On wavelet polynomials and Weyl multipliers

Abstract

For the wavelet type orthonormal systems ${\varphi }_n$, we establish a new bound \begin{equation} \left\|\max_{1\le m\le n}\left|\sum_{j\in G_m}\langle f,ϕ_j\rangle ϕ_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.

17 pages. arXiv admin note: text overlap with arXiv:2005.04017

References in corpus (2)

• On systems of non-overlapping Haar polynomials
• On Weyl multipliers of the rearranged trigonometric system