Necessary and sufficient conditions for the existence of compactly supported $L^p$-solutions for the two-dimensional two-scale dilation equations are given.

A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that ${𝒳}^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then ${𝒳}^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice...

Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure.As a byproduct, we also give...

In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces $L^p[0,1]$, 1 ≤ p < ∞. Although perhaps not probable, the latter result would...

We study series expansions for harmonic functions analogous to Hartogs series for holomorphic functions. We apply them to study conjugate harmonic functions and the space of square integrable harmonic functions.

Considering functions f on ℝⁿ for which both f and f̂ are bounded by the Gaussian $e^{-1/2a\left|x\right|²}$, 0 < a < 1, we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for O(n)-finite functions, thus extending a one-dimensional result of Vemuri.

We show that the functions in L2(Rn) given by the sum of infinitely sparse wavelet expansions are regular, i.e. belong to C∞L2 (x0), for all x0 ∈ Rn which is outside of a set of vanishing Hausdorff dimension.

In this paper we prove that the maximal operator $${\tilde{\sigma }}^{\kappa ,*}f:=\underset{n\in \mathbb{P}}{sup}\frac{|{\sigma }_n^{\kappa }f|}{{log}^2(n+1)},$$ where ${\sigma }_n^{\kappa }f$ is the $n$-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1/2}\left(G\right)$ to the space $L_{1/2}\left(G\right).$

We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_p$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

We prove and discuss some new $(H_p,L_p)$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients $\{q_k:k\ge 0\}$. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund...