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Derivative sampling expansions in shift-invariant spaces with error estimates covering discontinuous signals

Kumari Priyanka, A. Antony Selvan

TL;DR

The paper develops a derivative-sampling framework in shift-invariant spaces $V(\phi)$, introducing a new derivative-sampling polynomial structure and a Laurent-operator approach to characterize uniform sampling with derivatives. A core contribution is the equivalence between CIS (order $\rho-1$) and the invertibility of a block Laurent operator $U_{\kappa}$ with a uniformly bounded determinant $|\det\Psi_{\kappa}(t)|$, yielding reconstruction via $\Theta_{i,\kappa}$. It provides $L^p$ error bounds for derivative sampling expansions in terms of the $L^p$-averaged modulus of smoothness $\tau_r$, and extends to signals with discontinuities using modulus-based rates, giving convergence $\|S_W^{\kappa}f-f\|_p=\mathcal{O}(W^{-\alpha})$ under suitable smoothness. Through concrete examples with shift-invariant spline spaces $V(Q_m)$, the work derives explicit reconstruction functions, confirms reproduction orders (e.g., order 2 or 3 for specific $\kappa$), and demonstrates the practical behavior for smooth, piecewise-smooth, and discontinuous signals. The results collectively advance robust derivative-based sampling and interpolation in realistic, non-bandlimited settings.

Abstract

This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of polynomials based on derivative samples is introduced, which is different from the Euler-Frobenius polynomials for the multiplicity $r>1$. A complete characterization of uniform sampling with derivatives is given using Laurent operators. The rate of approximation of a signal (not necessarily continuous) by the derivative sampling expansions in shift-invariant spaces generated by compactly supported functions is established in terms of $L^p$- average modulus of smoothness. Finally, several typical examples illustrating the various problems are discussed in detail.

Derivative sampling expansions in shift-invariant spaces with error estimates covering discontinuous signals

TL;DR

The paper develops a derivative-sampling framework in shift-invariant spaces , introducing a new derivative-sampling polynomial structure and a Laurent-operator approach to characterize uniform sampling with derivatives. A core contribution is the equivalence between CIS (order ) and the invertibility of a block Laurent operator with a uniformly bounded determinant , yielding reconstruction via . It provides error bounds for derivative sampling expansions in terms of the -averaged modulus of smoothness , and extends to signals with discontinuities using modulus-based rates, giving convergence under suitable smoothness. Through concrete examples with shift-invariant spline spaces , the work derives explicit reconstruction functions, confirms reproduction orders (e.g., order 2 or 3 for specific ), and demonstrates the practical behavior for smooth, piecewise-smooth, and discontinuous signals. The results collectively advance robust derivative-based sampling and interpolation in realistic, non-bandlimited settings.

Abstract

This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of polynomials based on derivative samples is introduced, which is different from the Euler-Frobenius polynomials for the multiplicity . A complete characterization of uniform sampling with derivatives is given using Laurent operators. The rate of approximation of a signal (not necessarily continuous) by the derivative sampling expansions in shift-invariant spaces generated by compactly supported functions is established in terms of - average modulus of smoothness. Finally, several typical examples illustrating the various problems are discussed in detail.
Paper Structure (9 sections, 18 theorems, 160 equations, 6 figures)

This paper contains 9 sections, 18 theorems, 160 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Derivative sampling in shift-invariant spline spaces
  4. Approximation by means of derivative sampling expansions
  5. examples
  6. For $\kappa=(Q_3,0,2)$
  7. For $\kappa=(Q_4,0,3)$
  8. For $\kappa=(Q_4,\tfrac{1}{2},2)$
  9. Appendix

Key Result

Theorem 2.1

Let $\mathcal{L}=[L_{sj}]=[\widehat{\Psi}(s-j)]$ denote the matrix of a function $\Psi\in L^2_{m\times m}[0,1]$.Then $\mathcal{L}$ is an invertible block Laurent operator with the symbol $\Psi\in L^\infty_{m\times m}[0,1]$ if and only if there exist two positive constants $A$ and $B$ such that

Figures (6)

  • Figure 1: Approximation of $f_1$ based on $S_W^{\kappa}f_1$ for $\kappa=(Q_3,0,2).$
  • Figure 2: Approximation of $f_2$ based on $S_W^{\kappa}f_2$ for $\kappa=(Q_3,0,2).$
  • Figure 3: Approximation of $f_3$ based on $S_W^{\kappa}f_3$ for $\kappa=(Q_3,0,2).$
  • Figure 4: Approximation of $f_1$ based on $S_W^{\kappa}f_1$ for $\kappa=(Q_4,0,3).$
  • Figure 5: Approximation of $f_2$ based on $S_W^{\kappa}f_2$ for $\kappa=(Q_4,0,3).$
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • ...and 13 more