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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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