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Sampling Theorem and explicit interpolation formula for non-decaying unbounded signals

Nikolai Dokuchaev

TL;DR

The paper extends the classical sampling theorem to band-limited signals that may be unbounded or non-decaying, by introducing an explicit interpolation formula with coefficients $a_k(t)$ defined via a spectral kernel $E(t,\omega)$. For sublinear growth with $\alpha\in[0,1/2)$, it provides closed-form $a_k(t)$ yielding coefficients decaying as $|a_k(t)|\sim 1/k^2$, and it generalizes to larger growth rates through semi-explicit integral representations. The construction relies on a detailed spectral-framework using $\rho$-weighted function spaces and a kernel $E$ that matches $e^{i\omega t}$ on the spectrum, ensuring absolute convergence of the interpolation series. Numerical experiments corroborate the theory, showing improved truncation behavior compared to classical Shannon interpolation for non-decaying signals and validating the applicability to signals with polynomial growth.

Abstract

The paper establishes an analog Whittaker-Shannon-Kotelnikov sampling theorem for unbounded non-decaying band-limited signals. An explicit interpolation formula is obtained for signals sublinear growth with rate of growth less than 1/2. At any time, the rate of decay for the $k$th coefficients of this formula is $\sim 1/k^2$. In addition, the paper obtains a method for calculating the coefficients of the interpolation formula applicable to signals with arbitrarily high rate of polynomial growth.

Sampling Theorem and explicit interpolation formula for non-decaying unbounded signals

TL;DR

The paper extends the classical sampling theorem to band-limited signals that may be unbounded or non-decaying, by introducing an explicit interpolation formula with coefficients defined via a spectral kernel . For sublinear growth with , it provides closed-form yielding coefficients decaying as , and it generalizes to larger growth rates through semi-explicit integral representations. The construction relies on a detailed spectral-framework using -weighted function spaces and a kernel that matches on the spectrum, ensuring absolute convergence of the interpolation series. Numerical experiments corroborate the theory, showing improved truncation behavior compared to classical Shannon interpolation for non-decaying signals and validating the applicability to signals with polynomial growth.

Abstract

The paper establishes an analog Whittaker-Shannon-Kotelnikov sampling theorem for unbounded non-decaying band-limited signals. An explicit interpolation formula is obtained for signals sublinear growth with rate of growth less than 1/2. At any time, the rate of decay for the th coefficients of this formula is . In addition, the paper obtains a method for calculating the coefficients of the interpolation formula applicable to signals with arbitrarily high rate of polynomial growth.
Paper Structure (7 sections, 15 theorems, 64 equations, 2 figures)

This paper contains 7 sections, 15 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main result: Sampling theorem and interpolation formula
  3. Background for the proofs
  4. Proofs
  5. Proof of Proposition \ref{['prop1']}
  6. Proof of Theorem \ref{['ThM']}
  7. Some numerical experiments

Key Result

Proposition 2.2

For any $\alpha\ge 0$ and $\Omega\in (0,\pi)$, there exists a sequence of functions $\{a_k(\cdot)\}_{k\in{\mathbb{Z}}}\subset \rho^{} L_1({\bf R})$ such that $\sup_{t\in{\bf R}}\sum_{k\in{\mathbb{Z}}}(1+|k|)^{\alpha}|a_k(t)|<+\infty$ and (Gen) holds, i.e., $x(t)=\sum_{k\in{\mathbb{Z}} }a_k(t)x(k)$ f

Figures (2)

  • Figure 1: The differences $\widetilde{D}_k(t)$, $\bar{D}_k(t)$, and $D_k$, between the coefficients $a_k(t)$ used in numerical experiments with $\Omega=5\pi/6$, $t=-1.7$, and $N=6$, $m=-8$. Here big blue circles are for $\widetilde{D}_k(t)$, medium red circles are for $\bar{D}_k(t)$, and black dots are for $D_k$.
  • Figure 2: The sequences $L_k(t)=\log\left(|k|^{2.49}|a_k^{(1)}(t)|+1\right)$ (light) and $M_k=\log\left(|k|^{2.49}|a_k^{(2)}(t)|+1\right)$ (dark) for the coefficients calculated as in Theorem \ref{['ThM']} and Theorem \ref{['ThD2']} respectively.

Theorems & Definitions (18)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Proposition 3.6
  • ...and 8 more