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Compression using Quasi-Interpolation

Martin Buhmann, Feng Dai

TL;DR

The paper develops a comprehensive framework for compression and pointwise convergence in multivariate approximation via quasi-interpolation, with a focus on radial basis functions and wavelet-based nonlinear approximations. It establishes decay and polynomial-reproduction properties for quasi-interpolants, introduces a modified operator for handling rough integrands, and derives pointwise error bounds that scale with smoothness and local regularity. It also analyzes compression in both $C({\mathbb R}^d)$ and $L^p({\mathbb R}^d)$, showing negative results without decay at infinity and positive, rate-optimal results under decay conditions, aided by a wavelet/triebel-Lizorkin framework for nonlinear approximation. The results collectively provide practical guidance for N-term compression and high-dimensional approximation using radial basis functions and wavelet-based methods in spaces with varying smoothness.

Abstract

We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from their linear spaces and provide convergence estimates. The results include an error estimate for nonlinear approximation by quasi-interpolation, results about compression in the space of continuous functions and a pointwise convergence estimate for approximands of low smoothness.

Compression using Quasi-Interpolation

TL;DR

The paper develops a comprehensive framework for compression and pointwise convergence in multivariate approximation via quasi-interpolation, with a focus on radial basis functions and wavelet-based nonlinear approximations. It establishes decay and polynomial-reproduction properties for quasi-interpolants, introduces a modified operator for handling rough integrands, and derives pointwise error bounds that scale with smoothness and local regularity. It also analyzes compression in both and , showing negative results without decay at infinity and positive, rate-optimal results under decay conditions, aided by a wavelet/triebel-Lizorkin framework for nonlinear approximation. The results collectively provide practical guidance for N-term compression and high-dimensional approximation using radial basis functions and wavelet-based methods in spaces with varying smoothness.

Abstract

We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from their linear spaces and provide convergence estimates. The results include an error estimate for nonlinear approximation by quasi-interpolation, results about compression in the space of continuous functions and a pointwise convergence estimate for approximands of low smoothness.
Paper Structure (8 sections, 14 theorems, 111 equations)

This paper contains 8 sections, 14 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Quasi-interpolation: General assumptions and results
  3. Pointwise approximation by quasi-interpolation for functions of lower smoothness in $L^p$
  4. Proof of Theorem \ref{['thm-6-4-18']}
  5. Compression in the space $C({\mathbb R}^d)$
  6. Compression in $L^p$ with $1<p<\infty$
  7. Wavelet decomposition and the Triebel-Lizorkin spaces
  8. Nonlinear approximation

Key Result

Lemma 2.1

Given a quasi-uniformly distributed subset $A$ of ${\mathbb R}^d$ and a positive integer $k$, there exists a sequence of compactly supported functions $\{ N_{\alpha}\}_{{\alpha}\in A}\subset C_c^{k}({\mathbb R}^d)$ with the properties that each $N_{\alpha}$ is supported in the ball $B_{M/2}({\alpha} Here the constants $C$ and $M$ depend only on $k$, $d$.

Theorems & Definitions (26)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.3
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 3.4
  • ...and 16 more