We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V₀ is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as in computing the period of a Riesz wavelet, answering in the affirmative a conjecture of Daubechies and Han. Additionally, we completely characterize when the canonical dual of a quasi-affine frame has the...

We show that, given a set E ⊂ 𝕋 of measure zero, the set of continuous functions whose Fourier series expansion is divergent at any point t ∈ E is dense-algebrable, i.e. there exists an infinite-dimensional, infinitely generated dense subalgebra of 𝓒(𝕋) every non-zero element of which has a Fourier series expansion divergent in E.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar...

Bellow and Calderón proved that the sequence of convolution powers ${\mu }_{n}f\left(x\right)={\sum }_{k\in \mathbb{Z}}{\mu }^n\left(k\right)f\left(T^{k}x\right)$ converges a.e, when $\mu$ is a strictly aperiodic probability measure on $\mathbb{Z}$ such that the expectation is zero, $E\left(\mu \right)=0$, and the second moment is finite, $m_2\left(\mu \right)\<\infty$. In this paper we extend this result to cases where $m_2\left(\mu \right)=\infty$.

We prove the almost everywhere convergence of the Marcinkiewicz means of integrable functions σₙf → f for every f ∈ L¹(I²), where I is the group of 2-adic integers.

We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions ${\ell }_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., in $L^2({ℝ}_{+},x^{a}dx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f\in L^p\left(x^{a}dx\right)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.

Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{x}^{a/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].