Tn dt v 2s+2 (t) √ I(x) = − . t−x 1 − t2 −1
Then the following asymptotics hold if 2κ + 2 < μ: 2κ+1 (2κ)
(sin ψν )
(4.7) (2κ)
I
(xν ) = n
2κ
(2κ) v(xν )s2s+1 (Φ(ψν ))
1 1+O , n
where s2s+1 (Φ(ψν )) is given by (4.3), and, if 2κ + 3 < μ,
xν v (xν ) 1 (2κ+1) (2κ) + (κ + 1) (xν ) = (2κ + 1)I (xν ) 1+O (4.8) I . 2 v(xν ) 1 − xν n
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2653
Proof. First let us note that dj+1 j+1 (sm (Φ(ϕ))) = s(j+1) (Φ(ϕ)) (Φ (ϕ)) m dϕj+1 j j−1 + i s(j) Φ (ϕ) m (Φ(ϕ)) (Φ (ϕ)) i=0
+ O (Φ (ϕ))
j−3
.
˙ Since Φ (ϕ) = O(n), it follows that (4.9)
1 d2κ (2κ) 2κ (v(cos ϕ)sm (Φ(ϕ))) = sm (Φ(ϕ))n v(cos ϕ) 1 + O , dϕ2κ n (2κ+1)
and since sm
(4.10)
(Φ(ψν )) = 0, that
d2κ+1 (v(cos ϕ)sm (Φ(ϕ)))ϕ=ψν dϕ2κ+1
d d2κ (v(cos ϕ)) = (2κ + 1) (sm (Φ(ϕ)))ϕ=ψν 2κ dϕ ϕ=ψν dϕ 2κ−1 + O (Φ (ψν ))
1 = −(2κ + 1)v (xν ) sin ψν n2κ s(2κ) (Φ(ψ )) 1 + O . ν m n
Now by Theorem 4.1 dj dj (sin ϕ I(cos ϕ)) = (v(cos ϕ)sm (Φ(ϕ)))ϕ=ψν + O (4.11) ϕ=ψ ν dϕj dϕj
1 nμ−j−1
.
With the help of Remark 3.1, (4.9), (4.10), and induction arguments, it follows that j (4.12) I (j) (x)x=xν = O n2[ 2 ] which yields, by Remark 3.1, (4.11) in conjunction with (4.9), relation (4.7). Concerning relation (4.8), we observe that by Remark 3.1 and (4.12), (−1)
d2κ+1 (sin ϕ I(cos ϕ))ϕ=ψν = (sin ψν )2κ+2 I (2κ+1) (xν ) dϕ2κ+1 − (2κ + 1)(κ + 1)(sin ψν )2κ (cos ψν )I (2κ) (xν )
+ O n2κ−2 ,
which gives by (4.11), (4.10), and (4.7) the assertion. Again the asymptotics could be given up to higher order. 5. Asymptotics of Gauss–Tur´ an quadrature weights. The main result of the paper is now as follows. Theorem 5.1. Let s ∈ N0 , and suppose that w is positive on [−1, 1] and that m+α , with m + α > (2s + 2)2 , where m ∈ N, α ∈ [0, 1). Put w(cos ϕ) ∈ C2π 2−2s
s s
x + (2j)2 = aj,s xs−j , j=1
j=0
2654
FRANZ PEHERSTORFER
and let
n 2s w(x) f (x) √ dx = Aj,ν,n,s f (j) (xν,n ) + Rn,s (f ) π 1 − x2 −1 ν=1 j=0 +
be the Gauss–Tur´ an quadrature formula, i.e., Rn,s (p) = 0 for p ∈ P2(s+1)n−1 . Then the quadrature weights associated with nodes xν,n ∈ [−1+, 1−], where > 0 arbitrary but fixed, have the following asymptotic representation, j = 0, . . . , s:
s−j
aj,s 1 − x2ν,n w(xν,n ) 1 A2s−2j,ν,n,s = 1 + O (5.1) (s!)2 n2s+1−2j n and
A2s−2j−1,ν,n,s =
−(2s − 2j − 1) xν,n w (xν,n ) + b j,s 2 1 − x2ν,n w(xν,n )
(2s − 2j)A2s−2j,ν,n,s ,
(5.2) where bj,s = 1 for j = 0, 1 and all s ∈ N0 , and otherwise bj,s is an unknown constant which depends on j and s only, in particular not on the weight function w. Conjecture: We conjecture that for all s ∈ N0 (5.3)
bj,s = 1 for j = 0, . . . , s.
Corollary 5.1. For fixed s ∈ N0 let +1 n w(x) 2s G G (5.4) g(x)Tn,s (x; w) √ dx = Bν,n,s g(xν,n ) + Rn,s (g) π 1 − x2 −1 ν=1 G be the Gaussian quadrature formula, i.e., Rn,s (g) = 0 for g ∈ P2n−1 . Then, under the assumptions of Theorem 5.1 on the weight function w, the Gaussian weights associated with nodes xν,n ∈ [−1 + , 1 − ], > 0, satisfy
1 1 1 2s G nBν,n,s = 2s (w(xν,n )) s+1 1 + O . s 2 n
The connection between Gauss–Tur´ an quadrature formulas and (5.4) has been observed in [15]. 6. Proof of Theorem 5.1. First we need some auxiliaries concerning the representation of the quadrature weights, having in mind (1.11) and (1.14). Lemma 6.1. Let pn−1 (x) be a polynomial of degree n−1 such that for j = 1, . . . , s, (6.1)
(j) pn−1 (x) 1 2[ 2j ] = cj (x)n 1+O pn−1 (x) n
for x ∈ En ⊂ [−1, 1], where cj (x) is bounded on ∪n∈N En . Put ˜ r (x) = (pn−1 (x))r for r ∈ N. Ω Then for any r ∈ N, j = 1, . . . , s, (6.2)
j ˜ (j) Ω r (x) = O n2[ 2 ] ˜ r (x) Ω
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2655
on En , where the constant in the O() term does not depend on j and n. Now suppose that for every j = 2l, 1 ≤ j ≤ s, the function c2l (x) in (6.1) is of the form c2l (x) = cl /(c(x))2l ,
(6.3)
cl ∈ R\{0}. Then for any r ∈ N, 1 ≤ 2l ≤ s,
2l ˜ (2l) Ω n 1 r (x) = c˜l,r (6.4) 1+O for x ∈ En , ˜ r (x) c(x) n Ω where c˜l,r ∈ R. If in addition for every j = 2l + 1, 1 ≤ 2l + 1 ≤ s, c2l+1 (x) =
(6.5)
dl f (x) + el g(x) , (c(x))2l
then for any r ∈ N, 1 ≤ 2l + 1 ≤ s,
2l ˜ (2l+1) Ω 1 n (x) r ˜ (6.6) (dl,r f (x) + e˜l,r g(x)) 1 + O = for x ∈ En . ˜ c(x) n Ωr (x) Proof. Relation (6.2) follows immediately by
(6.7)
˜ (m) Ω 1 dm−1 r+1 = r+1 m−1 ((r + 1)prn−1 pn−1 ) ˜ r+1 p dx Ω m−1 m − 1 (prn−1 )(m−1−κ) p(κ+1) n−1 = (r + 1) r κ p p n−1 n−1 κ=0 =
m−1
m−(κ+1) (κ+1) O n2[ 2 ] O n2[ 2 ] ,
κ=0
where in the last equality we used the induction hypothesis with respect to m on (6.2) and (6.1). Concerning (6.4), since ˜ (2l)
2 Ω 1 d2l−2 r+1 r−1 r (r + 1)rp p = + rp p n−1 n−1 n−1 n−1 ˜ r+1 ˜ r+1 dx2l−2 Ω Ω 2l−2 κ (κ+1−ν) (ν+1) ˜ (2l−2−κ) Ω pn−1 κ pn−1 r−1 = r(r − 1) ˜ r−1 ν pn−1 pn−1 Ω κ=0 ν=0 (2l−2−κ) ˜r 2l−2 (κ+2) Ω pn−1 +r , ˜r pn−1 Ω κ=0 it follows by (6.2) and (6.1) that the summands in the first sum may reach order ˙ 2l ) only if all three indices κ, κ + 1 − ν, and ν + 1 are even. Recall that O( ˙ ) O(n denotes the precise order. Similarly the summands in the second sum may reach ˙ 2l ) only if κ is even. By induction hypothesis on (6.4) and the form of p(2j) /pn−1 O(n n−1 given by (6.1) and (6.2), the assertion follows. Equation (6.6) follows with the help of (6.7), (6.4), and induction arguments, observing that in (6.7) m − 1 − κ is odd and κ + 1 is even or conversely, since m is supposed to be odd.
2656
FRANZ PEHERSTORFER
Let
(6.8)
Ων (x) =
Tn (x; w) x − xν
2s+1 ,
where xν is a zero of Tn (x; w) and +1 dt Tn (x; w)2s+1 − Tn (t; w)2s+1 w(t) √ (6.9) . I(x) = x − t 1 − t2 −1 As already mentioned it has been shown by Stancu [25] that the weights Ai,ν are given by
I d2s−i ˜ Ai,ν := i!(2s − i)!Ai,ν = 2s−i (6.10) (xν ), dx Ων (j)
and thus if we know Ων (xν ) and I (j) (xν ) (or asymptotics of them) for j = 0, . . . , 2s, then (asymptotics of) the weights A˜2s−m,ν , m = 0, . . . , 2s, can be determined successively for 2s, 2s − 1, . . . with the help of the following relation (6.13) not observed so far. (j) Asymptotics for Ων (xν ) can be derived by Corollary 3.1 using the fact that by Taylor expansion of Tn (x; w) at the zero x = xν ,
(6.11)
Tn (x; w) x − xν
(j)
(j+1)
(xν ) =
Tn
(xν ; w) . j+1
Corollary 4.1 provides us with asymptotics for I (j) (xν ) since at the zeros xν of Tn (x; w), I (j) (xν ) = I (j) (xν ).
(6.12)
+1 1 w(t) dj √ Note that dx j (−−1 x−t 1−t2 dt) exists for j = 1, . . . , 2s, since w is supposed to be 2s-times continuously differentiable at least. Lemma 6.2. Put w(x) = v 2s+2 (x), and let m ∈ {0, . . . , 2s}, ν ∈ {1, . . . , n}, and set
I(x) dj ˜ A2s−j,ν (x) = j . dx Ων (x) Then m−1 (m−j) (x) ˜ I (m) (x) m Ων − A2s−j,ν (x). A˜2s−m,ν (x) = j Ων (x) Ω (x) ν j=0
(6.13) Furthermore,
A˜2s−m,ν (xν,n ) = O
(6.14)
1 m
n2s+1−2[ 2 ]
,
and, in particular, (6.15)
A˜2s−2l,ν (xν,n ) =
s−l
cl,s v 2s+2 (xν,n ) 1 − x2ν,n 1 1+O , 2s+1−2l n n
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2657
where the constant cs,l depends on s and l only. Furthermore,
v (xν,n ) xν,n A˜2s−2l−1,ν,n (xν ) = A˜2s−2l,ν,n (xν,n ) dl,s − el,s (6.16) , v(xν,n ) 1 − x2ν,n where the constants dl,s and el,s depend on s and l only. Proof. The relation (6.13) follows by induction arguments using the fact that −A˜2s−(m+1),ν = − A˜2s−m,ν m−1 (m−j) m Ων(m+1−j) Ω ν A˜2s−j,ν + A˜2s−(j+1),ν = j Ω Ω ν ν j=0 ⎛ ⎞
(m−j) m−1 m Ω I (x) ⎠ I (m+1) (x) m Ων − ν⎝ A˜2s−j,ν − − , Ων j=0 j Ων Ων Ων where the second expression is by induction hypothesis equal to (Ων /Ων )A˜2s−m,ν . Concerning the other statements, in Lemma 6.1 put pn−1 (x) = ˜ Ω2s+1 = Ων , and by (6.11) (j+1)
xν ; v 2s+2 Tn (j) pn−1 (xν ) = (6.17) . j+1
Tn (x;v 2s+2 ) ; x−xν
hence
Thus by Corollary√3.1 the assumptions of Lemma 6.1 are satisfied on [−1+, 1−], > 0, where c(x) = 1 − x2 . Hence, relation (6.2) from Lemma 6.1 holds. Now (6.14) follows by (6.13), using Corollary 4.1, Corollary 3.1, and induction arguments with respect to m. Concerning (6.15), let us first observe that by (6.14) and (6.2), the even terms in the sum (6.13) can be of maximal order 1/n2s+1−2l only. Now the assertion follows by relation (6.4) from Lemma 6.1, Corollary 3.1, and Corollary 4.1. The last statement follows with the help of (6.13), Corollary 4.1, (6.15), Lemma 6.1, and induction arguments using the facts that each summand in (6.13) is the product ˜ or, conversely, since m is odd now, and that, of an odd derivative and an even A, putting f (x) = v (x)/v(x) and g(x) = x/(1 − x2 ), (2l+1−2j)
Ων
Ων
A˜2s−2j,ν s−l
v 2s+2 (xν ) 1 − x2ν 1 ˜s,j f (xν ) + e˜s,j g(xν ) d = . 1 + O n2s+1−2l n
6.1. Proof of Theorem 5.1. By Lemma 6.2 only the constants in (6.15) and (6.16) remain to be determined. Recall that they do not depend on the weight function. Therefore, considering the Chebyshev weight function, i.e., v(x) ≡ 1, we obtain by (1.9) and (1.10) the constants we are looking for, up to the dl,s in front of v (xν,n )/v(xν,n ) in (6.16). To demonstrate that b0,s and b1,s in (5.2) are equal to one for arbitrary s ∈ N0 , we calculate with the help of (6.13) A˜2s,ν , . . . , A˜2s−3,ν explicitly. In fact, by what we know already, it would not be necessary to calculate the even A˜2s−2j,ν . First we note that
v(x ) (−1)ν−1 2s 1 I(xν ) ˜ ν A2s,ν = (6.18) , (x ))2s+1 2 s Ων (xν ) 22s (T 1 − xν n ν
2658
FRANZ PEHERSTORFER
where in the last equality we have used Corollary 4.1, (4.3), (6.12) and Remark 4.1. Concerning A˜2s−1,ν , we observe that cos ψν I(cos ψν ) − sin2 ψν I (cos ψν ) =
d v (xν ) (sin ϕI(cos ϕ))ϕ=ψν − sin2 ψν I(xν ), dϕ v(xν )
where in the last equality we used Theorem 4.1 in conjunction with (4.3). Hence
I (xν ) I(xν ) v (xν ) xν + , Ων (xν ) Ων (xν ) 1 − x2ν v(xν )
Ω (x ) v (xν ) xν ν) (− Ωνν (xνν ) + 1−x which yields by (6.13) A˜2s−1,ν ΩI(x 2 + v(x ) ). With the help of ν (xν ) ν ν (6.11) and Corollary 3.1, we get
2s + 1 xν v (xν ) Ων (xν ) − 2 , Ων (xν ) 2 1 − x2ν v(xν )
and by the first relation in (6.18) b0,s = 1 follows. Recall that w = v 2s+2 . Concerning A˜2s−2,ν , we note that with respect to n by the lower order of Ων (xν )/ Ων (xν ) and A˜2s−1,ν /Ων (xν ), we have by (6.13)
A˜2s−2,ν
(6.19)
I (2) (xν ) Ων (xν ) ˜ A2s,ν + Ων (xν ) Ων (xν )
,
which gives by Corollaries 3.1 and 4.1, (4.3), and Remark 4.1 the asymptotic value for A˜2s−2,ν . Finally, to calculate the asymptotic value of A˜2s−3,ν in a short way, we recall that by Corollary 4.1 I (3) (xν ) Ων (xν )
cos ψν v (xν ) I (2) (xν ) 6 2 +3 v(xν ) Ων (xν ) sin ψν
and insert for I (2) (xν )/Ων (xν ) the value from (6.19). Then using (6.13) and %
& (3) (2s + 1) −n2 Ων (xν ) v (xν ) xν = − (4s + 2) (2s + 3) , Ων (xν ) 2 1 − x2ν 1 − x2ν v(xν ) it follows, after surprising cancellation, that (3) xν Ων (xν ) v (xν ) ˜ ˜ A2s−2,ν , −6 −3 −A2s−3,ν 3 Ων (xν ) 1 − x2ν v(xν ) which yields b1,s = 1. 6.2. Proof of Corollary 5.1. It is well known that G Bν,n,s
1 = Tn (xν ; w)
+1
−1
Tn (t; w) 2s w(t) I(xν ) , T (t; w) √ dt = 2 t − xν n πT π 1−t n (xν ; w)
which gives, by (6.12), Corollaries 4.1 and 3.1, and Remark 4.1, the assertion.
´ QUADRATURE: ASYMPTOTICS OF WEIGHTS GAUSS–TURAN
2659
REFERENCES [1] N.I. Achieser, Vorlesungen u ¨ber Approximationstheorie, Akademie Verlag, Berlin, 1967. [2] S.N. Bernstein, Sur les polynomes orthogonaux relatifs a un segment fini, J. Math., 9 (1930), pp. 127–177, 10 (1931), pp. 219–286. [3] S.N. Bernstein, Sur une classe de polynomes orthogonaux, Commun. Soc. Math. de Kharkov, IV (1930), pp. 79–93. [4] B. Bojanov, On a quadrature formula of Micchelli and Rivlin, J. Comput. Appl. Math., 70 (1996), pp. 349–356. [5] W. Gautschi, The circle theorem and related theorem for Gauss-type quadrature rules, Electron. Trans. Numer. Anal., 25 (2006), pp. 129–137. ´, S-orthogonality and construction of Gauss-Tur´ [6] W. Gautschi and G.V. Milovanovic an-type quadrature formulae, J. Comput. Appl. Math., 86 (1997), pp. 205–218. [7] A. Ghizzetti and A. Ossicini, Quadrature Formulae, Birkh¨ auser, Basel, 1970. [8] L. Gori and C.A. Micchelli, On weight functions which admit explicit Gauss-Tur´ an quadrature formulas, Math. Comp., 65 (1996), pp. 1567–1581. [9] O. Kis, Remark on mechanical quadrature, Acta Math. Acad. Sci. Hungar., 8 (1957), pp. 473– 476 (in Russian). ´ and F. Peherstorfer, Asymptotic representation of Lp -minimal polynomials 1 < [10] A. Kroo p < ∞, Constr. Approx., 25 (2007), pp. 29–39. [11] G.G. Lorenz, Approximation of Functions, Chelsea, New York, 1986. [12] C.A. Micchelli, The Fundamental Theorem of Algebra for Monosplines with Multiplicities, ISNM 20, Birkh¨ auser, Basel, 1972, pp. 419–430. [13] C.A. Micchelli and T.J. Rivlin, Tur´ an formulae and highest precision quadrature rules for Chebyshev coefficients, IBM J. Res. Develop., 16 (1972), pp. 372–379. [14] C.A. Micchelli and A. Sharma, On a problem of Tur´ an: Multiple node Gaussian quadrature, Rend. Mat. Ser. VII, 3 (1983), pp. 529–552. ´, Construction of s-orthogonal polynomials and Tur´ [15] G.V. Milovanovic an quadrature formulae, in Proceedings of the Numerical Methods and Approximation Theory III Niˇs, 1987, pp. 311– 328. ´, Quadrature with multiple nodes, power orthogonality and moment pre[16] G.V. Milovanovic serving spline approximation, J. Comput. Appl. Math., 127 (2001), pp. 267–286. ´ and M.M. Spalevic ´, Error bounds for Gauss-Tur´ [17] G.V. Milovanovic an quadrature formulae of analytic functions, Math. Comp., 72 (2003), pp. 1855–1872. ´, M.M. Spalevic ´, and A.S. Cvetkovic ´, Calculation of Gaussian-type [18] G.V. Milovanovic quadratures with multiple nodes, Math. Comput. Modelling, 39 (2004), pp. 325–347. [19] A. Ossicini and F. Rosati, Sulla convergenza dei funzionali ipergaussiani, Rend. Mat., 11 (1978), pp. 97–108. [20] F. Peherstorfer, Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and Lq -minimal polynomials, q ∈ [1, ∞], J. Approx. Theory, to appear. [21] F. Peherstorfer, Positive quadrature formulas III: Asymptotics of weights, Math. Comp., 77 (2008), pp. 2241–2259. [22] R.D. Riess, Gauss-Tur´ an quadratures of Chebyshev type and error formulae, Computing, 15 (1975), pp. 173–179. [23] T.J. Rivlin, Chebyshev polynomials. From Approximation Theory to Algebra and Number Theory, 2nd ed., Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1990. [24] Y.G. Shi and G. Xu, Construction of sigma-orthogonal polynomials and Gaussian quadrature formulas, Adv. Comput. Math., 27 (2007), pp. 79–94. [25] D.D. Stancu, Asupra unor formule generale de integrare numeric˘ a, Acad. R.P. Romˆ ane Stud. Cerc. Mat., 9 (1958), pp. 209–216. [26] A.H. Stroud and D.D. Stancu, Quadrature formulas with multiple Gaussian nodes, J. SIAM Ser. B: Numer. Anal., 2 (1965), pp. 129–143. ¨ , Orthogonal Polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. [27] G. Szego Soc., Providence, RI, 1967. [28] V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math., 81 (2000), pp. 283–303. ´ n, On the theory of the mechanical quadrature, Acta Sci. Math. (Szeged), 12 (1950), [29] P. Tura pp. 30–37. ´ n, On some open problems of approximation theory, J. Approx. Theory, 29 (1980), [30] P. Tura pp. 23–85. [31] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge, University Press, London, 1959.
