Geodesic
An extremal problem for positive definite functions with support in a ball
Sbornik. Mathematics, Tome 215 (2024) no. 7, pp. 920-931 Cet article a éte moissonné depuis la source Math-Net.Ru

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The extremal problem under consideration is related to the set of continuous positive definite functions on $\mathbb{R}^n$ with support in a closed ball of radius $r>0$ and fixed value at the origin (the class $\mathfrak{F}_r(\mathbb{R}^n)$). Given $r>0$, the problem consists in finding the supremum on $\mathfrak{F}_r(\mathbb{R}^n)$ of a functional of a special form. A general solution to this problem is obtained for $n\neq2$. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type $\leqslant r$. Bibliography: 24 titles.
Keywords: positive definite functions, extremal problems, entire functions of exponential spherical type.
Mots-clés : Fourier transform 
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A. D. Manov. An extremal problem for positive definite functions with support in a ball. Sbornik. Mathematics, Tome 215 (2024) no. 7, pp. 920-931. http://geodesic.mathdoc.fr/item/SM_2024_215_7_a2/

[1] Z. Sasvári, Multivariate characteristic and correlation functions, De Gruyter Stud. Math., 50, Walter de Gruyter Co., Berlin, 2013, x+366 pp. | DOI | MR | Zbl

[2] C. L. Siegel, “Über Gitterpunkte in convexen Körpern und ein damit zusammenhängendes Extremalproblem”, Acta Math., 65:1 (1935), 307–323 | DOI | MR | Zbl

[3] R. P. Boas, Jr., and M. Kac, “Inequalities for Fourier transforms of positive functions”, Duke Math. J., 12:1 (1945), 189–206 | DOI | MR | Zbl

[4] D. V. Gorbachev, “Extremum problem for periodic functions supported in a ball”, Math. Notes, 69:3 (2001), 313–319 | DOI | MR | Zbl

[5] J.-P. Gabardo, “The Turán problem and its dual for positive definite functions supported on a ball in $\mathbb R^d$”, J. Fourier Anal. Appl., 30:1 (2024), 11, 31 pp. | DOI | MR | Zbl

[6] A. A. Arestov and E. E. Berdysheva, “The Turán problem for a class of polytopes”, East J. Approx., 8:3 (2002), 381–388 | MR | Zbl

[7] M. Kolountzakis and S. G. Révész, “On a problem of Turán about positive definite functions”, Proc. Amer. Math. Soc., 131:11 (2003), 3423–3430 | DOI | MR | Zbl

[8] G. Bianchi and M. Kelly, “A Fourier analytic proof of the Blaschke–Santaló inequality”, Proc. Amer. Math. Soc., 143:11 (2015), 4901–4912 | DOI | MR | Zbl

[9] D. V. Gorbachev and S. Yu. Tikhonov, “Wiener's problem for positive definite functions”, Math. Z., 289:3–4 (2018), 859–874 | DOI | MR | Zbl

[10] A. V. Efimov, “A version of the Turán's problem for positive definite functions of several variables”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 93–112 | DOI | MR | Zbl

[11] S. G. Révész, “Turán's extremal problem on locally compact abelian groups”, Anal. Math., 37:1 (2011), 15–50 | DOI | MR | Zbl

[12] A. D. Manov, “On an extremal problem for positive definite functions”, Chebyshevskii Sb., 22:5 (2021), 161–171 (Russian) | DOI | MR | Zbl

[13] O. Szász, “Über harmonische Funktionen und $L$-Formen”, Math. Z., 1:2–3 (1918), 149–162 | DOI | MR | Zbl

[14] Z. Sasvári, Positive definite and definitizable functions, Math. Top., 2, Akademie Verlag, Berlin, 1994, 208 pp. | MR | Zbl

[15] R. M. Trigub and E. S. Bellinsky, Fourier analysis and approximation of functions, Kluwer Acad. Publ., Dordrecht, 2004, xiv+585 pp. | DOI | MR | Zbl

[16] W. Rudin, “An extension theorem for positive-definite functions”, Duke Math. J., 37 (1970), 49–53 | DOI | MR | Zbl

[17] W. Ehm, T. Gneiting and D. Richards, “Convolution roots of radial positive definite functions with compact support”, Trans. Amer. Math. Soc., 356:11 (2004), 4655–4685 | DOI | MR | Zbl

[18] A. V. Efimov, “An analog of Rudin's theorem for continuous radial positive definite functions of several variables”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 79–86 | DOI | MR | Zbl

[19] M. G. Krein, “The problem of extension of Hermitian-positive continuous functions”, Dokl. Akad. Nauk SSSR, 26:1 (1940), 17–22 (Russian) | MR | Zbl

[20] L. Golinskii, M. Malamud and L. Oridoroga, “Radial positive definite functions and Schoenberg matrices with negative eigenvalues”, Trans. Amer. Math. Soc., 370:1 (2018), 1–25 | DOI | MR | Zbl

[21] S. M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems, Grundlehren Math. Wiss., 205, Springer-Verlag, New York–Heidelberg, 1975, viii+418 pp. | DOI | MR | Zbl

[22] I. I. Ibragimov, “Extremal problems in the class of entire functions of finite degree”, Izv. Akad. Nauk SSSR Ser. Mat., 23:2 (1959), 243–256 (Russian) | MR | Zbl

[23] J. Korevaar, “An inequality for entire functions of exponential type”, Nieuw Arch. Wiskunde (2), 23:2 (1949), 55–62 | MR | Zbl

[24] D. V. Gorbachev, “Sharp Bernstein–Nikol'skii inequalities for polynomials and entire functions of exponential type”, Chebyshevskii Sb., 22:5 (2021), 58–110 | DOI | MR | Zbl