On the estimation of wavelet coefficients. (English) Zbl 0955.42023

Ehrich, Sven

Adv. Comput. Math. 13, No. 2, 105-129 (2000).

In this paper the author studies the magnitude of wavelet coefficients by investigating the quantities \[ c_k(\psi)=\sup_{f\in A_k}{|(\psi, f)|\over \|\psi\|_2}. \] Here, the function classes \(A_k\) are defined by \[ A_k=\{f|\|f^{(k)}\|_2 < 1\}\quad k\in {\mathbb{N}}. \] In particular, the expressions \(\lim_{m\rightarrow\infty} c_k(\psi_m)\), for a fixed \(k\), and \(\lim_{m\rightarrow\infty} c_m(\psi_m)\) are explicitly computed for Daubechies orthonormal wavelets and for semiorthogonal spline wavelets, where \(m\) denotes the number of vanishing moments of \(\psi_m\).
It turns out that these constants are considerably smaller for spline wavelets.

Reviewer: Gerlind Plonka (Duisburg)

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
41A15	Spline approximation

Keywords:

wavelet coefficients; bounds; Daubechies wavelets; semiorthogonal spline wavelets