Cesaro Method Research Articles

Some Bounded Linear and Compact Operators into the Generalised Hahn Space

We characterise some classes of bounded linear and compact operators from the spaces of sequences that are strongly summable or strongly summable to zero with index p ≥ 1 by the Cesaro method of order 1 into the generalised Hahn space. Furthermore, we establish estimates for the norm and the Hausdorff measure of noncompactness of those operators. Our results are complementary to recently published dual corresponding results with the order of the sequence spaces reversed. We also apply our results to the graphical representations of related potential surfaces and the corresponding Wulff’s crystals.

SOME SUMMABILITY TECHNIQUES IN INFINITE SERIES & SEQUENCE

The study of summability techniques plays a vital role in the fields of Functional Analysis, Fourier series, and Engineering. This paper is mainly focused on the comparison between Cesaro summability and Able summability. Besides providing a theorem, and giving examples related to Able method and Cesaro method, we give a theorem to prove the completeness of a sequence space.

On The Summability Of Fourier Series By The Generalized Cesaro Method

The analogous of Lebesgue-Gergen convergence test for generalized Ces´aro means of Fourier trigonometric series is given.

ГЕОДИНАМІКА

Purpose. To investigate the features of the algorithm implementation for finding the derivatives of the spatial distribution function of the planet's masses with the use of high-order Stokes constants and, on the basis of this, to find its analytical expression. According to the given methodology, to carry out calculations with the help of which to carry on the study of dynamic phenomena occurring inside an ellipsoidal planet. The proposed method involves the determination of the derivatives of the mass distribution function by the sum, the coefficients of which are obtained from the system of equations, which is incorrect. In order to solve it, an error-resistant method for calculating unknowns was used. The implementation of the construction is carried out in an iterative way, while for the initial approximation we take the three-dimensional function of the density of the Earth's masses, built according to Stokes constants up to the second order inclusive, by dynamic compression by the one-dimensional density distribution, and we determine the expansion coefficients of the derivatives of the function in the variables to the third order inclusive. They are followed by the corresponding density function, which is then taken as the initial one. The process is repeated until the specified order of approximation is reached. To obtain a stable result, we use the Cesaro summation method (method of means).. The calculations performed with the help of programs that implement the given algorithm, while the achieved high (ninth) order of obtaining the terms of the sum of calculations. The studies of the convergence of the sum of the series have been carried out, and on this basis, a conclusion has been made about the advisability of using the generalized finding of the sums based on the Cesaro method. The optimal number of contents of the sum terms has been chosen, provides convergence both for the mass distribution function and for its derivatives. Calculations of the deviations of mass distribution from the mean value ("inhomogeneities") for extreme points of the earth's geoid, which basically show the total compensation along the radius of the Earth, have been performed. For such three-dimensional distributions, calculations were performed and schematic maps were constructed according to the taken into account values of deviations of three-dimensional distributions of the mean ("inhomogeneities") at different depths reflecting the general structure of the Earth's internal structure. The presented vector diagrams of the horizontal components of the density gradient at characteristic depths (2891 km - core-mantle, 700 km - middle of the mantle, also the upper mantle - 200, 100 km) allow us to draw preliminary conclusions about the global movement of masses. At the same time, a closed loop is observed on the “core-mantle” edge, which is an analogy of a closed electric circuit. For shallower depths, differentiation of vector motions is already taking place, which gives hope for attracting these vector-grams to the study of dynamic motions inside the Earth. In fact, the vertical component (derivative with respect to the z variable) is directed towards the center of mass and confirms the main property of mass distributions - growth when approaching the center of mass. The method of stable solution of incorrect linear systems is applied, by means of which the vector-gram of the gradient of the mass distribution function is constructed. The nature of such schemes provides a tool for possible causes of mass redistribution in the middle of the planet and to identify possible factors of tectonic processes in the middle of the Earth, i.e indirectly confirms the gravitational convection of masses. The proposed technique can be used to create detailed models of density functions and its characteristics (derivatives) of the planet's interior, and the results of numerical experiments - to solve tectonics problems.

Generalised regularity and compactness of matrix operators

A well-known result by Cohen and Dunford ([2], 1937) characterises the class of all regular compact linear operators. It follows that a regular matrix transformation cannot be compact. This means that if c denotes the set of all complex sequences of complex numbers, then an infinite matrix that maps c into c and preserves the limits cannot be compact. We obtained this result in a different way applying the theory of BK spaces from functional analysis and summability, and using the Hausdorff measure of noncompactness. Furthermore, we present the extension of this result to matrix transformations between the spaces c and the spaces of strongly summable sequences by the Cesaro method of order 1, and of strongly convergent sequences. We present new unified proofs for our main results.

Inclusion theorems for (C,k) summability method of sequences of fuzzy numbers

We introduce (C,k) summability method of sequences of fuzzy numbers and compare (C,k) summability method with Abel summability method, Holder summability method and with higher order Cesaro method of sequences of fuzzy numbers. Some Tauberian conditions under which (C,k) summability of sequences of fuzzy numbers imply convergence in fuzzy number space are also obtained as corollaries.

An Alternative Proof of a Tauberian Theorem for the Weighted Mean Method of Summability

Let $$(p_n)$$ be a sequence of nonnegative numbers such that $$p_0>0$$ and $$\begin{aligned} P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty, n\rightarrow \infty . \end{aligned}$$ Let $$(u_n)$$ be a sequence of real or complex numbers. The nth weighted mean of $$(u_n)$$ is defined by $$\begin{aligned} \sigma _n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k u_k \quad (n =0,1,2,\ldots ) \end{aligned}$$ We give an alternative proof of a Tauberian theorem stating that the existence of the limit $$\lim _{n \rightarrow \infty } u_n=s$$ follows from that of $$\lim _{n \rightarrow \infty } \sigma _n=s$$ and a Tauberian condition. If $$(u_n)$$ is a sequence of real numbers, then these Tauberian conditions are one-sided. If $$(u_n)$$ is a sequence of complex numbers, these Tauberian conditions are two-sided. Significance Statement: If a sequence converges, then its weighted means converge to the same number. But, the converse of this implication is not true in general and its partial converse might be valid. This manuscript presents an alternative proof of a well-known Tauberian theorem stating that convergence of a slowly decreasing sequence (in case of sequences of real numbers) or a slowly oscillating sequence (in case of sequences of complex numbers) follows from its weighted mean summability. Corollaries of the main results in this manuscript consist of well-known Tauberian theorems for Cesaro and logarithmic summability methods.

Polynomial in momenta invariants of Hamilton's equations, divergent series and generalized functions

We consider the problem of the motion of two identical interacting particles on a circle in a potential force field. We study conditions for the existence of an additional first integral in the form of a polynomial of degree 4 in momenta with periodic coefficients. These conditions are reduced to the solvability of a certain nonlinear functional-differential equation for the potential of the external force and the interaction potential. If the potentials are smooth, this equation has only trivial solutions: at least one of the potentials is a constant function. We classify generalized solutions of this equation when the interaction potential is a distribution while the external potential is a smooth function. Formal Fourier series for the interaction potential can be summed up by the (C, 2)-Cesaro method to periodic functions with singularities.

Approximation of a generalized Lipschitz class function by Euler - Cesàro means of Fourier series

In this paper, I have taken product of two summability methods, Euler and Cesaro; and establish a new theorem on the degree of approximation of the function f belonging to W(Lp, ?(t)) classes by Euler - Cesaro method. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7190 BIBECHANA 9 (2013) 151-158

Tauberian conditions connecting Cesaro and Riesz discrete means methods

Everywhere below, if not stated otherwise, we assume that {an} n=0 is a sequence of real numbers; ∑ an is the series corresponding to it (if summation limits are not specified, we assume summation from 0 to +∞). Let Ω and Λ be summation methods for number series. The summability of a series ∑ an to a number S by the method Ω is briefly denoted as ∑ an = S(Ω). We say that the method Λ is included into the method Ω (Λ ⊂ Ω) if the equality ∑ an = S(Λ) implies ∑ an = S(Ω). We say that the methods Ω and Λ are equivalent (Ω ∼ Λ) if both the inclusions Λ ⊂ Ω and Ω ⊂ Λ are valid. We say that the method Ω is stronger than the method Λ if Λ ⊂ Ω, but the methods Ω and Λ are not equivalent. For Ω and Λ we consider the summation method of Cesaro (C,α) and methods of summation by Riesz means denoted by (Rd, α), where α > −1. Definitions and basic properties of Cesaro methods can be found in [1, § 5.4–5.7]; Riesz methods were described in [2, 3]. Present here the corresponding definitions in the form convenient for further use. Definition 1. A series ∑ an is said to be summable by the Cesaro method of order α to a number S if

Interconnection of Cesaro methods and discrete Riesz means

Interconnection of Cesaro methods and discrete Riesz means

Continuity in Λ-variation and summation of multiple Fourier series by Cesàro methods

We construct new examples of functions of bounded Λ-variation not continuous in Λ-variation. Using these examples, we show that, in the problem of the summability of multiple Fourier series by the Cesaro method of negative order, the condition of continuity in Λ-variation, is essential in contrast to the one-dimensional case.

Regular methods of summability on tree-sequences in Banach spaces

Suppose that X is a Banach space, 〈a ij 〉 is a regular method of summability and (x s ) s∈S is a bounded sequence in X indexed by the dyadic tree S. We prove that there exists a subtree S' ⊆ S such that: either (a) for any chain β of S' the sequence (x s ) s∈β is summable with respect to 〈a ij 〉 or (b) for any chain β of S' the sequence (x s ) s∈β is not summable with respect to 〈a ij 〉. Moreover, in case (a) we prove the existence of a subtree T ⊆ S' such that if β is any chain of T, then all the subsequences of (x s ) s∈β are summable to the same limit. In case (b), provided that 〈a ij 〉 is the Cesaro method of summability and that for any chain β of S' the sequence (x s ) s∈β is weakly null, we prove the existence of a subtree T C S' such that for any chain β of T any spreading model for the sequence (x s ) s∈β has a basis equivalent to the usual l 1 -basis. Finally, we generalize the theory of spreading models to tree-sequences. This also allows us to improve the result of case (b).

A family of generalized Nörlund methods and related power series methods applied to double sequences

Inclusion, convexity and Tauberian theorems are proved for certain generalized Norlund methods and power series methods applied to double sequences. Families of summability methods are developed which form a hierarchy of methods and can be used to connect matrix and power series methods. This extends various results proved in the papers [15], [9], [17] and [14] for generalized Norlund summmability methods and power series methods applied to ordinary sequences (sn) to summability of double sequences (smn) in the sense of bounded Pringsheim convergence. As examples some special methods like Cesaro methods and extensions as generalized Cesaro methods and quasi Cesaro methods are considered (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the mean summability by Cesaro method of Fourier trigonometric series in two-weighted setting

The Cesaro summability of trigonometric Fourier series is investigated in the weighted Lebesgue spaces in a two-weight case, for one and two dimensions. These results are applied to the prove of two-weighted Bernstein's inequalities for trigonometric polynomials of one and two variables.

A New Method of Summation of Fourier Series Converging in C-Norm

A new method is based on some properties of a semigroup of fractional powers of some positive operator in Banach space. This method uses an essentially smaller deformation of an ordinary partial sum of the Fourier series than in Cesaro's method or in its generalizations.

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 Cesaro methods of positive order. The conditions imposed on the coefficients are of classical type.

SUMMATION OF DIVERGENT SERIES AND ERGODIC THEOREMS

In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.

Tauberian Type Conditions Connecting Cesàro and Riesz Summability Methods of Different Orders

In the paper conditions are studied which ensure the equivalence of two given summability methods on the class of numerical series satisfying these conditions. One of these methods is the regular Cesaro method (C,w), which replaces the role of the convergence (C,0) the latter occurring in the classical Tauberian theorems. The other summability method considered is either the summability method (Rd,α) by discrete Riesz means or the Cesaro method with (Rd,α) with α > w. New Tauberian type conditions are obtained under which the corresponding Tauberian theorems are proved.

Tauberian type gap conditions for Cesàro summation methods

In this paper, we study gap conditions ensuring, for the class of series satisfying them, that the Cesaro methods (C, k) and (C, α) with fixedk and α (0<k<α) are equivalent. We obtain a sufficient condition in which gaps of the corresponding sizes separate uniformly bounded sets ofk+1 nonzero elements rather than neighboring nonzero elements (as in the classical case ofk=0).