Uniform convergence and integrability of multiplicative Fourier transforms

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• 作者：S. S. Volosivets ; B. I. Golubov
• 关键词：multiplicative Fourier transform ; weighted integrability of Fourier transforms ; Lipschitz class ; Lipschitz—Besov space ; Herz space ; Dirichlet kernel ; H?lder’s inequality ; Hardy’s inequality ; Minkowski’s inequality
• 刊名：Mathematical Notes
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：98
• 期：1-2
• 页码：53-67
• 全文大小：717 KB
• 参考文献：1.M. Dyachenko, E. Liflyand, and S. Tikhonov, “Uniform convergence and integrability of Fourier integrals,-J. Math. Anal. Appl. 372 (1), 328-38 (2010).MATH MathSciNet CrossRef
  2.G. Sampson and G. Tuy, “Fourier transforms and their Lipschitz classes,-Pacific J. Math. 75 (2), 519-37 (1978).MATH MathSciNet CrossRef
  3.F. Moricz, “Best possible sufficient conditions for the Fourier transform to satisfy Lipschitz or Zygmund condition,-Studia Math. 199 (2), 199-05 (2010).MATH MathSciNet CrossRef
  4.E. Liflyand and S. Tikhonov, “Extended solution of Boas-conjecture on Fourier transforms,-C. R. Acad. Sci. Paris, Ser. I. 346 (21-22), 1137-142 (2008).MATH MathSciNet CrossRef
  5.S. S. Volosivets and B. I. Golubov, “Weighted integrability of multiplicative Fourier transforms,-in Trudy Mat. Inst. Steklov Vol. 269: Theory of Functions and Differential Equations (MAIK, Moscow, 2010), pp. 71-1 [Proc. Steklov Inst.Math. 269, 65-5 (2010)].
  6.B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Series and Walsh Transforms: Theory and Applications (Nauka, Moscow, 1987) [in Russian].MATH
  7.C. W. Onneweer, “Generalized Lipschitz spaces and Herz spaces on certain totally disconnected groups,-in Martingale Theory in Harmonic Analysis and Banach Spaces, Lecture Notes in Math. (Springer-Verlag, Berlin, 1982), Vol. 939, pp. 106-21.MathSciNet CrossRef
  8.B. I. Golubov and S. S. Volosivets, “On the integrability and uniform convergence of multiplicative Fourier transforms,-Georgian Math. J. 16 (3), 533-46 (2009).MATH MathSciNet
  9.G. Hardy, D. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934; Inostr. Lit., Moscow, 1948).
  10.J. S. Bradley, “Hardy inequalities with mixed norms,-Canad. Math. Bull. 21 (4), 405-08 (1978).MATH MathSciNet CrossRef
  11.E. M. Stein, “Interpolation of linear operators,-Trans. Amer.Math. Soc. 83, 482-92 (1956).MATH MathSciNet CrossRef
  12.C. W. Onneweer, “The Fourier transform of Herz spaces on certain groups,-Monatsch. Math. 97 (4), 297-10 (1984).MATH MathSciNet CrossRef
  13.S. S. Volosivets, “Fourier transforms and generalized Lipschitz classes in uniform metric,-J. Math. Anal. Appl. 383 (2), 344-52 (2011).MATH MathSciNet CrossRef
  14.S. S. Volosivets, “The modified multiplicative integral and derivative of arbitrary order on the semiaxis,-Izv. Ross. Akad. Nauk Ser.Mat. 70 (2), 3-4 (2006) [Izv.Math. 70 (2), 211-31 (2006)].MathSciNet CrossRef
  15.S. Fridli, “On the rate of convergence of Cesaro means of Walsh-Fourier series,-J. Approx. Theory 76 (1), 31-3 (1994).MATH MathSciNet CrossRef
  
• 作者单位：S. S. Volosivets (1)
  B. I. Golubov (2)
  
  1. Saratov State University, Saratov, Russia
  2. Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow Region, Russia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1573-8876

文摘

For multiplicative Fourier transforms, analogs of results obtained by R. P. Boas, F. Moricz, M. I. D’yachenko, I. P. Liflyand, and S. Yu. Tikhonov and dealing with conditions for the uniform convergence and weighted integrability with power weight of classical Fourier transforms as well as conditions for these transforms to belong to Lipschitz classes are proved. Certain results of C. W. Onneweer concerning conditions for multiplicative Fourier transforms to belong to Lipschitz—Besov and Herz spaces are also generalized. Keywords multiplicative Fourier transform weighted integrability of Fourier transforms Lipschitz class Lipschitz—Besov space Herz space Dirichlet kernel H?lder’s inequality Hardy’s inequality Minkowski’s inequality