TITLE: Does every locally positive-definite function have a positive-definite extension?
QUESTION [1 upvotes]: Let $B$ denote the unit ball in $\mathbb{R}^d$, and suppose $f\colon B\rightarrow\mathbb{C}$ has the property that for every $n\geq1$ and $x_1,\ldots,x_n\in\mathbb{R}^d$ with $\|x_i-x_j\|<1$, the $n\times n$ matrix $[f(x_i-x_j)]_{ij}$ is positive semidefinite. Can $f$ be extended to a positive definite function over $\mathbb{R}^d$?

REPLY [1 votes]: This is true for $d=1$ (M. Krein) but not true for $d>1$ (W. Rudin). For a criterion of extension see 
O. Jorgensen,  R. Niedzialomski,
Extension of positive definite functions, 
J. Math. Anal. Appl. 422 (2015), no. 1, 712–740.
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