Universally optimal wavelets

A complete description of wavelet bases generated by a fixed function whose Fourier transform is the characteristic function of a set is presented. In particular, for the case of Sobolev spaces, wavelet bases are constructed possessing the following property of universal optimality: the subspaces generated by these functions are extremal for projection lattice widths (in the univariate case also for Kolmogorov widths) of the unit ball in $W^m_2(E_n)$ in the metric of $W^s_2(E_n)$ simultaneously for the whole scale of Sobolev classes (that is, for all $s,m\in E_1$, such that $s<m$). En route, certain results concerning completeness and the basis property of systems of exponentials are established.