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Asymptotic evaluation of the Sinc transform of entire exponential type function resulting to exact polynomial asymptotic behavior

Nathalie Liezel R. Rojas, Eric A. Galapon

TL;DR

This work analyzes the Sinc transform $I(\\lambda) = \int_{0}^{\\infty} f(x) \frac{\\sin^{n}(\\lambda x)}{x^{n}} dx$ for an even, entire function $f$ of exponential type $\\tau>0$. Using contour integration that bypasses the origin and Hadamard finite-part regularization, the authors derive terminating Poincaré-type expansions that yield polynomial growth in $\\lambda$ for large $\\lambda$, with thresholds $\\lambda>\\tau/2$ when $n$ is even and $\\lambda>\\tau$ when $n$ is odd. They obtain explicit formulas for both even and odd $n$: a finite-part term plus a finite sum of odd powers of $\\lambda$ for $n=2m$, and a polynomial in even powers of $\\lambda$ for $n=2m+1$, with coefficients built from derivatives of $f$ at $0$ and combinatorial sums. The paper also provides concrete examples, including closed-form expressions involving Bessel functions, illustrating exact polynomial asymptotics rather than the usual inverse-power behavior, and suggesting potential applications in spectral analysis and transition-rate calculations in quantum problems.

Abstract

We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(λx)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $τ>0$, for large values of the parameter $λ$, where $n$ is a positive integer. We refer to this integral as the Sinc transform. Under the condition that $f(x)$ is even with respect to $x$, we derive a terminating asymptotic expansion of the Sinc transform which behave as a polynomial in positive powers of $λ$ as $λ$ grows large provided that the conditions $λ> τ/2$ for even $n$ and $λ>τ$ for odd n are satisfied.

Asymptotic evaluation of the Sinc transform of entire exponential type function resulting to exact polynomial asymptotic behavior

TL;DR

This work analyzes the Sinc transform for an even, entire function of exponential type . Using contour integration that bypasses the origin and Hadamard finite-part regularization, the authors derive terminating Poincaré-type expansions that yield polynomial growth in for large , with thresholds when is even and when is odd. They obtain explicit formulas for both even and odd : a finite-part term plus a finite sum of odd powers of for , and a polynomial in even powers of for , with coefficients built from derivatives of at and combinatorial sums. The paper also provides concrete examples, including closed-form expressions involving Bessel functions, illustrating exact polynomial asymptotics rather than the usual inverse-power behavior, and suggesting potential applications in spectral analysis and transition-rate calculations in quantum problems.

Abstract

We consider the asymptotic evaluation of the integral transform of an exponential type function of type , for large values of the parameter , where is a positive integer. We refer to this integral as the Sinc transform. Under the condition that is even with respect to , we derive a terminating asymptotic expansion of the Sinc transform which behave as a polynomial in positive powers of as grows large provided that the conditions for even and for odd n are satisfied.
Paper Structure (5 sections, 2 theorems, 66 equations, 4 figures)

This paper contains 5 sections, 2 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Case 1
  3. Example
  4. Case 2
  5. Example

Key Result

Theorem 1

Let $f(x)$ be an entire function of exponential type $\tau>0$ satisfying $f(-x)=f(x)$. For positive integer $m$ and $\lambda > \tau/2$, we have where $\,\,\, \;\backslash\!\!\!\!\!\!\!\backslash\!\!\!\!\!\!\int_{0}^{\infty} \, \frac{f(x)}{x^{2m}} \, \text{d}x$ is the finite part of the divergent integral $\int_{0}^{\infty} \, \frac{f(x)}{x^{2m}} \, \text{d}x$, and

Figures (4)

  • Figure 1: The contour $C^+$ consists of the path $\Gamma_+$, $L_1$, $L_2$ and $L_3$, while the contour $C^-$ consists of $\Gamma_-$, $L_1$, $L_2$ and $L_3$.
  • Figure 2: The contour $C'^+$ consists of the path $\Gamma_+$, $L_4$, $L_5$ and $L_6$, while the contour $C'^-$ consists of $\Gamma_-$, $L_4$, $L_5$ and $L_6$.
  • Figure 3: The plot of the integrand, $\frac{\sin\sqrt{\tau^2 x^2 + \sigma^2}}{\sqrt{\tau^2 x^2 + \sigma^2}} \, \frac{\sin^{4} (\lambda x)}{x^{4}}$ for $\lambda =2,3,4$.
  • Figure 4: The plot of the integrand, $\frac{\sin\sqrt{\tau^2 x^2 + \sigma^2}}{\sqrt{\tau^2 x^2 + \sigma^2}} \, \frac{\sin^{3} (\lambda x)}{x^{3}}$ for $\lambda =2,3,4$.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof