Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quarklet Characterizations for bivariate Bessel-Potential Spaces on the Unit Square via Tensor Products

Marc Hovemann

Abstract

In this paper we deduce new characterizations for bivariate Bessel-Potential spaces defined on the unit square via B-spline quarklets. For that purpose in a first step we use univariate boundary adapted quarklets to describe univariate Bessel-Potential spaces on intervals. To obtain the bivariate characterizations a recent result of Hansen and Sickel is applied. It yields that each bivariate Bessel-Potential space on a square can be written as an intersection of function spaces which have a tensor product structure. Hence our main result is a characterization of bivariate Bessel-Potential spaces on squares in terms of quarklets that are tensor products of univariate quarklets on intervals.

Quarklet Characterizations for bivariate Bessel-Potential Spaces on the Unit Square via Tensor Products

Abstract

In this paper we deduce new characterizations for bivariate Bessel-Potential spaces defined on the unit square via B-spline quarklets. For that purpose in a first step we use univariate boundary adapted quarklets to describe univariate Bessel-Potential spaces on intervals. To obtain the bivariate characterizations a recent result of Hansen and Sickel is applied. It yields that each bivariate Bessel-Potential space on a square can be written as an intersection of function spaces which have a tensor product structure. Hence our main result is a characterization of bivariate Bessel-Potential spaces on squares in terms of quarklets that are tensor products of univariate quarklets on intervals.
Paper Structure (12 sections, 14 theorems, 194 equations)

This paper contains 12 sections, 14 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions: Function Spaces and Quarklets
  3. Bessel-Potential Spaces on $\mathbb{R}^d$ and on Domains
  4. B-Splines, Quarks and Quarklets on the Real Line
  5. Quarks and Quarklets on the Interval
  6. Quarklet Characterizations for Bessel-Potential Spaces on Bounded Intervals
  7. Upper Estimates
  8. Lower Estimates
  9. Bivariate Bessel-Potential Spaces via Tensor Products
  10. Bivariate Quarklets via Tensor Product Methods
  11. Quarklet Characterizations for Bivariate Bessel-Potential Spaces $H^{s}_{r}((0,1)^2)$
  12. Summary and Further Issues

Key Result

Proposition 1

Let $\Omega = (0,1)^d$ with $d \in \mathbb{N}$. Let $1 < r < \infty$, $1 \leq v \leq \infty$, $N \in \mathbb{N}$ and Then $H^{s}_{r}(\Omega)$ is the collection of all $f \in L_{\max(r,v)}(\Omega)$ such that in the sense of equivalent norms. Here we use the abbreviation $V^{N}(x,t) := \{ h \in \mathbb{R}^d : |h| < t \; \hbox{and} \; x + \tau h \in \Omega \; \hbox{for} \; 0 \leq \tau \leq N \}$.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 3
  • Definition 4
  • Remark 1
  • ...and 36 more