Related papers: The $(abc,pqr)$-problem for Approximate Schauder F…

For $d = 2, 3, \ldots$ and $p \in [1, \infty),$ we define a class of representations $\rho$ of the Leavitt algebra $L_d$ on spaces of the form $L^p (X, \mu),$ which we call the spatial representations. We prove that for fixed $d$ and $p,$…

We discuss the problem on approximation by tight step wavelet frames on the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $G_n=\{x=\sum_{k=n}^\infty x_k p^k\}$, $X$ be a set of characters. We define a step function $\lambda({\chi})$ that is…

We show that, for suitable enumerations, the multivariate Haar system is a Schauder basis in the classical Sobolev spaces on $\mathbb R^d$ with integrability $1<p<\infty$ and smoothness $1/p-1<s<1/p$. This complements earlier work by the…

In this work we consider the Neumann problem for the Laplace operator and we prove an existence result in the H\"older spaces and obtain Schauder estimates. According to our knowledge this result is not explicitly proved in the several…

This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a…

For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em not} identified. This means that the Riesz isomorphism is not used as an identification, which, for example,…

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function…

The goal of this paper is to study band-dominated operators on Banach spaces with Schauder basis with respect to uniformly locally finite metric spaces as well as the Banach algebras generated by them: the so called uniform Roe algebras. We…

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism…

The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded,…

We propose a fast method for high order approximations of the solution of the Cauchy problem for the linear non-stationary Stokes system in $R^3$ in the unknown velocity $\bf u$ and kinematic pressure $P$. The density ${\bf f}({\bf x},t)$…

Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a…

We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main…

Recent research has shown that the properties of overcomplete Gabor frames and frames arising from shift-invariant systems form a precise match with certain conditions that are necessary for a frame in $L^2(\mathbf R)$ to have a…

The present paper is devoted to the semiclassical analysis of linear Schr\"odinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order…

This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense…

We study the quasilinear equation $(P)\qquad - {\rm div} (a(x,u,\nabla u)) +A_t(x,u,\nabla u) + |u|^{p-2}u\ =\ g(x,u) \qquad \hbox{in $\R^N$,} $ with $N\ge 3$ and $p > 1$. Here, we suppose $A : \R^N \times \R \times \R^N \to \R$ is a given…

We consider the quantum complexity of computing Schatten $p$-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of…

The Schauder estimates are among the oldest and most useful tools in the modern theory of elliptic partial differential equations (PDEs). Their influence may be felt in practically all applications of the theory of elliptic boundary-value…

In this paper we introduce the concept of Bochner pg-frames for Banach spaces. We characterize the Bochner pg-frames and specify the optimal bounds of a Bochner pg- frames. Then we define a Bochner qg-Riesz basis and verify the relations…