Ergodic properties of max-infinitely divisible processes

We prove that a stationary max-infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesàro summable to 0). These criteria are applied to some classes of max-infinitely divisible processes.     

Complete convergence and Cesàro summation for i.i.d. random variables

Summary Various results generalizing summation methods for divergent series of real numbers to analogous results for independent, identically distributed random variables have appeared during the last two decades. The main result of this paper provides necessary and sufficient conditions for the complete convergence of the Cesàro means of i.i.d random variables.     

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.     

Uniqueness for Cesàro summable double Walsh series

We prove a Tauberian theorem for Walsh series of two variables, and use it to obtain several results about uniqueness of Cesàro summable double Walsh series. Namely, we show that up to sets of measure zero, Cesàro summability of double Walsh series is the same as convergence of the square dyadic partial sums and, under a suitable growth condition, that uniqueness holds for Cesàro summable double Walsh series.     

Convergence rate of spherical harmonic expansions of smooth functions

We extend a well-known result of Bonami and Clerc on the almost everywhere (a.e.) convergence of Cesàro means of spherical harmonic expansions. For smooth functions measured in terms of φ-derivatives on the unit sphere, we obtained the sharp a.e. convergence rate of Cesàro means of their spherical harmonic expansions.     

Cesàro operators on Hardy spaces in the unit ball

This article establishes the boundedness of the generalized Cesàro operator on holomorphic Hardy spaces in the unit ball. The approach consists in writing the generalized Cesàro operator as a composition of certain integral operators.     

Rate of Cesàro summability of double orthogonal series

Unimprovable estimates (in the class of double orthogonal series) are obtained on the rate of almost everywhere summability of orthogonal expansions of square-integrable functions by Cesàro methods of positive order. The conditions imposed on the coefficients are of classical type.     

On the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its fourier series

In a recent paper Lal and Yadov [4] obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using the product of the Cesàro and Euler means of order one of its Fourier series. In this paper we extend this result to any regular Hausdorff matrix for the same class of functions.     

Cesàro wedge and weak Cesàro wedge FK-spaces

In this paper we deal with Cesàro wedge and weak Cesàro wedge FK-spaces, and give several characterizations. Some applications of these spaces to general summability domains are also studied.     

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.     

Cesàro α-Integrability and Laws of Large Numbers I

For a sequence of integrable random variables, we introduce a new set of conditions called Cesàro -Integrability and Strong Cesàro -Integrability and show that, for <1/2, these conditions that are strictly weaker than Cesàro Uniform Integrability and Strong Cesàro Uniform Integrability respectively, are sufficient for WLLN and SLLN to hold for a sequence of pairwise independent random variables. For some special kinds of dependent sequences of random variables also, Cesàro -integrability for appropriate is shown to be sufficient for WLLN to hold.     

Comparison of certain summability methods by speeds of convergence

We deal with families of summability methods which depend on a continuous parameter and where two different methods are connected either by a Cesàro-type or Euler–Knopp-type method. Extending and applying some results of [14], we are able to compare speeds of convergence in families of generalized Nörlund methods and to give certain Tauberian remainder theorems. Particular cases are the families of Cesàro, generalized Cesàro and Euler–Knopp methods.     

On the Approximation of Bounded Functions with Discontinuities of the First Kind by Generalized Shepard Operators

The authors investigate the approximation of bounded functions with discontinuities of the first kind by generalized Shepard operators. Estimate for the rate of convergence at a continuity point is obtained, while at the points of discontinuity it is shown that the sequence of generalized Shepard operators is almost always divergent. It is also shown that the sequence of Cesàro means of generalized Shepard operators is convergent everywhere for bounded functions which have only discontinuities of the first kind in [0,1].     

Resolvent estimates and decomposable extensions of generalized Cesàro operators

We determine the spectrum of generalized Cesàro operators with essentially rational symbols acting on various spaces of analytic functions, including Hardy spaces, weighted Bergman and Dirichlet spaces. Then we show that in all cases these operators are subdecomposable.     

Certain topological properties and duals of the domain of a triangle matrix in a sequence space

The matrix domain of the particular limitation methods Cesàro, Riesz, difference, summation and Euler were studied by several authors. In the present paper, certain topological properties and β- and γ-duals of the domain of a triangle matrix in a sequence space have been examined as an application of the characterization of the related matrix classes.     

On Sign Distribution in Relatively Convergent Series

We study conditions of convergence of series with alternating signs. This problem has already attracted attention of many authors; among the most prominent let us mention Cesàro who first discovered even distributionof positive and negative terms in such series. This observation has been rediscovered and reproved by many authors. One of the goals of this paper is to show that Cesàro's result is indeed the best possible. In order to achieve this we introduce several quite sophisticated constructions of series with desired properties. The other goal is to get an interesting generalization of Cesàro's result by introducing the notion of logarithmic density (and then further generalizing it).     

Another look at Cesàro sequence spaces

We consider the Cesàro sequence space cesp as a closed subspace of the infinite ?p-sum of finite dimensional spaces. We easily obtain many known results, for example, cesp has property (β) of Rolewicz, uniform Opial property, and weak uniform normal structure. We also consider some generalized Cesàro sequence spaces. Finally, we compute the von Neumann-Jordan and James constants of the two-dimensional Cesàro sequence space when 1<p?2.     

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.     

On Fourier–Haar coefficients of the function from the Marcinkiewicz space

The main objective of the paper is to describe asymptotic behaviour of Fourier–Haar coefficients of functions from Marcinkiewicz spaces. We also discuss the Cesàro summability and the almost convergence of sequences related to Fourier–Haar coefficients and generalise a result from [17]. Some analogues of Mercer's theorem for Fourier coefficients are proved.     

Some inclusion theorems for absolute summability

In this work the inclusion relations between absolute summability domains of a normal matrix A and certain factorable matrices are described. Thus, some classes of factorable matrices transforming the absolute summability domain of A into a set of convergent or absolutely convergent series are characterized. As an application, the special case where A is the Cesàro matrix is considered.