## Banach Journal of Mathematical Analysis

### Triangular summability and Lebesgue points of $2$-dimensional Fourier transforms

Ferenc Weisz

#### Abstract

We consider the triangular $\theta$-summability of $2$-dimensional Fourier transforms. Under some conditions on $\theta$, we show that the triangular $\theta$-means of a function $f$ belonging to the Wiener amalgam space $W(L_{1},\ell_{\infty})({\mathbb{R}}^{2})$ 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 $f\in W(L_{p},\ell_{\infty})({\mathbb{R}}^{2})$ whenever $1\lt p\lt \infty$. Some special cases of the $\theta$-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

Mathematical Reviews number (MathSciNet)
MR3582397

Zentralblatt MATH identifier
1354.42013

#### 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

#### 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.