Banach Journal of Mathematical Analysis

Triangular summability and Lebesgue points of 2-dimensional Fourier transforms

Ferenc Weisz

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Abstract

We consider the triangular θ-summability of 2-dimensional Fourier transforms. Under some conditions on θ, we show that the triangular θ-means of a function f belonging to the Wiener amalgam space W(L1,)(R2) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points for the so-called modified Lebesgue points of fW(Lp,)(R2) whenever 1<p<. Some special cases of the θ-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.

Article information

Source
Banach J. Math. Anal. Volume 11, Number 1 (2017), 223-238.

Dates
Received: 26 January 2016
Accepted: 24 March 2016
First available in Project Euclid: 9 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1481274116

Digital Object Identifier
doi:10.1215/17358787-3796829

Zentralblatt MATH identifier
06667696

Subjects
Primary: 42B08: Summability
Secondary: 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A24: Summability and absolute summability of Fourier and trigonometric series 42B25: Maximal functions, Littlewood-Paley theory

Keywords
Fourier transforms triangular summability Fejér summability $\theta$-summability Lebesgue points

Citation

Weisz, Ferenc. Triangular summability and Lebesgue points of $2$ -dimensional Fourier transforms. Banach J. Math. Anal. 11 (2017), no. 1, 223--238. doi:10.1215/17358787-3796829. https://projecteuclid.org/euclid.bjma/1481274116


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References

  • [1] H. Berens, Z. Li, and Y. Xu, On $l_{1}$ Riesz summability of the inverse Fourier integral, Indag. Math. (N.S.) 12 (2001), 41–53.
  • [2] H. Berens and Y. Xu, $l$-1 summability of multiple Fourier integrals and positivity, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 149–172.
  • [3] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation: One-Dimensional Theory, I, Birkhäuser, Basel, 1971.
  • [4] H. G. Feichtinger and F. Weisz, Wiener amalgams and pointwise summability of Fourier transforms and Fourier series, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 3, 509–536.
  • [5] G. Gát, Pointwise convergence of cone-like restricted two-dimensional $(C,1)$ means of trigonometric Fourier series, J. Approx. Theory 149 (2007), no. 1, 74–102.
  • [6] G. Gát, U. Goginava, and K. Nagy, On the Marcinkiewicz–Fejér means of double Fourier series with respect to Walsh–Kaczmarz system, Studia Sci. Math. Hungar. 46 (2009), no. 3, 399–421.
  • [7] U. Goginava, Marcinkiewicz–Fejér means of $d$-dimensional Walsh–Fourier series, J. Math. Anal. Appl. 307 (2005), no. 1, 206–218.
  • [8] U. Goginava, Almost everywhere convergence of (C, a)-means of cubical partial sums of d-dimensional Walsh–Fourier series, J. Approx. Theory 141 (2006), no. 1, 8–28.
  • [9] U. Goginava, The maximal operator of the Marcinkiewicz–Fejér means of $d$-dimensional Walsh–Fourier series, East J. Approx. 12 (2006), no. 3, 295–302.
  • [10] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ, 2004.
  • [11] H. Lebesgue, Recherches sur la convergence des séries de Fourier, Math. Ann. 61 (1905), 251–280.
  • [12] L.-E. Persson, G. Tephnadze, and P. Wall, Maximal operators of Vilenkin-Nörlund means, J. Fourier Anal. Appl. 21 (2015), no. 1, 76–94.
  • [13] P. Simon, $(C,\alpha)$ summability of Walsh–Kaczmarz–Fourier series, J. Approx. Theory 127 (2004), no. 1, 39–60.
  • [14] L. Szili and P. Vértesi, On multivariate projection operators, J. Approx. Theory 159 (2009), no. 1, 154–164.
  • [15] R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer, Dordrecht, 2004.
  • [16] F. Weisz, Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory 7 (2012), 1–179.
  • [17] F. Weisz, Triangular summability of two-dimensional Fourier transforms, Anal. Math. 38 (2012), no. 1, 65–81.
  • [18] F. Weisz, Lebesgue points of two-dimensional Fourier transforms and strong summability, J. Fourier Anal. Appl. 21 (2015), no. 4, 885–914.