Uniqueness of the Continuation of a Certain Function to a Positive Definite Function

In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function $f$ defined in an interval $(-a,a)$, $a>0$, to a positive definite function on the whole number axis $\mathbb R$. In addition, Krein showed that the function $1-|x|$, $|x|<a$, can be extended to a positive definite one on $\mathbb R$ if and only if $0<a\le 2$, and this function has a unique extension only in the case $a=2$. The present paper deals with the problem of uniqueness of the extension of the function $1-|x|$, $|x|\le a$, $a\in(0,1)$, for a class of positive definite functions on $\mathbb R$ whose support is contained in the closed interval $[-1,1]$ (the class $\mathfrak F$). It is proved that if $a\in[1/2,1]$ and $\operatorname{Re}\varphi(x)=1-|x|$, $|x|\le a$, for some $\varphi\in\mathfrak F$, then $\varphi(x)=(1-|x|)_+$, $x\in\mathbb R$. In addition, for any $a\in(0,1/2)$, there exists a function $\varphi\in\mathfrak F$ such that $\varphi(x)=1-|x|$, $|x|\le a$, but $\varphi(x)\not\equiv(1-|x|)_+$. Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.