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Special-Affine Wavelets: Multi-Resolution Analysis and Function Approximation in L^2(R)

Waseem Z. Lone, Vikash K. Sahu, Amit K. Verma

TL;DR

The paper develops a SAFT-domain multiresolution framework (SAMRA) to construct orthonormal bases in $L^2(\mathbb{R})$ by leveraging a SAFT-based sampling theory. It establishes a Shannon-type sampling theorem in the SAFT domain and defines SAFT-inspired nested subspaces $\mathscr{V}_S^k$ and wavelet spaces $\mathscr{W}_S^k$, deriving explicit refinement and wavelet coefficients that guarantee an orthonormal basis for $L^2(\mathbb{R})$. By relating SAMRA to classical MRA and other MRAs for special choices of the SAFT parameters, the work provides a flexible, generalized framework for SAFT-domain time-frequency analysis and function approximation. The methodology is illustrated with constructions of scaling and wavelet functions (e.g., sinc-based and Haar-like examples) and demonstrated through function-approximation experiments, highlighting practical impact for band-limited signal analysis in generalized Fourier domains.

Abstract

The multiresolution analysis (MRA) associated with the Special affine Fourier transform (SAFT) provides a structured approach for generating orthonormal bases in \( L^2(\mathbb R) \), making it a powerful tool for advanced signal analysis. This work introduces a robust sampling theory and constructs multiresolution structures within the SAFT domain to support the formation of orthonormal bases. Motivated by the need for a sampling theorem applicable to band-limited signals in the SAFT framework, we establish a corresponding theoretical foundation. Furthermore, a method for constructing orthogonal bases in $L^2(\mathbb R)$ is proposed, and the theoretical results are demonstrated through illustrative examples.

Special-Affine Wavelets: Multi-Resolution Analysis and Function Approximation in L^2(R)

TL;DR

The paper develops a SAFT-domain multiresolution framework (SAMRA) to construct orthonormal bases in by leveraging a SAFT-based sampling theory. It establishes a Shannon-type sampling theorem in the SAFT domain and defines SAFT-inspired nested subspaces and wavelet spaces , deriving explicit refinement and wavelet coefficients that guarantee an orthonormal basis for . By relating SAMRA to classical MRA and other MRAs for special choices of the SAFT parameters, the work provides a flexible, generalized framework for SAFT-domain time-frequency analysis and function approximation. The methodology is illustrated with constructions of scaling and wavelet functions (e.g., sinc-based and Haar-like examples) and demonstrated through function-approximation experiments, highlighting practical impact for band-limited signal analysis in generalized Fourier domains.

Abstract

The multiresolution analysis (MRA) associated with the Special affine Fourier transform (SAFT) provides a structured approach for generating orthonormal bases in \( L^2(\mathbb R) \), making it a powerful tool for advanced signal analysis. This work introduces a robust sampling theory and constructs multiresolution structures within the SAFT domain to support the formation of orthonormal bases. Motivated by the need for a sampling theorem applicable to band-limited signals in the SAFT framework, we establish a corresponding theoretical foundation. Furthermore, a method for constructing orthogonal bases in is proposed, and the theoretical results are demonstrated through illustrative examples.

Paper Structure

This paper contains 5 sections, 5 theorems, 93 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiresolution Analysis Associated with the SAFT
  4. Function Approximation
  5. Conclusion

Key Result

Lemma 2.3

Let $f \in L^2(\mathbb R)$ and $a\neq 0$. Then the following identity holds:

Figures (3)

  • Figure 1: Special affine wavelets $\psi_{S,0,0}(x)$ corresponding to the scaling function $\varphi(x) = \text{sinc}(\pi x)$, for two SAFT parameter matrices $S$ and $S'$.
  • Figure 2: Special affine wavelets $\psi_{S,0,0}(x)$ corresponding to the scaling function $\varphi(x) = \chi_{[0,1)}(x)$, for two SAFT parameter matrices $S$ and $S'$.
  • Figure 3: Affine wavelet approximation of $x^2$ on $[0,1]$ for $J=2$ and $J=3$ by using Imaginary part of $\psi$.

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 3.1
  • Remark 3.2
  • ...and 5 more