Convergence rate of Cesàro means of Fourier-Laplace series

The convergence rate of Fourier-Laplace series in logarithmic subclasses of L²(Σd) defined in terms of moduli of continuity is of interest. Lin and Wang [C. Lin, K. Wang, Convergence rate of Fourier-Laplace series of L²-functions, J. Approx. Theory 128 (2004) 103-114] recently presented a characterization of those subclasses and provided the almost everywhere convergence rates of Fourier-Laplace series in those subclasses. In this note, the almost everywhere convergence rates of the Cesàro means for Fourier-Laplace series of the logarithmic subclasses are obtained. The strong approximation order of the Cesàro means and the partial summation operators are also presented.     

Divergence of Cesàro means of spherical h-harmonic expansions

A divergence result for Cesàro means of spherical h-harmonics expansions with a product weight is proved in this short note.     

Iterates of a product of conditional expectation operators

Let (Ω,F,μ) be a probability space and let T=P₁P₂?Pd be a finite product of conditional expectations with respect to the sub σ-algebras F₁,F₂,…,Fd. We show that for every f∈Lp(μ), 1<p?2, the sequence {Tnf} converges μ-a.e., with

     

A condition under which slow oscillation of a sequence follows from Cesàro summability of its generator sequence

In this paper we retrieve slow oscillation of a real sequence from Cesàro summability of its generator sequence under some one-sided condition.     

On the James constant and B-convexity of Cesàro and Cesàro-Orlicz sequence spaces

The classical James constant and the nth James constants, which are measure of B-convexity for the Cesàro sequence spaces cesp and the Cesàro-Orlicz sequence spaces cesM, are calculated. These investigations show that cesp,cesM are not uniformly non-square and even they are not B-convex. Therefore the classical Cesàro sequence spaces cesp are natural examples of reflexive spaces which are not B-convex. Moreover, the James constant for the two-dimensional Cesàro space is calculated.     

Convergence rate of Fourier–Laplace series of L-functions

The almost everywhere convergence rates of Fourier–Laplace series are given for functions in certain subclasses of L²(Σ_n−1) defined in terms of moduli of continuity.     

Strong Convergence of Spherical Harmonic Expansions on H1(Sd-1)

Let $\sigma_k^\delta$ denote the Cesaro means of order $\delta > -1$ of the spherical harmonic expansions on the unit sphere $S^{d-1}$, and let $E_j(f, H^1)$ denote the best approximation of $f$ in the Hardy space $H^1(S^{d-1})$ by spherical polynomials of degree at most $j$. It is known that $\lambda:= (d-2)/2$ is the critical index for the summability of the Cesaro means on $H^1(S^{d-1})$. The main result of this paper states that, for $ f\in H^1(S^{d-1})$, $$\sum_{j=0}^N \f 1{j+1} \|\sigma_j^\lambda (f) -f\|_{H^1}\approx \sum_{j=0}^N \f 1{j+1} E_j(f, H^1),$$ where “$\approx$” means that the ratio of both sides lies between two positive constants independent of $f$ and $N$.     

On Cesàro means, Kaplan classes and a conjecture of S.P. Robinson

We establish special cases of a conjecture of S.P. Robinson [S.P. Robinson, Approximate identities for certain dual classes, DPhil thesis, University of York, UK, 1996] concerning Cesàro means of certain classes of analytic functions in the unit disk. This has applications, for instance, to the so-called Kaplan classes and subordination under ‘linearly accessible’ functions.     

A Komlós Theorem for abstract Banach lattices of measurable functions

Consider a Banach function space X(μ) of (classes of) locally integrable functions over a σ-finite measure space (Ω,Σ,μ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L¹[0,1] holds also in these spaces; i.e. for every bounded sequence n(fn) in X(μ), there exists a subsequence k(fnk) and a function f∈X(μ) such that for any further subsequence j(hj) of k(fnk), the series converges μ-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions — spaces L¹(ν) of integrable functions with respect to a vector measure ν on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L¹(ν) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.     

向径函数上的球面平均及其点态收敛性

设球面平均函数为,则当f∈LP(Rn)是向径函数, n≥3,1≤P≤n/n-1时,几乎处处成立.     

Convergence of series of dependent φ-subgaussian random variables

The almost sure convergence of weighted sums of φ-subgaussian m-acceptable random variables is investigated. As corollaries, the main results are applied to the case of negatively dependent and m-dependent subgaussian random variables. Finally, an application to random Fourier series is presented.     

On summability of eigenfunction expansions of piecewise smooth functions

We consider two forms of eigenfunction expansions associated with an arbitrary elliptic differential operator with constant coefficients and order m, that is the multiple Fourier series and integrals. For the multiple Fourier integrals, we prove the convergence of the Riesz means of order s?>?(N???3)/2 of piecewise smooth functions of N?≥?2 variables. The same result is proved in the case of the N?≥?3 dimensional multiple Fourier series.     

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

For a weighted L¹ space on the unit sphere of R^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 R^d.     

Ratio limit theorems and Tauberian theorems for vector-valued functions and sequences

We prove ratio limit theorems for (C,γ)-means (γ?0) and Abel means of functions and sequences in Banach spaces, and ratio Tauberian theorems for (C,γ)-means (γ?1) and Abel means of functions and sequences in Banach lattices.     

A random approximation of set valued càdlàg functions

In this paper we study the problem of estimating a càdlàg function f whose values are compact convex sets. For this purpose a random selection of points in the interval [0,1] is considered and for each selected point x, a random sample in f(x). On the basis of this a sequence of approximants f_n,m is constructed (where n and m are the respective sample sizes). Under general conditions, rates of convergence are obtained for Skorokhod's J₁ topology, and in case of continuity of the estimated function also for the uniform one.     

The role of restriction theorems in harmonic analysis on harmonic NA groups

We shall obtain inequalities for Fourier transform via moduli of continuity on NA groups. These results in particular settle the conjecture posed in a recent paper by W.O. Bray and M. Pinsky in the context of noncompact rank one symmetric spaces. These problems naturally demand versions of Fourier restriction theorem on these spaces which we shall prove. We shall also elaborate on the connection between the restriction theorem and the Kunze-Stein phenomena on NA groups. For noncompact Riemannian symmetric spaces of rank one analogues of all the results follow the same way.     

On rubin's harmonic analysis and its related positive definite functions

In this paper, a new formulation of the Rubin's q-translation is given, which leads to a reliable q-harmonic analysis. Next, related q-positive definite functions are introduced and studied, and a Bochner's theorem is proved.