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

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.