Related papers: On wavelet polynomials and Weyl multipliers

We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping Franklin polynomials. It will also be remarked that $\log n$ is the optimal sequence in this context.

We establish new quantitative estimates for general systems of functions with wavelet-type dyadic structure. These estimates are applied to obtain the optimal growth of various types of Weyl multipliers for certain wavelet-type systems.…

We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.

We prove that the condition \begin{equation} \sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation} is necessary for an increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for the…

We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim…

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…

Let $p_n(y)=\sum_k\hat{\alpha}_k\phi(y-k)+\sum_{l=0}^{j_n-1}\sum_k\hat {\beta}_{lk}2^{l/2}\psi(2^ly-k)$ be the linear wavelet density estimator, where $\phi$, $\psi$ are a father and a mother wavelet (with compact support),…

Let $E = \cup_{j = 1}^l [a_{2j-1},a_{2j}],$ $a_1 < a_2 < ... < a_{2l},$ $l \geq 2$ and set ${\boldmath$\omega$}(\infty) =(\omega_1(\infty),...,\omega_{l-1}(\infty))$, where $\omega_j(\infty)$ is the harmonic measure of $[a_{2 j - 1}, a_{2…

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series…

In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa\{\vphi_i\}_{i \in…

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that…

A novel method for learning optimal, orthonormal wavelet bases for representing 1- and 2D signals, based on parallels between the wavelet transform and fully connected artificial neural networks, is described. The structural similarities…

We provide a new proof of ``most" cases of the polynomial Wiener-Wintner theorem for $\sigma$-finite spaces, using hard-analytic methods. Specifically, we prove that whenever $(X,\mu,T)$ is a $\sigma$-finite measure-preserving system, and…

A maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We…

Given any $\varepsilon>0$, we construct an orthonormal system of $n_k$ uniformly bounded polynomials of degree at most $k$ on the unit sphere in $\mathbb R^{m+1}$ where $n_k$ is bigger than $1-\varepsilon$ times the dimension of the space…

Let G be a universal Chevalley group over an algebraically closed field and U^- be the subalgebra of Dist(G) generated by all divided powers X_{\alpha,m} with \alpha<0. We conjecture an algorithm to determine if Fe^+_\omega\ne0, where…

Let $\mu$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $\alpha_n$ its Verblunsky coefficients, and $\Phi_n$ its monic orthogonal polynomials. In this paper we…

We show the relevance of a multifractal-type analysis for pointwise convergence and divergence properties of wavelet series: Depending on the sequence space which the wavelet coefficients sequence belongs to, we obtain deterministic upper…

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials $\Phi_N$ for any $N\geq1$. These polynomials vanish at pairs of $j$-invariants of elliptic curves linked by cyclic isogenies of degree…