On the estimation of wavelet coefficients (Q1569742)

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.