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On the decay properties of the Franklin analyzing wavelet. (English) Zbl 0675.42021

The Franklin analyzing wavelet g: \({\mathbb{R}}\to {\mathbb{R}}\) generates bases for classical function spaces. A recent survey is given in Y. Meyer [Gaz. Math. Soc. Fr. 40, 31-42 (1989)]. By definition \[ g(\lambda):=\int_{R}e^{2\pi i\lambda x} e^{-i\pi x} \omega (x)\frac{\sin^ 2(\pi x/2)}{(\pi x/2)^ 2}dx, \] where \[ \omega (x)=\frac{s(x)}{1-(2/3)s(x)}\cdot [\frac{s^ 2(x)}{1- (2/3)s(x)}+\frac{c^ 2(x)}{1-(2/3)c(x)}]^{-1/2}, \] s(x):\(=\sin^ 2(\pi x/2)\), \(c(x):=\cos^ 2(\pi x/2)\). In particular, \(g(2^{- 1}+\lambda)=g(2^{-1}-\lambda)\) for all \(\lambda\). g has exponential decay at infinity and is piecewise linear with nodes in \(2^{-1}m\), \(m\in {\mathbb{Z}}\). The author proves the following result: \(sgn(g(n/2))=(- 1)^{n/2},\) for all even positive n; \(sgn(g(n/2))=(-1)^{\lfloor n/2\rfloor},\) for all sufficiently large \(n\in {\mathbb{N}}\); \(| g(n/2)| >| g((n+1)/2)|,\) for all sufficiently large \(n\in {\mathbb{N}}\); \(\sigma n^{-1/2}\beta^ n_ 1\leq | g(n/2)| \leq \lambda n^{-1/2}\beta^ n_ 1\), for all sufficiently large \(n\in {\mathbb{N}}\), where \(\sigma\) and \(\lambda\) are positive constants, and \(\beta_ 1=\sqrt{(2-\sqrt{3})}\).
Reviewer: W.Luther

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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References:

[1] Ahlfors, L. V., Complex analysis (1966), McGraw-Hill Book Co.: McGraw-Hill Book Co. New York · Zbl 0154.31904
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