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

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.