On asymptotic behavior of Battle-Lemarié scaling ... - Semantic Scholar
On asymptotic behavior of Battle-Lemari´e scaling functions and wavelets ∗ Hong Oh Kim†, Rae Young Kim‡and Ja Seong Ku§ January 4, 2005
Abstract We show that the ‘centered’ Battle-Lemari´e scaling function and wavelet of order n converge in Lq (2 ≤ q ≤ ∞), uniformly in particular, to the Shannon scaling function and wavelet as n tends to the infinity.
AMS 2000 Subject Classification: 41A60, 42A20, 42C40. Keywords: B-spline, Battle-Lemari´e wavelet, Shannon wavelet.
1
Introduction
The Battle-Lemari´e scaling function is obtained by applying the orthogonalization trick to the B-spline functions. In order to get the symmetry about the origin, we will take the centered B-spline of order n as B1 (x) := χ[−1/2,1/2) (x), Bn (x) := Bn−1 ∗ B1 (x), n = 2, 3, · · · . The Fourier transform of Bn then has the form µ ¶ sin w/2 n ˆ ˆn (w/2). Bn (w) = = (cos w/4)n B w/2 ∗
(1.1)
(1.2)
This work was supported by Korea Research Foundation Grant (KRF-2002-070C00004). † Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea ([email protected]) ‡ Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea ([email protected]) § Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea ([email protected])
1
2 We note Φn (w) :=
X
ˆn (w + 2πk)|2 |B
k∈Z
= (cos w/4)2n Φn (w/2) + (sin w/4)2n Φn (w/2 + π).
(1.3)
and apply the orthonormalization trick to Bn to get the Battle-Lemari´e scaling function ϕn of order n defined by ˆn (w) B ϕˆn (w) := p = mn (w/2)ϕˆn (w/2), Φn (w) where
(1.4)
s mn (w) = (cos w/2)n
Φn (w) . Φn (2w)
(1.5)
The filter mn is 2π-periodic if n is even and 4π-periodic if n is odd. We note that mn is a CQF filter in the sense that |mn (w)|2 + |mn (w + π)|2 = 1.
(1.6)
The corresponding wavelet is given by ψˆn (2w) = e−iw Mn (w)ϕˆn (w),
(1.7)
where s n
Mn (w) = |(sin w/2)|
Φn (w + π) = |mn (w + π)|. Φn (2w)
(1.8)
Note that Mn is 2π-periodic. Therefore, if n is even, the function ϕn defines an orthonormal scaling function for a multiresolution analysis. If n is odd, ϕn does not define a scaling function of a multiresolution analysis, but they have the same asymptotic behavior as will be seen in the main theorem in this article. See [1, 4, 7] for the standard Battle-Lemari´e wavelet. In this short article, we show that the Battle-Lemari´e scaling function ϕn and its corresponding wavelet ψn tend, in Lq (R)(2 ≤ q ≤ ∞), in particular uniformly, to the Shannon scaling function ϕSH and Shannon wavelet ψSH as n approaches to the infinity, where ϕˆSH (w) := χ[−π,π] (w)
3 and
ψˆSH (w) := e−iw/2 χ[−2π,−π] S[π,2π] (w).
It is known that the centered B-spline Bn tends to the Gaussian distribution as n → ∞ [8, 11]. For the asymptotic behavior of Daubechies filters and scaling functions, see [5, 9, 10]. The idea of the proof also appears in [3, 6] for the analogous asymptotic behaviors of other family of wavelets.
2
Main result
We need the following property of the Euler-Frobenius polynomials. Proposition 2.1 ([2]) Let n be any positive integer and let E2n−1 be the Euler-Frobenius polynomial of degree 2n − 2 defined by E2n−1 (z) := (2n − 1)!
2n−2 X
B2n (−n + k + 1)z k .
k=0
Then the 2n − 2 roots, {λn,j : j = 1, · · · , 2n − 2}, of E2n−1 has the properties that λn,2n−2 < λn,2n−3 · · · < λn,n < −1 < λn,n−1 < · · · < λn,1 < 0; λn,j λn,2n−1−j = 1, (j = 1, 2, · · · , n − 1) and n−1 Y 1 − 2λn,k cos w + λ2n,k eiw(n−1) 1 −iw Φn (w) = E2n−1 (e ) = . (2n − 1)! (2n − 1)! |λn,k | k=1
Therefore, Φ Sn (x + π) ≤ Φn (x) on [−π/2, π/2] and Φn (x) ≤ Φn (x + π) on [−π, −π/2) (π/2, π]. The 2π-periodic filters for the Shannon scaling function and wavelet are given, respectively, as ½ 1, |w| ≤ π/2; 0 mSH (w) := (2.1) 0, π/2 < |w| ≤ π, and ½ mH SH (w)
:=
0, |w| < π/2; 1, π/2 ≤ |w| ≤ π.
(2.2)
4 We also define a 4π-periodic filter m1SH ∈ L2 ([−2π, 2π]) by 1, |w| ≤ π/2; 0, π/2 < |w| ≤ π; m1SH (w) := −1, π < |w| < 3π/2; 0, 3π/2 ≤ |w| ≤ 2π.
(2.3)
Notice that ϕˆSH (w) := χ[−π,π] (w) =
∞ Y j=1
m0SH (w/2j )
=
∞ Y
m1SH (w/2j ).
(2.4)
j=1
Lemma 2.2 As n approaches ∞, (a) m2n (w) converges to m0SH (w) for every w ∈ [−π, π] \ {±π/2}; (b) m2n+1 (w) converges to m1SH (w) for every w ∈ [−2π, 2π]\{±π/2, ±3π/2}; and so, Mn (w) converges to mH SH (w) for every w ∈ [−π, π] \ {±π/2}. S Proof. For w ∈ (−3π/2, −π/2) (π/2, 3π/2), Φn (w) ≤ Φn (w + π) by Proposition 2.1. By use of (1.3), we see that (cos w/2)2n Φn (w) Φn (2w) (cos w/2)2n (sin w/2)2n Φn (w) = (sin w/2)2n (cos w/2)2n Φn (w) + (sin w/2)2n Φn (w + π) 1 (sin w/2)2n Φn (w) ≤ (tan w/2)2n (sin w/2)2n Φn (w + π) 1 ≤ → 0 as n → ∞, (tan w/2)2n S S Now, let w ∈ (−2π, −3π/2) (−π/2, π/2) (3π/2, 2π). Note that |mn (w)|2 + |mn (w + π)|2 = 1. Hence limn→∞ |mn (w)| = 1. Since m2n (w) is 2πperiodic and positive by the definition of m2n , limn→∞ m2n (w) = 1. Therefore, (a) is satisfied. For (b), note that m2n+1 is 4π-periodic. If w ∈ (−π/2, π/2), then m2n+1 S (w) is positive. Hence limn→∞ m2n+1 (w) = 1. If w ∈ (−2π, −3π/2) (3π/2, 2π), then m2n+1 (w) is negative. Therefore limn→∞ m2n+1 (w) = −1. ¤ |mn (w)|2 =
We define an auxiliary 2π-periodic continuous function M , via ½ 1, |w| ≤ π2 ; M (w) = 3/2 3 2 (cos x/2) , π2 < |w| ≤ π, for the domination of mn in the following Lemma.
(2.5)
5 Lemma 2.3 (a) 0 ≤ |mn (w)| ≤ M (w), n = 3, 4, · · · . (b) M (w) = (cos w/2)3 S(w), and supw |S(w)| = 23/2 , where ½ 1/(cos w/2)3 , |w| ≤ π/2, S(x) = 23/2 , π/2 < |x| ≤ π. Q Therefore, ϕ(w) ˆ := ∞ M (w/2j ) has the decay |ϕ(w)| ˆ ≤ C(1 + |w|)−3/2 . ½ j=1 2, for all w, (c) |mn (w) − 1| ≤ 2|w|/π, |w| ≤ π/2. Proof. The estimates of (a) and (b) are trivial. The decay of ϕ(w) ˆ follows from Theorem 5.5 of [2]. For (c), we note that |mn (w) − 1| ≤ |mn (w)| + 1 ≤ 2. For |w| ≤ π/2 and for n ≥ 1, | tan w/2|2n ≤ | tan w/2| ≤ 2|w|/π. Therefore, we have for |w| ≤ π/2, ¯ ¯s ¯ ¯ Φ (w) ¯ ¯ n (cos w/2)n − 1¯ |mn (w) − 1| = ¯ ¯ ¯ Φn (2w) ¯ ¯p p ¯ Φ (w)(cos w/2)n − Φ (2w) ¯ ¯ ¯ n n p =¯ ¯ ¯ ¯ Φn (2w) ¯ ¯ ¯ ¯ 2n − Φ (2w) Φ (w)(cos w/2) ¯ ¯ n n p p = ¯p ¯ ¯ Φn (2w)( Φn (w)(cos w/2)n + Φn (2w)) ¯ (sin w/2)2n Φn (w + π) Φn (2w) (sin w/2)2n (cos w/2)2n Φn (w + π) = (cos w/2)2n Φn (2w) (cos w/2)2n Φn (w + π) = (tan w/2)2n (cos w/2)2n Φn (w) + (sin w/2)2n Φn (w + π) Φn (w + π) ≤ (tan w/2)2n Φn (w) 2 ≤ |w|, π ≤
where we used the fact that Φn (w + π) ≤ Φn (w) on [−π/2, π/2]. ¤ Q j Lemma 2.4 (a) For each fixed w, ϕˆn (w) = ∞ j=1 mn (w/2 ) converges uniformly on n.
6 (b) ϕˆn (w) → ϕˆSH (w) pointwise a.e. as n → ∞. (c) ψˆn (w) → ψˆSH (w) pointwise a.e. as n → ∞. Proof. (a) Fix w and choose j0 so that |w/2j0 | ≤ π/2. By Lemma 2.3(c), ∞ X j=1
j0 ∞ X X w w w |mn ( j ) − 1| = |mn ( j ) − 1| + |mn ( j ) − 1| 2 2 2 j=1
j=j0 +1
∞ X
≤ 2j0 +
j=j0 +1
2 |w| 2 |w| = 2j0 + , j π 2 π 2j0
uniformly on n. Therefore, the product ϕn (w) converges uniformly on n. j (b) Fix w ∈ / ∪∞ j=1 2 (±π + 2πZ) and let ² > 0. By (a) we can choose j1 (independent of n) so that |ϕˆn (w) −
j1 Y
mn (
j=1
and |ϕˆSH (w) −
j1 Y
w )| < ², 2j
miSH (
j=1
w )| < ², 2j
for i = 0, 1. Therefore, we have |ϕˆn (w) − ϕˆSH (w)| ≤ |ϕˆn (w) −
j1 Y
mn (
j=1
+|
j1 Y j=1
+|
j1 Y
w )| 2j
j1
Y w w miSH ( j )| mn ( j ) − 2 2 j=1
miSH (
j=1
< 2² + |
j1 Y j=1
w ) − ϕˆSH (w)| 2j j1
Y w w mn ( j ) − miSH ( j )|. 2 2 j=1
We choose i := i(n) = 0 (n=even), 1 (n=odd). Note that w/2j ∈ / ±π/2 + 2πZ for any j ≥ 1. Since m2n (w/2j ) → m0SH (w/2j ) and m2n+1 (w/2j ) → m1SH (w/2j ) as n → ∞, we can choose n0 ∈ N so that |
j1 Y j=1
mn (w/2j ) −
j1 Y j=1
i(n)
mSH (w/2j )| < ² for n ≥ n0 .
7
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Figure 1: (a) ϕ4 (b) ϕ10 (c) ϕSH (d) ψ4 (e) ψ10 (f) ψSH . j Therefore, ϕˆn (w) → ϕˆSH (w) pointwise as n → ∞ for w ∈ / ∪∞ j=1 2 (±π+2πZ). (c) The proof follows from (b) in view of the definition of ψˆn in (1.7). It is also proved in [7] with a different proof. ¤ Now, we state and prove our main result.
Theorem 2.5 (a) For 1 ≤ p < ∞, ||ϕˆn − ϕˆSH ||Lp (R) → 0 and ||ψˆn − ψˆSH ||Lp (R) → 0 as n → ∞. (b) For 2 ≤ q ≤ ∞, ||ϕn − ϕSH ||Lq (R) → 0 and ||ψn − ψSH ||Lq (R) → 0, as n → ∞. In particular, ϕn → ϕSH and ψn → ψSH uniformly on R as n → ∞.
8 Proof. Note that |ϕˆn (w)| = ≤
∞ Y j=1 ∞ Y j=1
|mn ( |M (
w )| 2j
w )| = |ϕ(w)| ˆ ≤ C(1 + |w|)−3/2 , 2j
|ψˆn (w)| = |Mn (w/2)||ϕˆn (w/2)| ≤ C(1 + |w/2|)−3/2 . Therefore (a) follows from Lemma 2.4 by the dominated convergence theorem. (b) follows from (a) by Hausdorff-Young inequality: ||f ||Lq (R) ≤ ||fˆ||Lp (R) , for 1 ≤ p ≤ 2, where q is the conjugate exponent to p.
¤
Remark. We illustrate the convergence of the Battle-Lemari´e scaling functions and wavelets (for n = 4 and 10) to the Shannon scaling function and wavelet in Figure 1.
References [1] G. Battle, A block spin construction of ondelettes. I. Lemari´e functions, Comm. Math. Phys. 110 (1987) 601-615. [2] C.K. Chui, An Introduction to Wavelets, Academic Press, San Diego, 1992. [3] M. Cotronei, M.L. LoCascio, H.O. Kim, C.A. Micchelli and T. Sauer, Refinable functions from Blaschke Products, preprint. [4] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [5] D. Kateb and P.G. Lemarie-Rieusset, Asymptotic behavior of the Daubechies filters, Appl. Comput. Harmon. Anal. 2 (1995) 398-399. [6] H.O. Kim, R.Y. Kim and K.J. Ku, Wavelet frames from Butterworth filters, preprint. [7] P.G. Lemari´e, Ondelettes `a localisation exponentielle, J. Math. Pures Appl. (9), 67 (1988) 227-236.
9 [8] I.J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Theory, 2 (1969) 167-206. [9] J. Shen and G. Strang, Asymptotic analysis of Daubechies polynomials, Proc. Amer. Math. Soc. 124 (1996) 3819-3833. [10] J. Shen and G. Strang, Asymptotics of Daubechies filters, scaling functions, and wavelets, Appl. Comput. Harmon. Anal. 5 (1998) 312-331. [11] M. Unser, A. Aldroubi and M. Eden, On the Asymptotic Convergence of B-Spline Wavekets to Gabor Functions, IEEE Trans. Inform. Theory, 38 (1992) 864-872.
Abstract We show that the ‘centered’ Battle-Lemari´e scaling function and wavelet of order n converge in Lq (2 ≤ q ≤ ∞), uniformly in particular, to the Shannon scaling function and wavelet as n tends to the infinity.
AMS 2000 Subject Classification: 41A60, 42A20, 42C40. Keywords: B-spline, Battle-Lemari´e wavelet, Shannon wavelet.
1
Introduction
The Battle-Lemari´e scaling function is obtained by applying the orthogonalization trick to the B-spline functions. In order to get the symmetry about the origin, we will take the centered B-spline of order n as B1 (x) := χ[−1/2,1/2) (x), Bn (x) := Bn−1 ∗ B1 (x), n = 2, 3, · · · . The Fourier transform of Bn then has the form µ ¶ sin w/2 n ˆ ˆn (w/2). Bn (w) = = (cos w/4)n B w/2 ∗
(1.1)
(1.2)
This work was supported by Korea Research Foundation Grant (KRF-2002-070C00004). † Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea ([email protected]) ‡ Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea ([email protected]) § Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea ([email protected])
1
2 We note Φn (w) :=
X
ˆn (w + 2πk)|2 |B
k∈Z
= (cos w/4)2n Φn (w/2) + (sin w/4)2n Φn (w/2 + π).
(1.3)
and apply the orthonormalization trick to Bn to get the Battle-Lemari´e scaling function ϕn of order n defined by ˆn (w) B ϕˆn (w) := p = mn (w/2)ϕˆn (w/2), Φn (w) where
(1.4)
s mn (w) = (cos w/2)n
Φn (w) . Φn (2w)
(1.5)
The filter mn is 2π-periodic if n is even and 4π-periodic if n is odd. We note that mn is a CQF filter in the sense that |mn (w)|2 + |mn (w + π)|2 = 1.
(1.6)
The corresponding wavelet is given by ψˆn (2w) = e−iw Mn (w)ϕˆn (w),
(1.7)
where s n
Mn (w) = |(sin w/2)|
Φn (w + π) = |mn (w + π)|. Φn (2w)
(1.8)
Note that Mn is 2π-periodic. Therefore, if n is even, the function ϕn defines an orthonormal scaling function for a multiresolution analysis. If n is odd, ϕn does not define a scaling function of a multiresolution analysis, but they have the same asymptotic behavior as will be seen in the main theorem in this article. See [1, 4, 7] for the standard Battle-Lemari´e wavelet. In this short article, we show that the Battle-Lemari´e scaling function ϕn and its corresponding wavelet ψn tend, in Lq (R)(2 ≤ q ≤ ∞), in particular uniformly, to the Shannon scaling function ϕSH and Shannon wavelet ψSH as n approaches to the infinity, where ϕˆSH (w) := χ[−π,π] (w)
3 and
ψˆSH (w) := e−iw/2 χ[−2π,−π] S[π,2π] (w).
It is known that the centered B-spline Bn tends to the Gaussian distribution as n → ∞ [8, 11]. For the asymptotic behavior of Daubechies filters and scaling functions, see [5, 9, 10]. The idea of the proof also appears in [3, 6] for the analogous asymptotic behaviors of other family of wavelets.
2
Main result
We need the following property of the Euler-Frobenius polynomials. Proposition 2.1 ([2]) Let n be any positive integer and let E2n−1 be the Euler-Frobenius polynomial of degree 2n − 2 defined by E2n−1 (z) := (2n − 1)!
2n−2 X
B2n (−n + k + 1)z k .
k=0
Then the 2n − 2 roots, {λn,j : j = 1, · · · , 2n − 2}, of E2n−1 has the properties that λn,2n−2 < λn,2n−3 · · · < λn,n < −1 < λn,n−1 < · · · < λn,1 < 0; λn,j λn,2n−1−j = 1, (j = 1, 2, · · · , n − 1) and n−1 Y 1 − 2λn,k cos w + λ2n,k eiw(n−1) 1 −iw Φn (w) = E2n−1 (e ) = . (2n − 1)! (2n − 1)! |λn,k | k=1
Therefore, Φ Sn (x + π) ≤ Φn (x) on [−π/2, π/2] and Φn (x) ≤ Φn (x + π) on [−π, −π/2) (π/2, π]. The 2π-periodic filters for the Shannon scaling function and wavelet are given, respectively, as ½ 1, |w| ≤ π/2; 0 mSH (w) := (2.1) 0, π/2 < |w| ≤ π, and ½ mH SH (w)
:=
0, |w| < π/2; 1, π/2 ≤ |w| ≤ π.
(2.2)
4 We also define a 4π-periodic filter m1SH ∈ L2 ([−2π, 2π]) by 1, |w| ≤ π/2; 0, π/2 < |w| ≤ π; m1SH (w) := −1, π < |w| < 3π/2; 0, 3π/2 ≤ |w| ≤ 2π.
(2.3)
Notice that ϕˆSH (w) := χ[−π,π] (w) =
∞ Y j=1
m0SH (w/2j )
=
∞ Y
m1SH (w/2j ).
(2.4)
j=1
Lemma 2.2 As n approaches ∞, (a) m2n (w) converges to m0SH (w) for every w ∈ [−π, π] \ {±π/2}; (b) m2n+1 (w) converges to m1SH (w) for every w ∈ [−2π, 2π]\{±π/2, ±3π/2}; and so, Mn (w) converges to mH SH (w) for every w ∈ [−π, π] \ {±π/2}. S Proof. For w ∈ (−3π/2, −π/2) (π/2, 3π/2), Φn (w) ≤ Φn (w + π) by Proposition 2.1. By use of (1.3), we see that (cos w/2)2n Φn (w) Φn (2w) (cos w/2)2n (sin w/2)2n Φn (w) = (sin w/2)2n (cos w/2)2n Φn (w) + (sin w/2)2n Φn (w + π) 1 (sin w/2)2n Φn (w) ≤ (tan w/2)2n (sin w/2)2n Φn (w + π) 1 ≤ → 0 as n → ∞, (tan w/2)2n S S Now, let w ∈ (−2π, −3π/2) (−π/2, π/2) (3π/2, 2π). Note that |mn (w)|2 + |mn (w + π)|2 = 1. Hence limn→∞ |mn (w)| = 1. Since m2n (w) is 2πperiodic and positive by the definition of m2n , limn→∞ m2n (w) = 1. Therefore, (a) is satisfied. For (b), note that m2n+1 is 4π-periodic. If w ∈ (−π/2, π/2), then m2n+1 S (w) is positive. Hence limn→∞ m2n+1 (w) = 1. If w ∈ (−2π, −3π/2) (3π/2, 2π), then m2n+1 (w) is negative. Therefore limn→∞ m2n+1 (w) = −1. ¤ |mn (w)|2 =
We define an auxiliary 2π-periodic continuous function M , via ½ 1, |w| ≤ π2 ; M (w) = 3/2 3 2 (cos x/2) , π2 < |w| ≤ π, for the domination of mn in the following Lemma.
(2.5)
5 Lemma 2.3 (a) 0 ≤ |mn (w)| ≤ M (w), n = 3, 4, · · · . (b) M (w) = (cos w/2)3 S(w), and supw |S(w)| = 23/2 , where ½ 1/(cos w/2)3 , |w| ≤ π/2, S(x) = 23/2 , π/2 < |x| ≤ π. Q Therefore, ϕ(w) ˆ := ∞ M (w/2j ) has the decay |ϕ(w)| ˆ ≤ C(1 + |w|)−3/2 . ½ j=1 2, for all w, (c) |mn (w) − 1| ≤ 2|w|/π, |w| ≤ π/2. Proof. The estimates of (a) and (b) are trivial. The decay of ϕ(w) ˆ follows from Theorem 5.5 of [2]. For (c), we note that |mn (w) − 1| ≤ |mn (w)| + 1 ≤ 2. For |w| ≤ π/2 and for n ≥ 1, | tan w/2|2n ≤ | tan w/2| ≤ 2|w|/π. Therefore, we have for |w| ≤ π/2, ¯ ¯s ¯ ¯ Φ (w) ¯ ¯ n (cos w/2)n − 1¯ |mn (w) − 1| = ¯ ¯ ¯ Φn (2w) ¯ ¯p p ¯ Φ (w)(cos w/2)n − Φ (2w) ¯ ¯ ¯ n n p =¯ ¯ ¯ ¯ Φn (2w) ¯ ¯ ¯ ¯ 2n − Φ (2w) Φ (w)(cos w/2) ¯ ¯ n n p p = ¯p ¯ ¯ Φn (2w)( Φn (w)(cos w/2)n + Φn (2w)) ¯ (sin w/2)2n Φn (w + π) Φn (2w) (sin w/2)2n (cos w/2)2n Φn (w + π) = (cos w/2)2n Φn (2w) (cos w/2)2n Φn (w + π) = (tan w/2)2n (cos w/2)2n Φn (w) + (sin w/2)2n Φn (w + π) Φn (w + π) ≤ (tan w/2)2n Φn (w) 2 ≤ |w|, π ≤
where we used the fact that Φn (w + π) ≤ Φn (w) on [−π/2, π/2]. ¤ Q j Lemma 2.4 (a) For each fixed w, ϕˆn (w) = ∞ j=1 mn (w/2 ) converges uniformly on n.
6 (b) ϕˆn (w) → ϕˆSH (w) pointwise a.e. as n → ∞. (c) ψˆn (w) → ψˆSH (w) pointwise a.e. as n → ∞. Proof. (a) Fix w and choose j0 so that |w/2j0 | ≤ π/2. By Lemma 2.3(c), ∞ X j=1
j0 ∞ X X w w w |mn ( j ) − 1| = |mn ( j ) − 1| + |mn ( j ) − 1| 2 2 2 j=1
j=j0 +1
∞ X
≤ 2j0 +
j=j0 +1
2 |w| 2 |w| = 2j0 + , j π 2 π 2j0
uniformly on n. Therefore, the product ϕn (w) converges uniformly on n. j (b) Fix w ∈ / ∪∞ j=1 2 (±π + 2πZ) and let ² > 0. By (a) we can choose j1 (independent of n) so that |ϕˆn (w) −
j1 Y
mn (
j=1
and |ϕˆSH (w) −
j1 Y
w )| < ², 2j
miSH (
j=1
w )| < ², 2j
for i = 0, 1. Therefore, we have |ϕˆn (w) − ϕˆSH (w)| ≤ |ϕˆn (w) −
j1 Y
mn (
j=1
+|
j1 Y j=1
+|
j1 Y
w )| 2j
j1
Y w w miSH ( j )| mn ( j ) − 2 2 j=1
miSH (
j=1
< 2² + |
j1 Y j=1
w ) − ϕˆSH (w)| 2j j1
Y w w mn ( j ) − miSH ( j )|. 2 2 j=1
We choose i := i(n) = 0 (n=even), 1 (n=odd). Note that w/2j ∈ / ±π/2 + 2πZ for any j ≥ 1. Since m2n (w/2j ) → m0SH (w/2j ) and m2n+1 (w/2j ) → m1SH (w/2j ) as n → ∞, we can choose n0 ∈ N so that |
j1 Y j=1
mn (w/2j ) −
j1 Y j=1
i(n)
mSH (w/2j )| < ² for n ≥ n0 .
7
1.4
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−8
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−0.4 −10
−8
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(a)
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(d)
−4
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10
(c)
1
−8
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(b)
1
−10
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−1
−8
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0
(e)
2
4
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−10
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−2
0
2
4
6
(f)
Figure 1: (a) ϕ4 (b) ϕ10 (c) ϕSH (d) ψ4 (e) ψ10 (f) ψSH . j Therefore, ϕˆn (w) → ϕˆSH (w) pointwise as n → ∞ for w ∈ / ∪∞ j=1 2 (±π+2πZ). (c) The proof follows from (b) in view of the definition of ψˆn in (1.7). It is also proved in [7] with a different proof. ¤ Now, we state and prove our main result.
Theorem 2.5 (a) For 1 ≤ p < ∞, ||ϕˆn − ϕˆSH ||Lp (R) → 0 and ||ψˆn − ψˆSH ||Lp (R) → 0 as n → ∞. (b) For 2 ≤ q ≤ ∞, ||ϕn − ϕSH ||Lq (R) → 0 and ||ψn − ψSH ||Lq (R) → 0, as n → ∞. In particular, ϕn → ϕSH and ψn → ψSH uniformly on R as n → ∞.
8 Proof. Note that |ϕˆn (w)| = ≤
∞ Y j=1 ∞ Y j=1
|mn ( |M (
w )| 2j
w )| = |ϕ(w)| ˆ ≤ C(1 + |w|)−3/2 , 2j
|ψˆn (w)| = |Mn (w/2)||ϕˆn (w/2)| ≤ C(1 + |w/2|)−3/2 . Therefore (a) follows from Lemma 2.4 by the dominated convergence theorem. (b) follows from (a) by Hausdorff-Young inequality: ||f ||Lq (R) ≤ ||fˆ||Lp (R) , for 1 ≤ p ≤ 2, where q is the conjugate exponent to p.
¤
Remark. We illustrate the convergence of the Battle-Lemari´e scaling functions and wavelets (for n = 4 and 10) to the Shannon scaling function and wavelet in Figure 1.
References [1] G. Battle, A block spin construction of ondelettes. I. Lemari´e functions, Comm. Math. Phys. 110 (1987) 601-615. [2] C.K. Chui, An Introduction to Wavelets, Academic Press, San Diego, 1992. [3] M. Cotronei, M.L. LoCascio, H.O. Kim, C.A. Micchelli and T. Sauer, Refinable functions from Blaschke Products, preprint. [4] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [5] D. Kateb and P.G. Lemarie-Rieusset, Asymptotic behavior of the Daubechies filters, Appl. Comput. Harmon. Anal. 2 (1995) 398-399. [6] H.O. Kim, R.Y. Kim and K.J. Ku, Wavelet frames from Butterworth filters, preprint. [7] P.G. Lemari´e, Ondelettes `a localisation exponentielle, J. Math. Pures Appl. (9), 67 (1988) 227-236.
9 [8] I.J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Theory, 2 (1969) 167-206. [9] J. Shen and G. Strang, Asymptotic analysis of Daubechies polynomials, Proc. Amer. Math. Soc. 124 (1996) 3819-3833. [10] J. Shen and G. Strang, Asymptotics of Daubechies filters, scaling functions, and wavelets, Appl. Comput. Harmon. Anal. 5 (1998) 312-331. [11] M. Unser, A. Aldroubi and M. Eden, On the Asymptotic Convergence of B-Spline Wavekets to Gabor Functions, IEEE Trans. Inform. Theory, 38 (1992) 864-872.
