Two-weight norm inequalities for Cesàro means of Laguerre expansions

Two-weight norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted' cases, by including all values of for all positive orders of the Cesàro summation and all values of the Laguerre parameter -1$">. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.

     

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.     

Two-weight norm inequalities for the Cesàro means of Hermite expansions

An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted' cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.

     

Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

L 1-Convergence for Weighted Sums of Some Dependent Random Variables

In this article, the authors discuss the L ₁-convergence for weighted sums of some dependent random variables under the condition of h-integrability with respect to an array of weights. The dependence structure of the random variables includes pairwise lower case negative dependence and conditions on the mixing coefficient, the maximal correlation coefficient, or the ρ*-mixing coefficient. They prove that all the weighted sums have similar limiting behaviour.     

The equivalence of generalized Hölder and Cesáro matrices

We show that the generalized Hölder and Cesáro matrices of order α > −1 are equivalent. We also show that the corresponding is true for doubly infinite generalized Hölder and Cesáro matrices.     

单位球上Dirichlet型空间D_q到β~p空间的加权Cesàro算子

本文研究了单位球B上Dirichlet空间Dq到βp空间的加权Cesàro算子的有界性和紧性问题.利用泛函分析多复变的方法,获得了单位球上Dirichlet型空间Dq到βp空间的加权Cesàro算子为有界算子和紧算子的充要条件.