On integral estimates of non-negative positive definite functions

Abstract

Let $\ell >0$ be arbitrary. We introduce the extremal quantities $$G(\ell ):=\frac{{\sup }_f{\int }_{-\ell }^{\ell }fdx}{{\int }_{-1}^{1}fdx},C(\ell ):=\frac{{\sup }_f{\sup }_{a\in \mathbb{R}}{\int }_{a-\ell }^{a+\ell }fdx}{{\int }_{-1}^{1}fdx},$$ 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.

Cited by in corpus (1)

• Doubling condition at the origin for non-negative positive definite functions