## 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.)

### Keywords:

Franklin analyzing wavelet
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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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