Optimal Wavelets

Abstract

The complete description of wavelet bases is given such that each of them is generated by the fixed function whose Fourier image is the characteristic function of some set. In particular, for the case of Sobolev spaces wavelet bases with the following property of universal optimality are constructed: subspaces generated by these functions are extremal for the projection-net widths (if n = 1, then also for Kolmogorov widths) of the unit ball in $W^m_2({\mathbb{R}}^n)$ with $W^s_2({\mathbb{R}}^n)$-metric for the whole scale of Sobolev classes simultaneously (i.e., for all s,m ∈ ℝ such that s < m). Some results concerning completeness and basis property of exponential systems are established in passing.