On integral estimates of non-negative positive definite functions

Abstract

Let $\ell>0$ be arbitrary. We introduce the extremal quantities $$ G(\ell):=\frac{\sup_{f} \int_{-\ell}^{\ell} f\,dx}{\int_{-1}^1 f\,dx},\quad C(\ell):=\frac{\sup_{f} \sup_{a\in {\mathbb R}} \int_{a-\ell}^{a+\ell} f\,dx}{\int_{-1}^1 f\,dx}, $$ where the supremum is taken over all not identically zero non-negative positive definite functions. We are interested in the question: how large can the above extremal quantities be? This problem was originally posed by Yu. Shteinikov and S. Konyagin for the case $\ell=2$. In this note we obtain exact values for the right limits $G(k+0)$ and $C(k+0)$ at natural numbers $k$, and sufficiently close bounds for other values of $\ell$. We point out that the problem provides an extension of the classical problem of Wiener.