For weighted $L^1$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in $h$-harmonics. The result applies to various methods of summability, including the de la Vallée Poussin means and the Cesàro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of $\RR^d$.
We consider an aspect of the open problem: Does every square-integrable function on SU(2) have an almost everywhere convergent Fourier series? Let 0 < alpha < 1. We show that to each countable set E in SU(2) there corresponds an alpha-Holder continuous function on SU(2) whose Fourier series diverges on E. We also show that the Fourier series of each alpha-Holder continuous function on SU(2) converges almost everywhere.
Let $(X,\mathcal{B},m,τ)$ be a dynamical system with $\ds (X,\mathcal{B},m)$ a probability space and $\ds τ$ an invertible, measure preserving transformation. The present paper deals with the almost everywhere convergence in $\ds{L}^1(X)$ of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures $\ds\{ν_i\}$ defined on $\ds\mathbb{Z}$. We then exhibi…
This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach extends to, among other cases, functions that are defined on $\mathbb{T}^d$, which allows us to establish an analogue of Zygmund's theorem in higher dimensions.
We introduce a maximal operator for the natural analogue of the disc multiplier in the framework of Jacobi analysis and determine the mapping properties of said maximal operator, including weak type endpoint results. In particular we obtain sharp information on the almost everywhere convergence of the inverse Jacobi transform.
In this paper we investigate the principle of the generalised localisation for spectral expansions of the polyharmonic operator, which coincides with the multiple Fourier integrals summed over the domains corresponding to the surface levels of the polyharmonic polynomials. It is proved that the partial sums of the multiple Fourier integrals of a function f\inL_2 (R^N ) converge to zero almost-everywhere on R^N\supp(f).
We consider the open problem: Does every square-integrable function f on a compact, connected Lie group G have an almost everywhere convergent Fourier series? We prove a general theorem from which it follows that if the integral modulus of continuity of f is O(t^a) for some a > 0 then the Fourier series of f converges almost everywhere on G. In particular, the Fourier series of any Holder continuous function of degree a > 0 on G converges almost everywhere. On the other hand, we show that to each countable subset E of G = SU(2) and each 0 < a < 1 there corresponds an Holder continuous function…
Let $\mathcal L$ denote the twisted Laplacian in $\mathbb C^d$. We study almost everywhere convergence of the Bochner--Riesz mean $S^δ_{t}(\mathcal L) f$ of $f\in L^p(\mathbb C^d)$ as $t\to \infty$, which is an expansion of $f$ in the special Hermite functions. For $2\le p\le \infty$, we obtain the sharp range of the summability indices $δ$ for which the convergence of $S^δ_{t}(\mathcal L) f$ holds for all $f\in L^p(\mathbb C^d)$.
In this paper, we investigate very general approximation kernels with special properties, called an approximate identity, and prove almost everywhere and norm convergence of these general methods, which consists of a class of summability methods and provide norm and a.e. convergence of these summability methods with respect to the trigonometric system. Investigations of these summations can be used to obtain norm convergence of Fejér means with respect to the Vilenkin system also, but these methods are not useful to study a.e. convergence in this case, because of some special properties of the…
We generalize a theorem of Bellow and Calderón concerning the a.e. convergence of the convolution powers $\ds μ^nf(x)=\sum_{k}μ^n(k)f(T^k x)$ where $T$ is a measure preserving transformation of a probability space and $μ$ is a probability measure on the integers.
In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence $σ_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n-1}S_{k, n-k}f\to f$ ($n\to\infty$) for each integrable two-variable function $f$.
In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to be bounded in $L^p$ for any $1\leq p<\infty$.
We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit $n$-dimensional cube onto itself, whose Jacobian is equal to $-1$ a.e. Moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.
We study a.e. convergence on $L^p$, and Lorentz spaces $L^{p,q}$, $p>\tfrac{2d}{d-1}$, for variants of Riesz means at the critical index $d(\tfrac 12-\tfrac 1p)-\tfrac12$. We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on $L^2$ spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted $L^2$ spaces, and a sharp endpoint bound for Stein's square-function associated with the Riesz means.
In this paper, we will first show that the maximal operator $S_*^α$ of spherical partial sums $S_R^α$, associated to Dunkl transform on $\mathbb{R}$ is bounded on $L^p(\mathbb{R}, |x|^{2α+1} dx)$ functions when $\frac{4(α+1)}{2α+3}<p<\frac{4(α+1)}{2α+1}$, and it implies that, for every $L^p(\mathbb{R}, |x|^{2α+1} dx)$ function $f(x)$, $S_R^αf(x)$ converges to $f(x)$ almost everywhere as $R\to \infty$. On the other hand we obtain a sharp version by showing that $S_*^α$ is bounded from the Lorentz space $L^{p_i,1}(\mathbb{R}, |x|^{2α+1})$ into $L^{p_i,\infty}(\mathbb{R}, |x|^{2α+1}),\quad i=0,1$…
The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of $z$ are replaced by products of other Möbius transforms of the disc. The system is also inherently connected to the so-called nonlinear phase unwinding decomposition which has been in the center of some recent activity. We prove $L^p$ bounds for the maximal partial sum operator of the Malmquist-Takenaka series under additional assumptions on the zeros of the Möbius transforms. We locate the problem in the time-frequency setting and, in particular, we connect it to the polynomial Carleson theo…
Let $G$ be a connected, simply connected, compact semisimple Lie group of dimension $n$. It has been shown by Clerc \cite{Clerc1974} that, for any $f\in L^1(G)$, the Bochner-Riesz mean $S_R^δ(f)$ converges almost everywhere to $f$, provided $δ>(n-1)/2$. In this paper, we show that, at the critical index $δ=(n-1)/2$, there exists an $f\in L^1(G)$ such that $$\limsup_{R\rightarrow\infty} \big|S_{R}^{(n-1)/2}(f)(x)\big|=\infty, \ \text{a.e.}\ x\in G.$$ This is an analogue of a well-known result of Kolmogorov \cite{Kolmogoroff1923} for Fourier series on the circle, and a result of Stein \cite{Stei…
The following problem is studied in this paper: Which multipliers $\{λ_{k, n}\}$ ensure the convergence, as $n\to \infty$, of the linear means of the Fourier series of functions $f\in L_1[-π, π]$ $$ \sum_{k=-\infty}^\infty λ_{k, n}\hat{f}_k e^{ikx}, $$ where $\widehat{f}_k$ is the $k$-th Fourier coefficient, at a point at which the derivative of the function $\int_0^x f$ exists. A criterion for the convergence of the $(C, 1)$-means ($λ_{k, n}=(1-\frac {|k|}{n+1})_+$) is found, while in the general case $λ_{k, n}=ϕ(\frac {k}{n+1})$ a sufficient condition is derived for the convergence at all su…
We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Cesàro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence theorems, we also obtain the corresponding results for Riesz and Bochner-Riesz means. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres.
We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.
The weighted Lebesgue spaces of initial data for which almost everywhere convergence of the heat equation holds was only very recently characterized. In this note we show that the same weighted space of initial data is optimal for the heat--diffusion parabolic equations involving the harmonic oscillator and the Ornstein--Uhlenbeck operator.
Let $Ω\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(Ω,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For $p\leq n-1$ we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and a topological image of a point is a continuum for every point in a set of positive measure in the domain.
Unlike the classical theory of Fourier series which deals with decomposition of a function into sinusoidal waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series but there are a lot of differences also. The aim of my master thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate Nörlund means but only in the case when their coefficients are monotone and p…
A uniformly bounded complete orthonormal system of functions $Θ=\{ θ_n\}_{n=1}^{\infty},$ $ \|θ_n\|_{L^\infty_{[0,1]} } \leq M $ is constructed such that $\sum_{n=1}^{\infty} a_{n}θ_{n}$ converges almost everywhere on $[0,1]$ if $\{ a_n\}_{n=1}^{\infty} \in \, l^2$ and $\sum_{n=1}^{\infty} a_{n}θ_{n}$ diverges a. e. for any $\{ a_n\}_{n=1}^{\infty} \not\in \, l^2$. Thus Menshov's theorem on the representation of measurable, almost everywhere finite, functions by almost everywhere convergent trigonometric series cannot be extended to the class of uniformly bounded complete orthonormal systems.
Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure
School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）；School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation $(\frac{1}{n}\sum\limits_{m=0}^{n-1}\left\vert S_{mm}f - f \right\vert^{p})^{1/p}\rightarrow 0$ for every two-dimensional functions belonging to $L\log L$ and $0<p\le 2$. From the theorem of Getsadze \cite{Gets} it follows that the space $L\log L$ can not be enlarged with preserving this strong summability property.
If a function $f$, acting on a Euclidean space $\mathbb{R}^n$, is "almost" orthogonally additive in the sense that $f(x+y)=f(x)+f(y)$ for all $(x,y)\in\bot\setminus Z$, where $Z$ is a "negligible" subset of the $(2n-1)$-dimensional manifold $\bot\subset\mathbb{R}^{2n}$, then $f$ coincides almost everywhere with some orthogonally additive mapping.
Kostyrko and Salat showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. We give a topological analogue of this theorem and provide some examples.
Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems established by Rodin in 1990. We prove, that if the growth of a function $Φ(t):[0,\infty)\to[0,\infty)$ is bigger than the exponent, then the strong $Φ$-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.
In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley-Wiener spaces $PW_c^p\subset L^p(\mathcal{R})$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley-Wiener like spaces $B_{α,c}^p\subset L^p(0,\infty)$ of functions whose Hankel transform $\mathcal{H}^α$ is supported in $[0,c]$.As a side product, we show the continuity of the projection oper…
Let $g(x)=χ_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then there does not exist $S \subset {\Bbb R}^{2d}$ such that ${ \{g(x-a)e^{2 πi x \cdot b} \}}_{(a,b) \in S}$ is an orthonormal basis for $L^2({\Bbb R}^d)$.
Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} λdE_{ L}(λ)$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R dE_{ L}(λ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$ for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domai…
In 1927 Littlewood constructed an example of bounded holomorphic function on the unit disk, which diverges almost everywhere along rotated copies of any given curve in the unit disk ending tangentially to the boundary. This theorem was the complement of a positive theorem of Fatou 1906, establishing almost everywhere nontangential convergence of bounded holomorphic functions. There are several generalizations of the Littlewood's theorem which proofs are based on the specific properties of Poisson kernel. We generalize Littlewood's theorem for operators having general kernels.
This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizsäker concerning almost everywhere convergence properties of series of the form $\sum_{n=1}^{\infty}f(nx)$. A more general, additive version of this problem is the following: Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is of type 1 if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the …
We define $λ(r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels $\{φ_r\}$, forming an approximation of identity. We prove that the bound \md0 \limsup_{r\to 1}λ(r) \|φ_r\|_\infty<\infty \emd is necessary and sufficient for almost everywhere $λ(r)$-convergence of the integrals \md0 \int_\ZT φ_r(t-x)dμ(t). \emd
Inspired by Menshov's representation theorem, we prove that there exists a sequence of frequecies such that any measurable (complex valued) function on R can be represented as a sum of almost everywhere convergent trigonometric series with these frequencies.
We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for…
We show that L^2-bounded singular integral in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.
We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if $f$ is locally distributionally integrable over the real line and $ψ\in\mathcal{D}(\mathbb{R}%) $ is a test function, then $fψ$ is distributionally integrable, and the formula% [<\mathsf{f},ψ> =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) ψ(x) \,\mathrm{d}% x\,,] defines a distr…
Every measurable function f on the circle can be represented as a sum of harmonics with positive spectrum, converging in measure. For convergence almost everywhere this is not true. We discuss several other subsets of Z for which one might get a Menshov type representation converging almost everywhere or in measure.
Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence gen…
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type, where the Lebesgue measure is replaced by an arbitrary measure $μ$. In particular we show that the differentiability properties of Lipschitz functions at $μ$-almost every point are related to the decompositions of $μ$ in terms of rectifiable one-dimensional measures. As a consequence we obtain a differentiability result for Lipschitz functions with respect …
