Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.07466

The Properties of Ces\'aro General Fourier Sums of Functions with Derivatives of Lipschitz Class Functions

G. Tutberidze·V. Tsagareishvili·G. Cagareishvil
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we investigate the Ces\'aro means of Fourier series with respect to general orthonormal systems (ONS), when the function \( f \) belongs to a certain differentiable class of functions. It is well known that the membership of a function \( f \not\equiv 0 \) in a differentiable class does not, in general, guarantee the summability of its Fourier series with respect to an arbitrary ONS. Therefore, in order for the Fourier series with respect to a given ONS to be summable, one must impose additional conditions on the system functions \( \{\varphi_n\} \). The main objective of this work is to determine such conditions on the functions \( \varphi_n \) of the ONS under which the Ces\'aro means of the Fourier series of any function whose derivative belongs to the Lipschitz class \( \mathrm{Lip}_1 \) are uniformly bounded. The results obtained are sharp in the sense that the conditions cannot be essentially weakened.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

G. Tutberidze, V. Tsagareishvili, G. Cagareishvil. 2025-09-09. The Properties of Ces\'aro General Fourier Sums of Functions with Derivatives of Lipschitz Class Functions. https://arxiv.org/abs/2509.07466

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

math.CA↗
arXiv2026

Riesz capacity ratios with negative exponents

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

math.CA↗
arXiv2026

Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, P\'erez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wr\'obel and Zienkiewicz.

math.CA↗