Abstract
Positive definite kernels and functions are considered. The key tool in the paper is the well-known main inequality for such kernels, namely, the Cauchy–Bunyakovskii inequality for the special inner product generated by a given positive definite kernel. It is shown that Ingham’s inequality (and, in particular, Hilbert’s inequality) is, essentially, the main inequality for the positive definite function \(\sin(\pi x)/x\) on \(\mathbb{R}\) and for a system of integer points. Using the main inequality, we prove new inequalities of Krein–Gorin type and Ingham’s inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces (Nauka, Moscow, 1985) [in Russian].
M. G. Krein, “A ring of functions on a topological group,” C. R. (Doklady) Acad. Sci. URSS (N. S.) 29 (4), 275–280 (1940).
M. G. Krein, “Sur le problème du prolongement des fonctions hermitiennes positives et continues,” C. R. (Doklady) Acad. Sci. URSS (N. S.) 26 (1), 17–22 (1940).
E. A. Gorin, “Inequalities for positive definite functions,” Funct. Anal. Appl. 49 (4), 301–303 (2015).
E. A. Gorin, “Universal symbols on locally compact Abelian groups,” Bull. Polish Acad. Sci. Math. 51 (2), 199–204 (2003).
E. A. Gorin, “Positive definite functions as an instrument of mathematical analysis,” J. Math. Sci. 197 (4), 492–511 (2014).
A. B. Pevnyi and S. M. Sitnik, “On generalizations of M. G. Krein, E. A. Gorin, and Yu. V. Linnik inequalities for positive definite functions for multipoint case,” Sib. Èlektron. Mat. Izv. 16, 263–270 (2019).
Z. Sasvári, Positive Definite and Definitizable Functions (Akademie Verlag, Berlin, 1994).
Z. Sasvári, Multivariate Characteristic and Correlation Functions (Walter de Gruyter, Berlin, 2013).
N. Aronszajn, “Theory of reproducing kernels,” Trans. Amer. Math. Soc. 68 (3), 337–404 (1950).
R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation Functions (Kluwer Acad. Publ., Dordrecht, 2004).
V. P. Zastavnyi, “On positive definiteness of some functions,” J. Multivariate Anal. 73, 55–81 (2000).
A. E. Ingham, “A note on Hilbert’s inequality,” J. London Math. Soc. 11 (3), 237–240 (1936).
G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934).
A. Weil, L’intégration dans les groupes topologiques et ses applications (Hermann & Cie, Paris, 1940).
Funding
This work was supported by the Russian Science Foundation under grant 17-11-01377.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zastavnyi, V.P. Inequalities for Positive Definite Functions. Math Notes 108, 791–801 (2020). https://doi.org/10.1134/S000143462011022X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143462011022X