Comparison of Speeds of Convergence in Riesz-Type Families of Summability Methods. II

Certain summability methods for functions and sequences are compared by their speeds of convergence. The authors are extending their results published in paper [9] for Riesz-type families {Aα} (α > α0) of summability methods Aα. Note that a typical Riesz-type family is the family formed by Riesz methods Aα = (R,α), α > 0. In [9] the comparative estimates for speeds of convergence for two methods Aγ and Aβ in a Riesz-type family {Aα} were proved on the base of an inclusion theorem. In the present paper these estimates are improved by comparing speeds of three methods Aγ , Aβ and Aδ on the base of a Tauberian theorem. As a result, a Tauberian remainder theorem is proved. Numerical examples given in [9] are extended to the present paper as applications of the Tauberian remainder theorem proved here.


Introduction and Basic Notions
We continue comparing speeds of convergence in Riesz-type families of summability methods started in paper [9]. In the mentioned paper any two methods in a Riesz-type family were compared by speed of convergence. In the present paper we improve our estimates comparing by speed of convergence any three methods in a Riesz-type family.
1.1. We begin our paper recalling the basic notions used in [9]. Let us consider functions x = x(u) defined for u ≥ 0, bounded and Lebesgue-measurable on every finite interval [0, u 0 ]. Let us denote the set of all such functions by X.
Suppose that A is a transformation of functions x = x(u) (or, in particular, of sequences x = (x n )) into functions Ax = y = y(u) ∈ X. If the limit lim u→∞ 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). If y = y(u) is bounded then we say that x is bounded with respect to A, and write x(u) = O(A). We denote by ωA the set of all these functions x, where the transformation A is applied, and by cA and mA the set of all functions x which are convergent and bounded with respect to the method A, respectively. The method A is said to be regular if lim u→∞ x(u) = s implies lim u→∞ y(u) = s whenever x ∈ X. Further we use the notation c 0 for the set of all functions x ∈ X having lim u→∞ x(u) = 0.
One of the most common summability method for functions x is an integral method A is defined with the help of transformation where a(u, v) is a certain function of two variables u ≥ 0 and v ≥ 0. We say also that the integral method A is defined by the function a(u, v). An example of an integral summability method is the generalized integral Nörlund method (N, P (u), Q(u)) defined with the help of transformation where P = P (u) and Q = Q(u) are non-negative functions from X such that In particular, if Q(u) = 1 and P (u) = u α−1 for u > 0 and α > 0, we get the Riesz method (R, α).
For sequences x = (x n ) we focus ourselves on certain semi-continuous summability methods A defined by transformations where a n (u) (n = 0, 1, . . .) are some functions from X. An example of a semicontinuous method is the Borel method B defined by the transformation (1.1)

1.2.
One of the basic notions in this paper is the "speed of convergence". We use here definitions based on the definitions for sequences (see [4] and [5]) and extended for functions in [8] and [12]. Let λ = λ(u) be a positive function from X such that λ(u) → ∞ as u → ∞. It is said that a function x = x(u) is convergent to s with speed λ (shortly: λ-convergent) if the finite We use the notations c λ and m λ for the sets of all λ-convergent and λbounded functions x, respectively. It is said that x is convergent or bounded with speed λ with respect to the summability method A if Ax ∈ c λ or Ax ∈ m λ , respectively.
1.3. The main subject of the paper is a Riesz-type family of summability methods ( [8,13]). Let {A α } be a family of summability methods A α where 1 α > (−) α 1 and which are defined by transformations of functions where r γ = r γ (u) and r β = r β (u) are some positive functions from X and M γ,β is a constant depending on γ and β.

Example 3. Consider the family of generalized Nörlund methods
(see [9], Example 1) and therefore it is a Riesz-type family. In particular, if

Preliminary Results
We need some results proved in [9]. 2.1. Speeds of convergence of any two methods in a Riesz-type family were compared in [9] on the base of an inclusion theorem which will be formulated as the following proposition.
be a Riesz-type family. Then we have for functions x = x(u) and numbers s and The next theorem (see [9], Theorem 1) describes how the speed of convergence changes if we go from one summability method in the family to a stronger one.
i) Then we have for functions x = x(u) and numbers s and β > γ that where the speeds are related through the formulas ii) Moreover, we have that Under restriction (2.4) the condition λ(u) → ∞ implies λ β (u) → ∞ in Theorem A (see [9], Remark 2). We note also that Theorem A can be considered as a generalization of case A) of Theorem 1 from [12], which was proved for matrix case. Certain evaluations for speed of convergence for Riesz and Nörlund matrix methods in Banach spaces were proved in recent papers [6] and [7].
Let a = a(u) and b = b(u) be two positive functions from X. If there exist positive numbers c 1 , c 2 and u 0 such that the condition is nondecreasing and condition (2.5) is satisfied with some positive c 1 and c 2 for any u > 0, then we say that a = a(u) is almost nondecreasing.
Previous result state that switching to a stronger method, the speed of convergence can not be improved but also it cannot become too much worse. This is consistent with results known for matrix methods (see e.g. [4,6,12]).

Main Results. A Tauberian Remainder Theorem
First we prove a convexity theorem.

be a Riesz-type family satisfying the condition
for all β > α > α 1 . Then we have for functions x = x(u) and numbers s and β > δ > γ > (−) α 1 that

2)
Proof. Suppose first that γ > α 1 . Without a loss of generality we may take β = γ + 1 and s = 0. Suppose that for a function x = x(u) and some value γ of the parameter, and show that for any δ such that γ < δ < γ + 1. By relation (1.3) we have that Choose some θ ∈ (1/2; 1) and divide y δ (u) into two parts: Thus we have the equality y δ (u) = I 1 (u, θ) + I 2 (u, θ). Note that I 1 (u, θ) and I 2 (u, θ) depend also on γ, δ. Integrating by parts, we get for I 1 (u, θ) the following form: and

Using conditions (3.1) and (3.3) we get
Thus we have I ′ 1 (u, θ) = o θ (1) as u → ∞. Let us show that also I ′′ we will show that the integral transformation defined by c ′ γ,δ (u, v) is a c 0 → c 0 type transformation. We use Theorem 6 from [3] which gives the sufficient conditions for the regularity of integral methods. Let us prove first that assuming that v 0 is a fixed positive number and v < v 0 < θu. We get: Following Theorem 6 from [3] it remains to show that the condition is also fulfilled. With the help of (3.1) we get: Thus we have shown that the integral transformation defined by c ′ γ,δ (u, v) is of type c 0 → c 0 for every θ ∈ (1/2; 1), and therefore condition I ′′ 1 (u, θ) = o θ (1) is satisfied. By the obtained relations we have that Next we evaluate the quantity I 2 (u, θ) using relations (3.1) and (3.3): So we have the estimate Now we are able to complete our proof showing that (3.4) is true for every γ < δ < γ + 1. We choose ε > 0 and afterwards θ ε ∈ (1/2, 1) so, that for any u > 0 (see (3.7)). Next we choose U = U θε so, that |I 1 (u, θ ε )| < ε/2 for all u > U (see (3.6)). It follows from (3.5) that |y δ (u)| < ε when u > U, i.e., (3.4) holds.
If γ = α 1 , then we choose some γ < γ 1 < δ and get that To finish the proof, it remains to apply implication (3.2), already proved, with γ 1 instead of γ. ⊓ ⊔ Note that Theorem 1 was formulated (but not proved) in [13] as Proposition 4 with a hint on analogy with matrix case (see [10,11]). The following Tauberian remainder theorem extends Theorem A.

⊓ ⊔
An analogous Tauberian remainder theorem for "matrix case" was proved in [12] as Theorem 2. Some Tauberian remainder theorems for Nörlund and Riesz matrix methods in Banach spaces were proved recently in [6] and [7]. Some estimates for speeds in a Riesz-type family (weaker than here) can be found also in [8].

Examples on Comparison of Speeds of Convergence
Here we give some numerical examples on application of Theorem 2 for comparison of speeds of convergence in special Riesz-type families. More precisely, we extend Examples 5, 7 and 9 from [9], where Theorem A was applied. In mentioned examples comparative evaluations (2.1) and (2.3) for speeds of any two methods A γ and A β in Riesz-type families {A α } are presented. Here we improve these results, comparing any three methods A γ , A β and A δ with the help of implication (3.9).