## Comparison of speeds of convergence in some families of summability methods for functions

#### Citation

Sheletski, A. (2009). Comparison of speeds of convergence in some families of summability methods for functions. Maltepe Üniversitesi. s. 105.#### Abstract

Speeds of convergence in certain family of summability methods for functions are compared in the talk. The results
introduced here extend the results proved in [1] for ”matrix case” to ”integral case” and are partly published in [2].
1. Let us denote by X the set of all functions x = x(u) defined for u ≥ 0, bounded and measurable by Lebesgue on
every finite interval [0, u0]. Suppose that A is a transformation of functions x = x(u) (or, in particular, of sequences
x = (xn)) into functions Ax = y = y(u) ∈ X. If the limit limu→∞ y(u) = s exists then we say that x = x(u) is
convergent to s with respect to the summability method A, and write x(u) → s(A).
One of the basic notions in our talk is the notion of speed of convergence. Let λ = λ(u) be a positive function from X
such that λ(u) → ∞ as u → ∞. We say that a function x = x(u) is convergent to s with speed λ if the finite limit
limu→∞ λ(u) [x(u) − s] exists. We say that x is convergent with speed λ with respect to the summability method A if
the function Ax = y = y(u) ∈ X is convergent with speed λ.
2. We discuss a Riesz-type family {Aα} of summability methods Aα where α > α0 and α0 is some fixed number and
which transform functions x = x(u) into functions Aαx = yα(u). This family is defined with the help of relation Aβ =
Cγ,β ◦ Aγ (β > γ > α0), where Cγ,β is certain integral transformation (see e.g. [2]). For example, the Riesz methods
(R, α) and certain generalized N¨orlund methods (N, pα(u), q(u)) form Riesz-type families.
It is important to be able to compare the speed of convergence of x = x(u) with respect to different methods in family
{Aα} For a given speed λ = λ(u) and a fixed number γ > α0 the speeds λβ = λβ(u) and λδ = λδ(u) can be found (see
[2]) such that for all β > δ > γ the next implications are true:
λ(u) [yγ (u) − s] → t =⇒ λβ(u) [yβ(u) − s] → t,
λ(u) [yγ (u) − s] = O(1), λβ(u) [yβ(u) − s] → t =⇒ λδ(u) [yδ(u) − s] → t.

#### Source

International Conference of Mathematical Sciences#### URI

https://www.maltepe.edu.tr/Content/Media/CkEditor/03012019014112056-AbstractBookICMS2009Istanbul.pdf#page=331https://hdl.handle.net/20.500.12415/5798

#### Collections

- Makale Koleksiyonu [586]

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