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Convolution and product in finitely generated shift-invariant spaces

Aleksandar Aksentijević, Suzana Aleksić, Stevan Pilipović

TL;DR

This work develops the structural theory of finitely generated shift-invariant spaces within Sobolev spaces $H^s$, focusing on convolution and product operations. By linking FGSI spaces with their periodic and dual structures, it provides duality characterizations, relations via regular matrices, and a comprehensive treatment of convolution equations and convolutors in this setting. The results include a precise description of Fourier multipliers acting as FGSI-convolutors, deconvolution mechanisms, and wave-front analyses for both convolution and product, along with conditions ensuring products remain within appropriate FGSI spaces. The findings advance understanding of shift-invariant analysis in Sobolev contexts, with implications for sampling theory, deconvolution, and microlocal analysis in FGSI frameworks.

Abstract

We discuss some structural properties of finitely generated shift-invariant (FGSI) spaces and subspaces of Sobolev spaces, particularly those related to convolution and the product within these spaces. We find shift-invariant solutions in FGSI spaces for a class of differential-difference equations with constant coefficients. Additionally, we analyze the Fourier multipliers in FGSI spaces and the wave fronts for the convolution and product in FGSI spaces.

Convolution and product in finitely generated shift-invariant spaces

TL;DR

This work develops the structural theory of finitely generated shift-invariant spaces within Sobolev spaces , focusing on convolution and product operations. By linking FGSI spaces with their periodic and dual structures, it provides duality characterizations, relations via regular matrices, and a comprehensive treatment of convolution equations and convolutors in this setting. The results include a precise description of Fourier multipliers acting as FGSI-convolutors, deconvolution mechanisms, and wave-front analyses for both convolution and product, along with conditions ensuring products remain within appropriate FGSI spaces. The findings advance understanding of shift-invariant analysis in Sobolev contexts, with implications for sampling theory, deconvolution, and microlocal analysis in FGSI frameworks.

Abstract

We discuss some structural properties of finitely generated shift-invariant (FGSI) spaces and subspaces of Sobolev spaces, particularly those related to convolution and the product within these spaces. We find shift-invariant solutions in FGSI spaces for a class of differential-difference equations with constant coefficients. Additionally, we analyze the Fourier multipliers in FGSI spaces and the wave fronts for the convolution and product in FGSI spaces.
Paper Structure (9 sections, 10 theorems, 63 equations)

This paper contains 9 sections, 10 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Periodic test function spaces
  4. FGSI spaces
  5. Duals for FGSI spaces
  6. Relations between FGSI spaces
  7. Convolution equations
  8. Convolutors
  9. On the product in FGSI spaces

Key Result

Proposition 1

$(a)$ Let $\{\phi_1,\ldots,\phi_m\}$ satisfy condition $(A)$ and let them be a basis for $V_s(\phi_1,\ldots,\phi_m)$. Assume that $\mathcal{B}$ is the set of all elements $(f_1,\ldots,f_m)\in(H^{-s})^m$ such that Then, $(\bigoplus$ denotes the direct sum$)$. $(b)$ Let $\{\phi_1,\ldots,\phi_m\}$ satisfy condition $(A)$ and let them be a semi-basis for $V_s(\phi_1,\ldots,\phi_m)$. Assume that $\ma

Theorems & Definitions (26)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 16 more