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

TL;DR

This paper derives an asymptotic polynomial expansion for the Sinc transform of exponential type functions as the parameter grows large, under specific conditions on the function and parameters.

Contribution

It provides a new asymptotic expansion for the Sinc transform of exponential type functions, revealing polynomial behavior for large parameters.

Findings

Asymptotic expansion behaves as a polynomial in positive powers of or large or specific conditions.

Conditions or or even and odd n ensure the validity of the expansion.

The expansion is explicit and terminates, providing precise asymptotic behavior.

Abstract

We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(\lambda x)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $\tau>0$, for large values of the parameter $\lambda$, 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 $\lambda$ as $\lambda$ grows large provided that the conditions $\lambda > \tau/2$ for even $n$ and $\lambda>\tau$ for odd n are satisfied.