Poisson summation for functions of bounded variation on ℝd
Tóm tắt
A one-dimensional version of the Poisson summation formula for functions of bounded variation due to R.M. Trigub is extended to the multivariate case under minimal assumptions on functions.
Tài liệu tham khảo
C. R. Adams and J. A. Clarkson, Properties of functions f(x, y) of bounded variation, Trans. Amer. Math. Soc., 36 (1934), 711–730.
E. S. Belinsky, On asymptotic behavior of integral norms of trigonometric polynomials, Metric Questions of the Theory of Functions and Mappings, Nauk. Dumka, Kiev, 6 (1975), 15–24 (in Russian).
Yu. Brudnyi, Multivariate functions of bounded (k, p)-variation, Banach Spaces and their Applications in Analysis, de Gruyter, 2007, 37–57.
J.A. Clarkson and C.R. Adams, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc., 35 (1934), 824–854.
A. Cordoba, La formule sommatoire de Poisson, C. R. Acad. Paris, Ser. I, 306 (1988), 373–376.
D. Faifman, A characterization of Fourier transform by Poisson summation formula, C. R. Acad. Paris, Ser. I, 348 (2010), 407–410.
E. Liflyand, U. Stadtmüller and R. Trigub, An interplay of multidimensional variations in Fourier Analysis, J. Fourier Anal. Appl., 17 (2011), 226–239.
E. Liflyand and U. Stadtmüller, A multidimensional Euler-Maclaurin formula and application, Proceedings of an international conference Complex Analysis and Dynamical Systems V, Israel Mathematical Conference Proceedings sub-series of Contemporary Mathematics, 2013, 181–191.
F. Móricz, On the regular convergence of multiple integrals of locally Lebesgue integrable functions over ̄ℝ m+ , C. R. Math. Acad. Sci. Paris, 350 (2012), 459–464.
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971.
L. Tonelli, Series trigonometriche, Bologna, 1928 (in Italian).
R. M. Trigub, Multipliers of Fourier series and approximation of functions by polynomials in spaces C and L, Dokl. Akad. Nauk SSSR, 306 (1989), 292–296 (in Russian); English transl.: Soviet Math. Dokl., 39 (1989), 494–498.
R. M. Trigub, A Generalization of the Euler-Maclaurin formula, Mat. Zametki, 61 (1997), 312–316 (in Russian); English transl.: Math. Notes, 61(1997), 253–257.
R. M. Trigub and E. S. Belinsky, Fourier Analysis and Appoximation of Functions, Kluwer, 2004.
A. Zygmund, Trigonometric series, Vol. I, II, Cambridge Univ. Press, Cambridge, U. K., 1966.
