Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness

In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to these spaces. In case these functions are not locally integrable, the authors also consider their generalized Lebesgue points defined via the γ-medians instead of the classical ball integral averages and establish the corresponding zero-capacity property of the exceptional sets.

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

We 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.

Cesàro summability and Lebesgue points of higher dimensional Fourier series

<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>

Function Polynomization Methods and their Applications

A class of semicontinuous quasiconvex methods of summation of Fourier – Chebyshev series is studied. Upper bounds are obtained for the norms of the corresponding operators in the space of continuous functions. The convergence of means in the metric of space is established. The summability at break points of the first kind is also considered. Processes for restoring functions from a given sequence of power moments are proposed. Ways of generalizing the results and extending them to the case of summability at Lebesgue points are indicated.