Some remarkable properties of sinc and related integrals
The paper deals with the evaluations of integrals of the form \[ \int_0^\infty \prod_{k=0}^n \text{sinc}(a_k x) dx \] where the function sinc is defined by sinc\((t)= {\sin t\over t}\) for \(t\neq 0\) and \(\{a_k\}_{k=0}^\infty\) is any sequence of positive numbers. Related integrals are also concerned with. One of the main results is Theorem 1. Suppose that \(\{a_k\}_{k=0}^\infty\) is a sequence of positive numbers. Let \(s_n=\sum_{k=1}^n\) and \(\tau_n=\int_0^\infty \Pi_{k=0}^n \text{ sinc }(a_k x) dx\). (i) Then \(0<\tau_n\leq {\pi\over 2a_0}\) with equality if \(n=0,\) or if \(a_0\geq s_n\) when \(n\geq 1\); (ii) If \(a_{n+1}\leq a_0<s_n\) with \(n\geq 1\), then \(0<\tau_{n+1}\leq\tau_n< {\pi\over 2a_0}\); (iii) If \(a_0< S_{n_0}\) with \(n_0\geq 1\) and \(\sum_{k=0}^\infty a_k^2 <\infty,\) then there is an integer \(n_1\geq n_0\) such that \[ \tau_n \geq \int_0^\infty \prod_{k=0}^\infty \text{sinc}(a_k x) dx \geq \int_0^\infty \prod_{k=0}^\infty \text{sinc}^2(a_k x) dx>0\quad \text{for\;all\;} n\geq n_1. \] Fourier transform techniques are employed.
scientific article; zbMATH DE number 1647669