On integral estimates of non-negative positive definite functions

TL;DR

This paper investigates extremal integral ratios of non-negative positive definite functions, providing exact values at natural numbers and bounds for other cases, extending classical Wiener problems.

Contribution

It introduces new extremal quantities for positive definite functions and computes their exact limits at natural numbers, advancing understanding of integral estimates.

Findings

Exact values for G(k+0) and C(k+0) at natural numbers k.

Bounds for extremal quantities for non-integer .

Extension of classical Wiener problem.

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.