On wavelet polynomials and Weyl multipliers

TL;DR

This paper establishes a new bound for wavelet orthonormal systems, demonstrating that the logarithmic factor is optimal for almost everywhere convergence of wavelet polynomial systems.

Contribution

It introduces a novel bound involving the maximum of wavelet partial sums and proves the optimality of the logarithmic Weyl multiplier for wavelet polynomials.

Findings

Established a new bound for wavelet partial sums involving a logarithmic factor.

Proved that the logarithmic sequence is an almost everywhere convergence Weyl multiplier.

Showed that the logarithmic factor is optimal in this context.

Abstract

For the wavelet type orthonormal systems $\phi_n$, we establish a new bound \begin{equation} \left\|\max_{1\le m\le n}\left|\sum_{j\in G_m}\langle f,\phi_j\rangle \phi_j\right|\right\|_p\lesssim \sqrt{\log (n+1)}\cdot \|f\|_p,\quad 1<p<\infty, \end{equation} where $G_m\subset N$ are arbitrary sets of indexes. Using this estimate, we prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping wavelet polynomials. It will also be remarked that $\log n$ is the optimal sequence in this context.