Divergence Of Fourier Series Research Articles

On the divergence of Fourier series

On the divergence of Fourier series

Divergence of Fourier series in the general Haar system

For any countable set $D \subset [0,1]$, we construct a bounded measurable function $f$ such that the Fourier series of $f$ with respect to the regular general Haar system is divergent on $D$ and convergent on $[0,1]\backslash D$.

Divergent Fourier series in function spaces near L1[0;1

Bochkariev's theorem states that for any uniformly bounded orthonormal system Φ, there is a Lebesgue integrable function such that the Fourier series of it with respect to the system Φ diverges on the set of positive measure. In this paper, we extended Bochkariev's theorem for some class of variable exponent Lebesgue spaces. We characterized the class of variable exponent Lebesgue spaces Lp(⋅)[0;1], 1<p(x)<∞ a.e. on [0;1], such that above mentioned Bochkarev's theorem is valid.

On universality and convergence of the Fourier series of functions in the disc algebra

We construct functions in the disc algebra whose Fourier series are pointwise universal on countable and dense sets and their sets of divergence contain Gδ and dense sets and have Hausdorff dimension zero. We also see that some classes of closed sets of measure zero do not accept uniformly universal Fourier series, although all such sets accept divergent Fourier series.

On the absolute divergence of Fourier series on the infinite-dimensional torus

On the absolute divergence of Fourier series on the infinite-dimensional torus

An abstract Kolmogorov theorem, and an application to metric spaces and topological groups

Using the method of averaging over the supports of -functions, we generalise Kolmogorov’s fundamental theorem on the existence of a trigonometric Fourier series that diverges almost everywhere to any bounded biorthonormal systems of complex-valued functions on an arbitrary measurable space. The abstract Kolmogorov theorem thus obtained is applied to construct divergent Fourier series on metric spaces and topological groups. We establish the existence of a Fourier series in the system of characters of an arbitrary compact Abelian group that is divergent almost everywhere. Bibliography: 37 titles.

Construction of Function Spaces Close to $$L^\infty $$ L ∞ with Associate Space Close to $$L^1$$ L 1

The paper introduces a variable exponent space X which has in common with $$L^{\infty }([0,1])$$ the property that the space C([0, 1]) of continuous functions on [0, 1] is a closed linear subspace in it. The associate space of X contains both the Kolmogorov and the Marcinkiewicz examples of functions in $$L^{1}$$ with a.e. divergent Fourier series.

On Exceptional Sets of the Hilbert Transform

We prove several theorems concerning the exceptional sets of the Hilbert transform on the real line. In particular, it is proved that any null set is an exceptional set for the Hilbert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.

On the divergence of Fourier series in the spaces ϕ(L) containing L

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than Lp = Lp[−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ(L) of Kolmogorov’s theorem on the convergence of Fourier series in Lp, p ∈ (0, 1), cannot be improved.

On the Divergence of Fourier Series in the Spaces $\varphi(L)$ Containing $L$

Исследуется вопрос о расходимости рядов Фурье в функциональных пространствах более широких, чем $L=L[-\pi,\pi]$, но более узких, чем $L^p=L^p[-\pi,\pi]$ при всех $p\in(0,1).$ Доказана неулучшаемость недавних результатов В. И. Филиппова об обобщении на пространства $\varphi(L)$ теоремы Колмогорова о сходимости рядов Фурье в $L^p$ при $p\in(0,1)$. Библиография: 3 названия.

Erratum to our paper On the divergence of Fourier series of functions in several variables

Erratum to our paper On the divergence of Fourier series of functions in several variables

Multifractal analysis of the divergence of Fourier series: The extreme cases

We study the size, in terms of the Hausdorff dimension, of the subsets of T such that the Fourier series of a generic function in L1(T), Lp(T), or C(T) may behave badly. Genericity is related to the Baire Category Theorem or the notion of prevalence. This paper is a continuation of [3].

Estimates of the Dirichlet kernel and divergent fourier series in the Walsh-Kaczmarz system

A new lower bound for the growth of the Dirichlet kernel for the Walsh-Kaczmarz system is obtained and an example of an almost everywhere divergent Fourier series with respect to this system from a class narrower than that examined earlier is constructed.

Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejer on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.

On the divergence of Fourier series of functions in several variables

The paper deals with the problems of divergence of the series from absolute values of the Fourier coefficients of functions in several variables. It is proved that as the dimension of the space increases, the absolute convergence of Fourier series with respect to any complete orthnormal system (ONS) of functions with continuous partial derivatives becomes worse. For instance, for any ɛ ∈ (0, 2) there exists a function in variables \(k > \frac{{2(2 - \varepsilon )}} {\varepsilon }\) having all the continuous partial derivatives, however the series of absolute values of its coefficients with respect to any complete orthnormal system diverges in power 2 − ɛ.

Examples of divergent Fourier series for a wide class of rearranged Walsh-Paley system

A rearranged Walsh system is considered in the paper. An example of a Fourier series from the class \(Lo\left( {\sqrt {\ln ^ + } } \right)L\) divergent almost everywhere is constructed.

Lineability of Universal Divergence of Fourier Series

It is proved that large vector spaces of continuous complex functions on the unit circle exist such that all their nonzero members satisfy that, for many small subsets E of the circle, the partial sums of their Fourier series approximate any prescribed function on E. This completes or improves a number of results by several authors.

On Divergence of Fourier Series by Some Methods of Summability

A new summability method of series is introduced and studied. The particular cases of this method are, for example, variable-order Cesaro and Riesz methods. Applications to divergence problem of Fourier series are given. An extension of Kolmogorov, Schipp, and Bočkarev’s well-known theorems on divergence of Fourier trigonometric, Walsh, and orthonormal series is established.

Multifractal analysis of the divergence of Fourier series

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without exceeding $O(n^{1/p})$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_nf(x)=O(n^\beta)$ and we are interested in the size of the exceptional sets $E_\beta$, namely the sets of $x\in\mathbb T$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p(\TT)$ have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in $L^p(\mathbb T)$, for all $\beta\in[0,1/p]$, $E_\beta$ has Hausdorff dimension equal to $1-\beta p$. We also investigate the same problem in $\mathcal C(\mathbb T)$, replacing polynomial divergence by logarithmic divergence. In this context, the results that we get on the size of the exceptional sets are rather surprizing.

Example of a divergent Fourier series in the Vilenkin system

We construct an example of a function from the class $Lo(\sqrt {\ln ^ + L} )$ whose Fourier-Vilenkin series diverges almost everywhere.