The purpose of this paper is to provide a method of reduction of some problems concerning families $A_t={\left(A\left(t\right)\right)}_{t\in }$ of linear operators with domains ${{(}_t)}_{t\in }$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${\left({\Psi }_t\right)}_{t\in }$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t\mapsto {\Psi }_t$ is sufficiently regular and (ii) ${\Psi }_t\left(\right){=}_t$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition with a parameter;...

The relationship between the spectral properties of the transfer operator corresponding to a wavelet refinement equation and the $L^p$-Sobolev regularity of solution for the equation is established.

We prove inclusion relations between generalizing Waterman's and generalized Wiener's classes for functions of two variable.

We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${\left|\phi ₙ\right|}^p$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series...

We prove that ridgelet transform $R:𝒮\left({ℝ}^2\right)\to 𝒮\left(𝕐\right)$ and adjoint ridgelet transform $R^{*}:𝒮\left(𝕐\right)\to 𝒮\left({ℝ}^2\right)$ are continuous, where $𝕐={ℝ}^{+}\times ℝ\times [0,2\pi ]$. We also define the ridgelet transform $ℛ$ on the space ${𝒮}^{\text{'}}\left({ℝ}^2\right)$ of tempered distributions on ${ℝ}^2$, adjoint ridgelet transform ${ℛ}^{*}$ on ${𝒮}^{\text{'}}\left(𝕐\right)$ and establish that they are linear, continuous with respect to the weak${}^{*}$-topology, consistent with $R$, $R^{*}$ respectively, and they satisfy the identity $({ℛ}^{*}\circ ℛ)\left(u\right)=u$, $u\in {𝒮}^{\text{'}}\left({ℝ}^2\right)$.

Given a set of positive measure on the circle and a set Λ of integers, one can ask whether $E\left(\Lambda \right):=e_{\lambda \in \Lambda }^{i\lambda t}$ is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step $\ell =O\left(N^{1-\epsilon }\right)$ with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain...

We propose a definition of Riesz transforms associated to the Ornstein-Uhlenbeck operator based on the Dunkl Laplacian. In the case related to the group ℤ ₂ it is proved that the Riesz transform is bounded on the corresponding $L^p$ spaces, 1 < p < ∞.