Divergence Of Fourier Series Research Articles

Example of divergent Fourier series over rearranged Walsh-Paley system

A specially chosen class of Shipp’s rearrangements of the Walsh system is considered in the paper. An example of a Fourier series from the class L(ln+ ln+)1−eL divergent almost everywhere is constructed for the systems obtained here.

Continuous functions with universally divergent Fourier series on small subsets of the circle

It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality properties.

Examples of divergent fourier series for classes of functions of bounded Λ-variation

The author has shown earlier that the requirement that a continuous function belong to the class HBV ([−π, π]m) for m ≥ 3 is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.

On Divergence of Fourier Series with Respect to Multiplicative Systems on the Sets of Measure Zero

Abstract For any multiplicative system of bounded type and any set of measure zero there exists a bounded measurable function whose Fourier series with respect to this system diverges on this set.

A note on rearrangement of Fourier series

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.

Divergent Fourier series of integrable functions with respect to orthonormal systems

There exist two orthonormal systems such that the Fourier series of each functionƒ ∈ L[0, 1],ƒ ≠ 0, with respect to at least one of these systems diverges on a set of positive measure.

On the set of all continuous functions with uniformly convergent Fourier series

In this article we calculate the exact location in the Borel hierarchy ofUCF,UCF,the set of all continuous functions on the unit circle with uniformly convergent Fourier series. It turns out to be completeFσδ.F_{\sigma \delta }.Also we prove that anyGδσG_{\delta \sigma }set that includesUCFUCFmust contain a continuous function with divergent Fourier series.

A Function in the Dirichlet Space such that its Fourier Series Diverges Almost Everywhere

An analytic function $F$ on the disc belongs to $B$ if $||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta dr < \infty } }$. Notice that $B \varsubsetneqq {H^1} \varsubsetneqq {L^1}$, where ${H^1}$ is the Hardy space of all analytic functions $F$ so that \[ ||F|{|_{{H^1}}} = \sup \limits _{0 < r < 1} \int _0^{2\pi } {|F(r{e^{i\theta }})| d\theta < \infty ,} \] ${L^1}$ is the Lebesgue space of integrable functions on $[0,2\pi ]$, and the inclusion ${H^1} \varsubsetneqq {L^1}$ is taken in the sense of boundary values, that is, $F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1}$. Kolmogorov in 1923 showed that there exists an $f$ in ${L^1}$ so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an $f$ in ${H^1}$ with an almost everywhere divergent Fourier series. The purpose of this note is to announce.

A function in the Dirichlet space such that its Fourier series diverges almost everywhere

An analytic function F F on the disc belongs to B B if | | F | | B = ∫ 0 1 ∫ 0 2 π | F ′ ( r e i θ ) | d θ d r &gt; ∞ ||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta \,dr &gt; \infty } } . Notice that B ⫋ H 1 ⫋ L 1 B \varsubsetneqq {H^1} \varsubsetneqq {L^1} , where H 1 {H^1} is the Hardy space of all analytic functions F F so that \[ | | F | | H 1 = sup 0 &gt; r &gt; 1 ∫ 0 2 π | F ( r e i θ ) | d θ &gt; ∞ , ||F|{|_{{H^1}}} = \sup \limits _{0 &gt; r &gt; 1} \int _0^{2\pi } {|F(r{e^{i\theta }})|\,d\theta &gt; \infty ,} \] L 1 {L^1} is the Lebesgue space of integrable functions on [ 0 , 2 π ] [0,2\pi ] , and the inclusion H 1 ⫋ L 1 {H^1} \varsubsetneqq {L^1} is taken in the sense of boundary values, that is, F ∈ H 1 ⇒ lim r → 1 − ℜ F ( r e i θ ) ∈ L 1 F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1} . Kolmogorov in 1923 showed that there exists an f f in L 1 {L^1} so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an f f in H 1 {H^1} with an almost everywhere divergent Fourier series. The purpose of this note is to announce. Theorem 1. There exists an f ∈ B f \in B , whose Fourier series diverges a.e. This problem was mentioned to the first author by Professor Guido Weiss.

The Complexity of Everywhere Divergent Fourier Series

AbstractA natural rank function is defined on the set DS of everywhere divergent sequences of continuous functions on the unit circle T. The rank function provides a natural measure of the complexity of the sequences in DS, and is obtained by associating a well-founded tree with each such sequence. The set DF of everywhere divergent Fourier series, and the set DT of everywhere divergent trigonometric series with coefficients that tend to zero, can be viewed as natural subsets of DS. It is shown that the rank function is a coanalytic norm which is unbounded in ω1 on DF. From this it follows that DF, DT and DS are not Borel subsets of the Polish space SC(T) of all sequences of continuous functions on T. Finally an alternative definition of the rank function is formulated by using nested sequences of closed sets.

A simple proof of the divergence of Fourier series on compact connected semisimple lie groups

We give an elementary proof of the result proved in [5] on theLp divergence of Fourier series on compact connected semisimple Lie groups.

Divergence of Fourier Series

AbstractThis note contains a strengthened version of the following well-known theorem: there exists a continuous function whose Fourier series diverges at a point.

A Cohen-type inequality for Jacobi expansions and divergence of Fourier series on compact symmetric spaces

A Cohen-type inequality for Jacobi expansions and divergence of Fourier series on compact symmetric spaces

On the Dini Test and Divergence of Fourier Series

On the Dini Test and Divergence of Fourier Series

Almost everywhere divergent Fourier series with respect to arbitrary bounded orthonormal system of the form {ϕ(n k x)}

Almost everywhere divergent Fourier series with respect to arbitrary bounded orthonormal system of the form {ϕ(n k x)}

Condensation of singularities and divergence results in approximation theory

Given a family A of linear continuous mappings of a topological vector space X into another topological vector space Y, the set A of singularities for A is defined as the set of all x in X for which A is an unbounded set in Y. The following general principle of condensation of singularities for nonequicontinuous families A is obtained: If either X is Hausdorff barrelled and Y is seminormable, or X is Hausdorff ultrabarrelled and Y is locally bounded, then A is an uncountably infinite dense G δ-set in X. A principle of double condensation of singularities in Banach spaces is also obtained. These principles are applied to prove the dense unbounded divergence of Fourier series, biorthogonal systems, Lagrange interpolation processes and some quadrature formulas.

On the Dini test and divergence of Fourier series

In this note we prove that no condition weaker than the Dini assures the pointwise convergence of a Fourier series in a set of positive measure.

Everywhere divergent Fourier series

Everywhere divergent Fourier series

A Cohen Type Inequality for Compact Lie Groups

The following theorem is proved: let G denote a compact connected semisimple Lie group. There exists $\theta = \theta (G)(3 \leqslant \theta < 4)$ such that, if ${\chi _1}, \ldots ,{\chi _N}$ are N distinct characters of G, ${d_1}, \ldots ,{d_N}$ their dimensions, ${c_1}, \ldots ,{c_N}$ complex numbers of modulus greater than or equal to one, then, for all $p > \theta ,|||\Sigma _{j = 1}^N{c_j}{d_j}{\chi _j}||{|_p} \geqslant \text {const}_p N^{\alpha _p}$ where $||| \cdot ||{|_p}$ denotes the ${L^p}(G)$ convolutor norm and $\text {const}_p$ and ${\alpha _p} = {\alpha _p}(G)$ are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.

A Cohen type inequality for compact Lie groups

The following theorem is proved: let G denote a compact connected semisimple Lie group. There exists θ = θ ( G ) ( 3 ⩽ θ &gt; 4 ) \theta = \theta (G)(3 \leqslant \theta &gt; 4) such that, if χ 1 , … , χ N {\chi _1}, \ldots ,{\chi _N} are N distinct characters of G, d 1 , … , d N {d_1}, \ldots ,{d_N} their dimensions, c 1 , … , c N {c_1}, \ldots ,{c_N} complex numbers of modulus greater than or equal to one, then, for all p &gt; θ , | | | Σ j = 1 N c j d j χ j | | | p ⩾ const p N α p p &gt; \theta ,|||\Sigma _{j = 1}^N{c_j}{d_j}{\chi _j}||{|_p} \geqslant \text {const}_p N^{\alpha _p} where | | | ⋅ | | | p ||| \cdot ||{|_p} denotes the L p ( G ) {L^p}(G) convolutor norm and const p \text {const}_p and α p = α p ( G ) {\alpha _p} = {\alpha _p}(G) are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.