The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables

R. M. Trigub

Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 759-779

Résumé

Given an $L_1(\mathbb{R}^2)$-function $f(x_1,x_2)=f_0(\max\{|x_1|,|x_2|\})$, necessary conditions and sufficient conditions for its Fourier transform $\widehat{f}$ to lie in $L_1(\mathbb{R}^2)$ and for the function $t\mapsto t\sup_{y_1^2+y_2^2\geqslant t^2}|\widehat{f}(y_1,y_2)|$ to be in $L_1(\mathbb{R}_{+})$ are indicated. The problem of the positivity of $\widehat{f}$ on $\mathbb{R}^2$ is shown to be completely reducible to the same problem for the function $\displaystyle f_1(x)=|x|f_0(x)+\int_{|x|}^\infty f_0(t)\,dt$ in $\mathbb{R}$. Bibliography: 20 titles.

Détail

Keywords: Wiener Banach algebra, positive definiteness, Bernstein's theorem on completely monotone functions, Marcinkiewicz sums of a double Fourier series, Wiener approximation theorem.
Mots-clés : Lebesgue points

Bibliographie

[1] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., 32, Princeton Univ. Press, Princeton, NJ, 1971, x+297 pp. | MR | Zbl | Zbl

[2] E. Liflyand, S. Samko, R. Trigub, “The Wiener algebra of absolutely convergent Fourier integrals: an overview”, Anal. Math. Phys., 2:1 (2012), 1–68 | DOI | MR | Zbl

[3] R. M. Trigub, “Almost everywhere summability of Fourier series with indication of the set of convergence”, Math. Notes, 100:1 (2016), 139–153 | DOI | DOI | MR | Zbl

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

[5] R. M. Trigub, “On Fourier multipliers and absolute convergence of Fourier integrals of radial functions”, Ukrainian Math. J., 62:9 (2010), 1487–1501 | DOI | Zbl

[6] A. Beurling, “On the spectral synthesis of bounded functions”, Acta Math., 81 (1949), 225–238 | DOI | MR | Zbl

[7] E. S. Belinskii, E. R. Liflyand, R. M. Trigub, “The Banach algebra $A^*$ and its properties”, J. Fourier Anal. Appl., 3:2 (1997), 103–129 | DOI | MR | Zbl

[8] E. Liflyand, R. Trigub, “Conditions for the absolute convergence of Fourier integrals”, J. Approx. Theory, 163:4 (2011), 438–459 | DOI | MR | Zbl

[9] A. N. Podkorytov, “Summation of multiple Fourier series over polyhedra”, Vestn. Leningr. Univ. Math., 13 (1981), 69–77 | MR | Zbl

[10] R. M. Trigub, “Integriruemost i asimptoticheskoe povedenie preobrazovaniya Fure radialnoi funktsii”, Metricheskie voprosy teorii funktsii i otobrazhenii, Naukova dumka, Kiev, 1977, 142–163 | MR | Zbl

[11] Yu. G. Kuritsyn, “Multivariate versions and two problems of Schoenberg”, J. Soviet Math., 59:4 (1992), 939–945 | DOI | MR | Zbl

[12] H. Berens, Yuan Xu, “$l-1$ summability for multiple Fourier integrals and positivity”, Math. Proc. Cambridge Philos. Soc., 122:1 (1997), 149–172 | DOI | MR | Zbl

[13] V. P. Zastavnyi, “On positive definiteness of some functions”, J. Multivariate Anal., 73:1 (2000), 55–81 | DOI | MR | Zbl

[14] T. Gneiting, “Criteria of Pólya type for radial positive definite functions”, Proc. Amer. Math. Soc., 129:8 (2001), 2309–2318 | DOI | MR | Zbl

[15] M. D. Buhmann, “A new class of radial basis functions with compact support”, Math. Comp., 70:233 (2001), 307–318 | DOI | MR | Zbl

[16] A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956, vi+108 pp. | MR | Zbl | Zbl

[17] L. V. Zhizhiashvili, “On the summation of double Fourier series”, Siberian Math. J., 8:3 (1967), 402–414 | DOI | MR | Zbl

[18] M. A. Skopina, “On almost everywhere convergence of Marcinkiewicz sums of double Fourier series”, J. Math. Sci. (N. Y.), 71:1 (1994), 2263–2268 | DOI | MR | Zbl

[19] F. Weisz, “Strong summability of Fourier transforms at Lebesque points and Wiener amalgam spaces”, J. Funct. Spaces, 2015 (2015), 420750, 10 pp. | DOI | MR | Zbl

[20] R. M. Trigub, “Summability of trigonometric Fourier series at $d$-points and a generalization of the Abel–Poisson method”, Izv. Math., 79:4 (2015), 838–858 | DOI | DOI | MR | Zbl