Geodesic
Positive Definiteness of a Family of Functions
Matematičeskie zametki, Tome 101 (2017) no. 2, pp. 215-225

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General necessary conditions on the real parameters $\alpha$, $\beta$, $C$, $D$ for the function $$ e^{-\alpha\rho(x)}(C\cos\beta\rho(x)+D\sin\beta\rho(x)), $$ where $\rho$ is the norm on $\mathbb R^n$, to be positive definite on $\mathbb R^n$, are obtained. For $\rho(x)=\|x\|_p$, a criterion on these parameters is obtained in the following cases: (i) $p=1$ or $p=2$; (ii) $3$ and $n=2$.
Keywords: positive definite function, Bochner's theorem.
Mots-clés : Fourier transform 
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     author = {V. P. Zastavnyi},
     title = {Positive {Definiteness} of a {Family} of {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {215--225},
     publisher = {mathdoc},
     volume = {101},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_101_2_a6/}
}
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V. P. Zastavnyi. Positive Definiteness of a Family of Functions. Matematičeskie zametki, Tome 101 (2017) no. 2, pp. 215-225. http://geodesic.mathdoc.fr/item/MZM_2017_101_2_a6/