ON INTEGRAL ESTIMATES OF NONNEGATIVE POSITIVE DEFINITE FUNCTIONS

Abstract

Let $\ell >0$ be arbitrary. We introduce the extremal quantities $$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$ where the supremum is taken over all not identically zero nonnegative positive definite functions. We investigate how large these extremal quantities can be. This problem was originally posed by Yu. Shteinikov and S. Konyagin (for the case $\ell =2$) and is an extension of the classical problem of Wiener. In this note we obtain exact values for the right limits $\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$ and $\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$$(k\in \mathbb{N})$ taken over doubly positive functions, and sufficiently close bounds for other values of $\ell$.