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Matematicheskie Zametki, 2019, Volume 106, Issue 2, Pages 222–240
DOI: https://doi.org/10.4213/mzm12130
(Mi mzm12130)
 

This article is cited in 2 scientific papers (total in 2 papers)

Some Problems Related to Completely Monotone Positive Definite Functions

V. P. Zastavnyi

Donetsk National University
Full-text PDF (593 kB) Citations (2)
References:
Abstract: This paper deals with several problems related to functions of the class CM of completely monotone functions and functions of the class Φ(E) of positive definite functions on a real linear space E. Theorem 1 verifies some conjectures of Moak related to the complete monotonicity of the function xμ(x2+1)ν. Theorem 2 states that if fC(0,+) and δR, then
f(x)aδf(ax)CMfor alla>1
if and only if δf(x)xf(x)CM. A similar result for functions in Φ(E) is obtained in Theorem 9: if εR and a function h:[0,+)R is continuous on [0,+) and differentiable on the interval (0,+) and satisfies the condition xh(x)0 as x+0, then
h(ρ(u))aεh(aρ(u))Φ(E)for alla>1
if and only if ψε(ρ(u))Φ(E), where ψε(x):=εh(x)xh(x) for x>0 and ψε(0):=εh(0). Here ρ is a nonnegative homogeneous function on E and ρ(u)0. It is proved (Example 6) that:
  • eαu(1βu)Φ(Rm) if and only if αβα/m;
  • eαu2(1βu2)Φ(Rm) if and only if 0β2α/m.
Here u is the Euclidean norm on Rm. Theorem 11 deals with the case of radial positive definite functions hμ,ν.
Keywords: completely monotone functions, positive definite functions, Hausdorff–Bernstein–Widder theorem, Fourier transform, Bochner–Khinchine theorem.
Received: 10.07.2018
English version:
Mathematical Notes, 2019, Volume 106, Issue 2, Pages 212–228
DOI: https://doi.org/10.1134/S0001434619070253
Bibliographic databases:
Document Type: Article
UDC: 517.5+519.213
Language: Russian
Citation: V. P. Zastavnyi, “Some Problems Related to Completely Monotone Positive Definite Functions”, Mat. Zametki, 106:2 (2019), 222–240; Math. Notes, 106:2 (2019), 212–228
Citation in format AMSBIB
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Linking options:
  • https://www.mathnet.ru/eng/mzm12130
  • https://doi.org/10.4213/mzm12130
  • https://www.mathnet.ru/eng/mzm/v106/i2/p222
  • This publication is cited in the following 2 articles:
    1. Henry J. Brown, Yury Grabovsky, “On Feasibility of Extrapolation of Completely Monotone Functions”, SIAM J. Math. Anal., 56:6 (2024), 7713  crossref
    2. V. Zastavnyi, A. Manov, “Some generalizations of the problem of positive definiteness of a piecewise linear function”, Journal of Mathematical Analysis and Applications, 519:2 (2023), 126864  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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