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

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