Strong summability methods in a Riesz-type family

We continue our studies on Riesz-type families of summability methods for functions and sequences, started in Proc. Estonian Acad. Sci., 2008, 57, 70–80 and Math. Model. Anal., 2010, 15, 103–112. Strong summability methods defined on the basis of a given Riesz-type family {Aα} are considered here. Inclusion theorems for these methods are proved. Our inclusion theorems allow us to compare the summability fields and speeds of convergence. The strong summability methods are also compared with ordinary summability methods Aα and with certain methods of statistical convergence. The proved theorems generalize different results that have already been published and are applied, in particular, to Riesz methods, generalized integral Nörlund methods, and Borel-type methods.


INTRODUCTION AND PRELIMINARIES
Let x = x(u) be the functions defined for u ≥ 0, bounded and Lebesgue-measurable on every finite interval [0, u 0 ].Denote the set of all these functions by X.
If the limit lim u→∞ x(u) = s exists, we say that x = x(u) is convergent to s. 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, we say that x = x(u) is convergent to s with respect to the summability method A (or x is summable to s by the method A) and write x(u) → s(A).If the function y = y(u) is bounded, we say that x is bounded with respect to the method A, and write x(u) = O(A).We denote by ωA the set of all these functions x, where the transformation A can be applied.The summability method A is said to be regular if for each x ∈ X lim The most common summability method for functions x is an integral method A defined by the transformation where a(u, v) is a certain function of two variables (u ≥ 0, v ≥ 0) with a(u, v) = 0 for v ≥ u.We also say that the integral method A is defined by the function a(u, v).
The notion of a statistically convergent function is also used in this paper.According to [9] we say that x is statistically convergent to s and write x(u) → s(st) if for any ε > 0 where |K ε,u | is the Lebesgue measure of the set Generalizing the given notion of statistical convergence, we come to the following definition.
Definition 1.Let A be a regular integral method defined by transformation (1.1), where a(u, v) is some non-negative function.We say that x = x(u) is A-statistically convergent to s and write x(u) → s(st A ) if for any ε > 0 where K ε,u is the set defined by (1.2).
In particular, the notion of A-statistical convergence for the matrix case was first defined in [3] and later generalized and discussed in different papers (see [4] and [8] for references).
For converging sequences x = (x n ) we focus on certain semi-continuous summability methods A defined by transformations where a n (u) (n = 0, 1, 2, . ..) are some functions from X.
One of the basic notions in this paper is the notion of the speed of convergence (see [10] and [12] and, in particular for sequences, [6] and [7]).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 lim u→∞ λ (u) [x(u) − s] exists.Note that the limit can be zero.If we have λ (u) [x(u) − s] = O(1) as u → ∞, then x is said to be bounded with speed λ .We say that x is convergent or bounded with speed λ with respect to the summability method A if Ax = y(u) is convergent or bounded with speed λ , respectively.
In our paper we study Riesz-type families of summability methods defined in [10] and [13].
Let {A α } be a family of summability methods A α , where1 α > (−) α 0 and which are defined by trans- formations of functions 3) holds and the methods A γ and A β are connected by the relation 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 .
In other words, a Riesz-type family is a family where every two methods are connected through the connection formula where C γ,β is the integral method defined by the function Next we introduce some examples of Riesz-type families (see, e.g., [15]).
Proposition 1.The methods C γ,β defined by (1.6) and (1.5) are regular for all β > γ > α 0 .These mehods are regular also for all β > γ = α 0 , provided that In the present paper the authors continue their studies on Riesz-type families started in [14] and [15].The main idea of the paper is to define the family of strong summability methods [A α+1 ] k on the basis of a given Riesz-type family {A α } (α > (−) α 0 ) and to describe it with the help of different inclusion theorems.These strong summability methods are compared with each other by summability fields, i.e., by sets of functions x they converge, and by the speed of convergence.The strong summability methods are also compared with methods A α and A α+1 and with certain methods of statistical convergence.

INCLUSION THEOREMS FOR STRONG SUMMABILITY METHODS [A α+1 ] k
We start this section with the definition of strong summability methods for converging functions x, supposing that where y α (u) = A α x (x ∈ ωA α ) and r α (u) and r α+1 (u) are defined by the family {A α }.
) be a Riesz-type family and k = k(u) be a positive function from X.We say that a function x = x(u) is strongly convergent to s with respect to the method A α+1 (in short, We say that a function x = x(u) is strongly bounded with respect to the method A α+1 (in short, (2.2) Thus we have defined the methods , then r α (u) and r α+1 (u) were defined in Examples 1, 2 or 3, respectively.
We begin proving some inclusion theorems.
be a Riesz-type family.Let k = k(u) and k = k (u) be two functions from X. Then the following statements are true for functions x = x(u) and numbers s and Proof.Take w.l.o.g.s = 0. (i) The quantity σ k γ+1 (u) can be written in the form and Thus we have the relations Using these relations, we get with the help of the Hölder inequality and (1.5) .
Thus we have got the relation which implies our statement (i).(ii) As using statement (i), get The part of statement (ii) about boundednes follows from (2.4) in an analogous way.(iii) According to (2.1) we have Now we get with the help of (1.4) and the Hölder inequality Thus we have shown that and we get the relations Further we use also the notation (2.6) Developing (2.5) with help of the last relations, we get So we have proved the relation It follows from (2.7) that if σ k γ+1 (u) → 0, then σ k β +1 (u) → 0 (as u → ∞), because σ k γ+1 (u) → 0 implies σ 1 γ+1 (u) → 0 by statement (i) and C γ+1,β +1 is a regular method by Proposition 1.The part of statement (iii) about boundedness follows from (2.7) analogously.

Remark 1.
(i) If we weaken the restrictions on k and k allowing also the case k(u) k (u) → ∞, then statement (i) of Theorem 1 is not true in general (see Remark 2 in [11]).(ii) If 0 < k(u) < 1, then statement (ii) is not true in general (see Remark 4 in [11]).(iii) In particular, if k(u) ≡ r, then relation (2.5) takes the form and completes the proof of statement (iii) of Theorem 1.
Then the following statements are true for functions x = x(u) and numbers s and γ > (−) α 0 : ), where the method C γ,γ+1 is defined by (1.6).
Thus we have proved the inequality ) due to Theorem 1 (ii), and thus the right side of inequality (2.11) tends to zero.Then also the left side of (2.11) tends to zero and therefore δ k γ+1 (u) → 0 as u → ∞. (i) Sufficiency.Using the same technique as in the proof of necessity, we get the inequality Statement (ii) can be proved in an analogous way with the help of (2.11) and (2.12) if s = 0.
Remark 2. In particular, if k(u) ≡ r, Theorems 1-3 are formulated with some hints at proofs in [13] as Theorems 4-6.Theorems analogous to Theorems 1-3 for the matrix case are proved in [11] (as Theorems 4 and 5), where also references for partial cases can be found.

COMPARATIVE ESTIMATIONS FOR SPEEDS OF [A ] k -CONVERGENCE
Let {A α } (α > (−) α 0 ) be a Riesz-type family and k = k(u) be a positive function from X. Suppose that λ = λ (u) is a positive function from X such that λ (u) → ∞ as u → ∞.Definition 4. We say that a function x = x(u) is [A α+1 ] k -convergent to s with speed λ if there exists the finite limit lim where σ k α+1 (u) is defined by (2.1).In this paper mainly the limit lim The following Theorems 4-6 help us to estimate the speed of (−) α 0 ) be a Riesz-type family.Let there be given some positive function λ = λ (u) → ∞ from X. Then the following statements are true for any γ > α 0 : Proof.Take w.l.o.g.s = 0. (i) By (2.3) we have the relation which implies statement (i) immediately.
(ii) By (2.4) and (2.6) we have the inequality Statement (i) and the last inequality complete the proof of (ii): Remark 3. Theorem 4 remains true if we replace o(1) by O(1) everywhere in it.
In papers [14] and [15] the speeds λ γ and λ β of convergence x = x(u) with respect to methods A γ and A β (β > γ) are compared in a Riesz-type family (see [14], Theorem 1 and [15], Theorem 2).Speed λ γ = λ is supposed to be a given speed and λ β = λ β (u) is defined by the relations Further we see that these speeds can be compared also for strong summability methods.
In order to see how statistical convergence is related to ordinary convergence in statements (i) and (ii) of Theorem 7, we formulate the proposition which can be proved in the same way as Theorem 7 (take k(u) ≡ 1 in its proof).Proposition 2. Let {A α } (α > (−) α 0 ) be a Riesz-type family satisfying (1.8) if α 0 is included.Then the following statements are true for functions x = x(u) and numbers s and γ > (−) α 0 : (i) if x(u) → s(A γ ), then A γ x → s(st C γ,γ+1 ); (ii) if x(u) = O(A γ ) and A γ x → s(st C γ,γ+1 ), then x(u) → s(A γ+1 ).

CONCLUSIONS
In this paper a Riesz-type family of summability methods A α (α > (−) α 0 ) is considered (see Definition 2).The strong summability methods [A α+1 ] k are defined (see Definition 3) and described with the help of inclusion theorems.These theorems give the conditions for comparing the methods [A α+1 ] k with each other and with methods A α (for different values of α) by summability fields (see Theorems 1-3) and by speed of convergence (see .The methods [A α+1 ] k are compared also with certain methods of statistical convergence (see Theorem 7).Theorems 1-3 generalize the theorems known earlier, in particular, for the case k = k(u) ≡ r (see [13]), showing that the methods [A α+1 ] k defined here are more flexible.In the authors' view the notion of methods [A α+1 ] k can be further generalized with the help of a modulus function f .A convexity theorem can also be proved for these methods.