Degree of approximation of functions by their fourier series in the generalized Hölder metric

The paper studies the degree of approximation of functions by matrix means of their Fourier series in the generalized Hölder metric, generalizing many known results in the literature     

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

     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas)

Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ _n ^(α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w ^(a,b) σ _n ^(α,β) ‖_∞ ≦ C‖w ^(a,b) f‖_∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.      

Cesàro α-Integrability and Laws of Large Numbers-II

For a sequence of random variables, a new set of properties called Cesàro α-Integrability and Strong Cesàro α-Integrability was recently introduced in an earlier paper and these properties were used to prove several new laws of large numbers, namely both Strong and Weak Laws of Large Numbers for pairwise-independent random variables as well as WLLN for some dependent sequences of random variables. In this paper, a set of weaker conditions called Residual Cesàro α-Integrability and Strong Residual Cesàro α-Integrability are introduced and significant improvements over earlier results are obtained. In addition, new results on L _p -convergence, for 0 < p < 2, and SLLN for some dependent sequences are proved.      

关于Szász算子的强逆不等式

In this paper,we obtain the strong converse inequality for Szász operators with K-functional by introducing a new K-functional of the form Kαλ(f,t2) = infg∈C2λ{‖f-g‖0 t2‖g‖2}(0≤λ≤1,0＜α＜2),where ‖·‖0,‖·‖2,C2λ are defined in the paper.As for its applications,we have extended some results before this paper.     

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.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .