scholarly journals Unrestricted Ces\`aro summability of $d$-dimensional Fourier series and Lebesgue points

2021 ◽  
Author(s):  
Ferenc WEİSZ
Keyword(s):  
Fourier Series ◽  
Lebesgue Points

scholarly journals Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Cesáro Means

AbstractWe generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

scholarly journals Strong Summability of Fourier Transforms at Lebesgue Points and Wiener Amalgam Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Fourier Transforms ◽  
Lebesgue Point ◽  
Strong Summability ◽  
Wiener Amalgam Spaces ◽  
Wiener Amalgam Space ◽  
Almost Everywhere ◽  
Lebesgue Points ◽  
Amalgam Spaces

We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, iffis in the Wiener amalgam spaceW(L1,lq)(R)andfis almost everywhere locally bounded, orf∈W(Lp,lq)(R)  (1<p<∞,1≤q<∞), then strongθ-summability holds at each Lebesgue point off. The analogous results are given for Fourier series, too.

scholarly journals Cesàro summability and Lebesgue points of higher dimensional Fourier series

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Maximal Functions ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
New Concepts ◽  
Higher Dimensional ◽  
Different Types

<p style='text-indent:20px;'>We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means <inline-formula><tex-math id="M1">\begin{document}$ \sigma_n^{\alpha}f $\end{document}</tex-math></inline-formula> of the Fourier series of a multi-dimensional function converge to <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> at each Lebesgue point as <inline-formula><tex-math id="M3">\begin{document}$ n\to \infty $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Approximation of Conjugate Functions by General Linear Operators of Their Fourier Series at the Lebesgue Points

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Włodzimierz Łenski ◽  
Bogdan Szal
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
General Linear ◽  
Conjugate Functions ◽  
Moduli Of Continuity ◽  
Pointwise Estimates ◽  
Lebesgue Points ◽  
Norm Approximation

AbstractThe pointwise estimates of the deviations r T͂n,A,Bf (·) - f͂͂ (·) and T͂n,A,Bf (·) - f͂͂ (·,ε) in terms of moduli of continuity ω̃f and r ω̃f are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal [3, Theorem 1, p. 437].

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