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Mat. Zametki, 2017, Volume 101, Issue 2, Pages 215–225 (Mi mz10850)  

Positive Definiteness of a Family of Functions

V. P. Zastavnyi

Donetsk National University

Abstract: 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) $3<p\le\infty$ and $n=2$.

Keywords: positive definite function, Fourier transform, Bochner's theorem.

DOI: https://doi.org/10.4213/mzm10850

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English version:
Mathematical Notes, 2017, 101:2, 250–259

Bibliographic databases:

UDC: 517.5+519.213
Received: 24.06.2015
Revised: 26.03.2016

Citation: V. P. Zastavnyi, “Positive Definiteness of a Family of Functions”, Mat. Zametki, 101:2 (2017), 215–225; Math. Notes, 101:2 (2017), 250–259

Citation in format AMSBIB
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