A generalization of the Riesz–Fischer theorem and linear summability methods (original) (raw)

Orlicz–Pettis Theorem through Summability Methods

Mathematics

This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect to an ideal, and other variations of summability methods.

Approximation of functions by linear summation methods in the Orlicz type spaces

Ukrainian Mathematical Bulletin, 2020

Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces S_M. In particular, in terms of approximations by such methods, constructive characteristics are obtained for classes of functions whose moduli of smoothness do not exceed a certain majorant.

A few remarks on Riesz summability of orthogonal series

Proceedings of the American Mathematical Society, 1991

We study convergence behavior of some sequences and series related to a given orthogonal series. Following the developed technique we define (in terms of fourth mixed moments only) a class of orthonormal functions {X¡}¡>1 such that the condition: 3k e N D,>i^,(ml ' ') < °° implies almost everywhere convergence of the series £,>i /*/^¡ > here for every i = 1, 2, ... ,j = 1, ... ,k, ln(1) ; = ln2 ;, ln(;) i = ln2(max(l, ln(;_l) «')).

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

2021

Let N be the set of positive integers, R and C also be the real and complex fields as usual, respectively. By ω, we denote the space of all real (or complex) valued sequences and any vector subspace of ω is also called as a sequence space. A K space is a locally convex sequence space X containing φ on which coordinate functionals πk(x) = xk are continuous for every k ∈ N, the set of positive integers, where φ is the space of finitely non-zero sequences spanned by the set {e : k ∈ N}. e is the sequence whose only non-zero term is 1 in the k place for all k ∈ N, and e is also the sequence with e = (1, 1, ...). A complete linear metric (or complete normed) K space is called an FK (or a BK) space. Let X ⊃ φ be a BK space and x = (xk) ∈ X. Then, by x [n] = ∑n k=1 xke k for all n ∈ N, we denote the n section of x. It is said that X ⊃ φ is an AK space if ∥

Regular methods of summability and the Banach-Saks property for double sequences

Filomat, 2021

ABanach space B is said to satisfy the Banach-Saks property with respect to a regular summability method if every bounded subsequence has a summable subsequence. We show that if a Banach space satisfies the Banach-Saks property with respect to a Robison-Hamilton regular summability method, for every bounded double sequence there exists a ?-subsequence whose subsequences are all summable to the same limit.

On translativity of the product of Riesz Norlund summability methods

Applied Mathematical Sciences, 2013

Necessary and Sufficient conditions for the product of Riesz Nörlund summability methods () (N, q) to be translative have been established. Two interesting examples will be given to show in the first that the product () (N, q) is translative and in the other example to show that () (N, q) need not be translative.

A note on regular methods of summability and the Banach-Saks property

Proceedings of the American Mathematical Society, 1976

Using the Galvin-Prikry partition theorem from set theory it is proved that every bounded sequence in a Banach space has a subsequence such that either every subsequence of which is summable or no subsequence of which is summable.