On the norms of polynomials in systems of periodic wavelets in the spaces \(L_p\)
In accordance with the multiresolution analysis on \(\mathbb{R}\) with a scaling function \(\varphi\), the author considers the following three systems of wavelets: (a) The wavelets generated by a function \(\varphi\) with compact support; (b) The Battle-Lemarié spline-wavelets; (c) The wavelets generated by a function \(\varphi\) whose Fourier transform has a compact support. For these systems of wavelets one can define polynomials \(T(y)\) which (in a sense) are similar to trigonometric polynomials. A number of statements concerning validity of the estimate of the form \[ K_1\Biggl({1\over M} \sum^M_{k= 1}| T(y_k)|^p\Biggr)^{1/p}\leq \| T\|_p\leq K_2\Biggl({1\over M} \sum^M_{k= 1}| T(y_k)|^p\Biggr)^{1/p} \] for certain points \(y_k\), \(k= 1,\dots, M\), \(M\leq K_3N\), and every positive integer \(N\), are announced.
\(L^p\)-estimates of polynomials of wavelets
scientific article; zbMATH DE number 1037766