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Hyperbolic trigonometric functions as approximation kernels and their properties II: Wavelets

M. Buhmann, J. Jódar, M. Rodríguez

TL;DR

The paper develops a general multiresolution framework to build prewavelets from hyperbolic radial basis functions (RBFs), enabling highly localized time-frequency decompositions for function approximation and signal analysis. It extends quasi-interpolation techniques to wavelet-like constructions by forming bi-orthogonal bases from scaling and cardinal functions and computing corresponding coefficients via Wiener's lemma, ensuring stability and localization. The approach is shown to be broadly applicable to other RBF families such as multiquadrics and thin-plate splines, with a concrete set of examples (cubic B-splines, multiquadrics, hyperbolic, thin-plate splines, generalized multiquadrics) illustrating the method. The results offer a flexible toolkit for quasi-interpolation and wavelet-based analysis with potentially impactful applications in filtering and signal processing. The appendix provides explicit Fourier-transform formulas for hyperbolic kernels used in the derivations.

Abstract

In a previous paper we have introduced a new class of radial basis functions that are powerful means to approximate functions by quasi-interpolation. In this article we extend the results to create new ways of approximating functions by prewavelets that are constructed from spaced spanned of the new hyperbolic radial basis functions. They consist of highly localised time-frequency decompositions that are suitable for analysis and filtering. The construction is sufficiently general to apply for large classes of other radial basis functions too - such as multiquadrics and their generalisations and thin-plate splines -, as well as, for example, polynomial splines.

Hyperbolic trigonometric functions as approximation kernels and their properties II: Wavelets

TL;DR

The paper develops a general multiresolution framework to build prewavelets from hyperbolic radial basis functions (RBFs), enabling highly localized time-frequency decompositions for function approximation and signal analysis. It extends quasi-interpolation techniques to wavelet-like constructions by forming bi-orthogonal bases from scaling and cardinal functions and computing corresponding coefficients via Wiener's lemma, ensuring stability and localization. The approach is shown to be broadly applicable to other RBF families such as multiquadrics and thin-plate splines, with a concrete set of examples (cubic B-splines, multiquadrics, hyperbolic, thin-plate splines, generalized multiquadrics) illustrating the method. The results offer a flexible toolkit for quasi-interpolation and wavelet-based analysis with potentially impactful applications in filtering and signal processing. The appendix provides explicit Fourier-transform formulas for hyperbolic kernels used in the derivations.

Abstract

In a previous paper we have introduced a new class of radial basis functions that are powerful means to approximate functions by quasi-interpolation. In this article we extend the results to create new ways of approximating functions by prewavelets that are constructed from spaced spanned of the new hyperbolic radial basis functions. They consist of highly localised time-frequency decompositions that are suitable for analysis and filtering. The construction is sufficiently general to apply for large classes of other radial basis functions too - such as multiquadrics and their generalisations and thin-plate splines -, as well as, for example, polynomial splines.

Paper Structure

This paper contains 12 sections, 3 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. General construction
  4. Obtaining the $a_{\cdot, i}$ coefficients
  5. Obtaining the $b_{\cdot, i}$ coefficients
  6. Examples
  7. Wavelet based on the truncated power of order 3
  8. Wavelet based on the multiquadric function
  9. Wavelet based on a hyperbolicus function
  10. Wavelet based on the Thin-Plate Spline
  11. Wavelet based on the Generalized Multiquadric
  12. Appendix: Fourier transform of $|x| \tanh |x|$

Key Result

Theorem 2.1

[Strang and Fix conditions] Let $\psi:\mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function such that Then the quasi-interpolant is well-defined and exact on $\mathbb{P}_m$. The approximation error for $0<h<1$ can be estimated by for $h\rightarrow 0$ and a bounded function $f\in C^{m+1}(\mathbb{R}^n)$ with bounded derivatives.

Figures (3)

  • Figure 1: Wavelet based on the truncated power of order 3.
  • Figure 2: Wavelet based on the multiquadric function.
  • Figure 3: Wavelet based on $\phi(r)=- r \tanh r$.

Theorems & Definitions (6)

  • Theorem 2.1
  • definition 1
  • Lemma 3.1
  • Theorem 3.2
  • definition 2
  • definition 3