Lebesgue points and Cesàro summability of higher dimensional Fourier series over a cone

2021 ◽  
Vol 87 (34) ◽  
pp. 505-515
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Higher Dimensional

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 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.

Sign in / Sign up

Export Citation Format

Share Document