Comparison of speeds of convergence in Riesz-type families of summability methods

We deal with Riesz-type families (see Proc. Estonian Acad. Sci. Phys. Math., 2002, 51, 18–34 and Acta Sci. Math. (Szeged), 2004, 70, 639–657) of summability methods Aα for converging functions and sequences. The methods Aα in a Riesztype family depend on a continuous parameter α, and are connected through certain generalized integral Nörlund methods. By extending and applying the results of Stadtmüller and Tali (Anal. Math., 2003, 29, 227–242), we compare speeds of convergence in a Riesz-type family. As expected, the speed of convergence cannot increase if we switch from one summability method to a stronger one. Comparative estimations for speeds are found. In particular, the families of integral Riesz methods, generalized integral Nörlund methods, and Abeland Borel-type summability methods are considered. Numerical examples are given.


INTRODUCTION AND PRELIMINARIES
Let us consider the functions x = x(u) defined for u ≥ 0, bounded and measurable in the sense of Lebesgue on every finite interval [0, u 0 ].Let us denote the set of all these 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 the function y = y(u) is bounded, then 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 is applied, and by cA and mA the set of all functions x which are, respectively, convergent and bounded with respect to the method A. The summability method A is said to be regular if lim u→∞ x(u) = s =⇒ lim u→∞ y(u) = s whenever x ∈ X.The most common summability method for functions x is an integral method A, defined with the help of the transformation where a(u, v) is a certain function of two variables u ≥ 0 and v ≥ 0. We also say that the integral method A is defined by the function a(u, v).An example of the integral summability method is the generalized integral Nörlund method (N, P(u), Q(u)), defined with the help of the transformation where P = P(u) and Q = Q(u) are nonnegative functions from X such that R(u) = u 0 P(u − v) Q(v) dv = 0 for u > 0.
For sequences x = (x n ) we do not consider in our paper matrix methods (which are the most common summability methods), but focus ourselves on certain semi-continuous summability methods A, defined by transformations where a n (u) (n = 0, 1, 2, ...) are some functions from X.
As examples on semi-continuous methods the Abel-type methods A α = (A, α) with α > −1 (see [1]) and the Borel-type methods A α = (B, α) with α > α 0 (where α 0 is some fixed number) can be considered (see [2,3]).The Abel-type methods (A, α) are defined by the transformation of x = (x n ) into y α = y α (u) with where A α n are the Cesàro numbers.In particular, if α = 0, we have the Abel method A = (A, 0).The Borel-type methods (B, α) are defined by the transformation where Γ(•) is the Gamma-function and N is the smallest integer satisfying the inequality N > max{−α 0 , −1/2}.In particular, if α = 1, we have the Borel method B = (B, 1).One of the basic notions in this paper is the notion of the "speed of convergence".We follow here the definitions based on the definitions for sequences (see [4,5]) and extended for functions in [6,7].
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 exists.Note that the limit can be zero.If we have as u → ∞, then x is said to be bounded with speed λ .We use the notations c λ and m λ for the sets of all these functions x = x(u) which are convergent to some s with speed λ and bounded with speed λ , respectively.In the obvious manner the notion of speed can be transferred to summability methods.We say 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.

RIESZ-TYPE FAMILIES OF SUMMABILITY METHODS
Here we discuss and extend the notion of a Riesz-type family of summability methods given in papers [6,8].
A. Let us start with some examples.
Example 1.Consider the generalized Nörlund methods A α = (N, u α−1 , q(u)), where α > 0 and q = q(u) is a positive function from X.These methods are defined with the help of the transformation of x into A α x = y α = y α (u) with where It can be easily shown that any two methods A γ and A β with β > γ > 0 are connected through the relation and where Let us prove first relation (2.2), starting from its right side and using the substitutions where B(., .)denotes the Beta-function.The verification of (2.1) follows along the same lines; we just have to replace r γ (u) by r γ (u) y γ (u) and r β (u) by r β (u) y β (u).
We have only to exchange places of y γ (u) and y β (u) in it.More precisely, we have the relation (see [1]) where M γ,β is defined by (2.5).
Example 2. Connection formula (2.6), together with (2.5), appears also if we consider the methods A α = (D, α) (α > −1), defined with the help of the integral transformation (see [10]) (2.7) As there exist many other families with the connection formulas analogous to (2.1) and, in particular, to (2.4), we next consider a more general notion, the notion of a Riesz-type family defined in [6,8], and extend it.

B.
Let {A α } be a family of summability methods A α where1 α > (−) α 1 and which are defined by trans- formations of functions where r γ = r γ (u) and r β = r β (u) are some positive functions from X related through (2. 2) 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 in case A ), and in case B), where C γ,β is the integral method defined with the help of the function Note that Definition 1 in case A ) was given in [6,8].We see that the methods (N, u α−1 , q(u)) (α > 0) and (A, α) and (D, α) (α > −1) discussed above form Riesz-type families.The first of them is a Riesz-type family of case A ), and the other two are Riesz-type families of case B).
Let us consider some more examples of Riesz-type families.
Example 3. Let {A α } be the family of generalized Nörlund methods (N, p α (u), q(u)) (α > α 0 ), defined with the help of positive functions p = p(u) ∈ X and q = q(u) ∈ X and number α 0 such that It is known that relation (2.1), together with (2.2) and (2.3), holds for any β > γ > α 0 (see [11]), and thus this family is a Riesz-type family of case A ).
C. We discuss here the property of monotony of a Riesz-type family.
be a Riesz-type family.The methods C γ,β are regular for all β > γ > α 1 .These methods are regular also for all β > γ = α 1 , provided that the condition holds.
Proof.For the case β > γ > α 1 , this result was proved in [1] as Proposition 1.It remains to prove our statement if β > γ = α 1 .Because of the relation (which follows from (2.1)) in case A ) and the relation (which follows from (2.10)) in case B), it suffices to verify our statement only for α 1 < β < α 1 + 1.We use Theorem 6 from [9], which gives the sufficient conditions for the regularity of integral methods.Since the methods C α 1 ,β are defined by positive functions and u 0 c α 1 ,β (u, v) dv = 1 by (2.2), it remains to show that for every finite v 0 > 0. Supposing that v ≤ v 0 < u, we get with the help of (2.11) that Hence condition (2.12) is satisfied for every v 0 > 0.
Remark 1.As we can see from the previous proof, the transformations C γ,β (β > γ > (−) α 1 ) transform all bounded functions of X into bounded functions of X again.Proposition 1.Let {A α } (α > (−) α 1 ) be a Riesz-type family.Then we have for functions x = x(u) and numbers s and and in case B) that Proof.This result follows directly from Definition 1 because the methods C γ,β are regular by Lemma 1.

COMPARISON OF SPEEDS OF CONVERGENCE IN A RIESZ-TYPE FAMILY
Theorem 1 below describes how the speed of convergence changes if we go from one summability method in the family to a stronger one.
Theorem 1.Let {A α } (α > α 0 ) be a Riesz-type family.Let there be given some positive function λ = λ (u) → ∞ from X and some number γ > α 0 such that r γ (u) λ (u) ∈ X. (i) Then we have for functions x = x(u) and numbers s and and in case B) that where the speeds are related through the formulas and in case B) that Proof.
Case A ). Set α 1 = γ and consider another family of summability methods B α (α ≥ γ), defined by the transformations of x into η α = η α (u) with η α (u) = λ α (u) y α (u) , where λ α = λ α (u) is given according to (3.1).The methods B α obey the relation and form therefore a Riesz-type family.Notice that we have for α ≥ γ: where λ γ (u) = λ (u).Now Proposition 1 in case A ) (apply it to B α and x(u) − s instead of A α and x(u)) yields the desired result.Notice that relation (3.3) defines the connection methods C * γ,β such that Define the methods B β and B γ by transformations of x into the functions η β and η γ , respectively, where η β (u) = λ (u) y β (u) and η γ (u) = λ β (u) y γ (u).Now we have the relation which yields the desired result due to the regularity of connection methods C * γ,β which have in case of (3.6) the same shape as in case of (3.3).We note that case A ) of Theorem 1 can be considered as a generalization of case A) of Theorem 1 of [7], which was proved for matrix case.Certain evaluations of the speed of convergence for matrix Nörlund methods in Banach spaces were proved in a recent paper [12].
Next we will compare the speeds λ = λ (u) and λ β = λ β (u) described in Theorem 1 by proving some inequalities.
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 holds for every u > u 0 , we write a(u) ≈ b(u).
If the function b = b(u) is nondecreasing and condition (3.7) is satisfied with some positive numbers c 1 and c 2 for any u > 0, then we say that the function a = a(u) is almost nondecreasing.
Proposition 2. Let there be given a Riesz-type family {A α } (α > α 0 ) and an almost nondecreasing function where M is some positive constant independent of u.
Proof.By the relation r γ (u) = b γ (u) λ (u) and the other formulas (3.1) we have for any u > 0.
This result says that the speed of convergence cannot be improved by switching to a stronger summability method.It is consistent with the results known for matrix methods (see [4,12]), which say that a regular triangular matrix method cannot improve the speed of convergence (see also Proposition 2 in [7]).However, the speed cannot become much worse if we switch to a stronger method.Proposition 3. Let there be given a Riesz-type family {A α } (α > α 0 ) and a positive function λ = λ (u).Suppose that λ β = λ β (u) (β > γ > α 0 ) is defined through (3.1).If b γ (u) = r γ (u)/λ (u) is almost nondecreasing, then for β > γ > α 0 we have where K is some constant independent of u.
Proof.With the help of formulas (3.1) we find that where the coefficients M γ,β are determined by the given Riesz-type family, and N and K depend on γ and β but not on u.Remark 3. If both λ (u) and b γ (u) are almost nondecreasing, then for β > γ > α 0 we have by Propositions 2 and 3 K r β (u) r γ (u) u β −γ λ (u) ≤ λ β (u) ≤ M λ (u) (u > 0), where K and M are positive constants independent of u.

EXAMPLES ON THE COMPARISON OF SPEEDS OF CONVERGENCE
Applying Theorem 1, we find here comparative evaluations of speeds of convergence for summability methods in some special Riesz-type families.
Suppose that x = x(u) is a function having the given speed of convergence λ (u) with respect to the method A γ = (R, γ).Determine with the help of Theorem 1 the speed of convergence λ β (u) of x = x(u) with respect to the methods A β = (R, β ) for β > γ.
a) If ρ < γ + 1, then (4.1) yields due to Theorem 42 of [9] the equivalence Calculating the last integral with the help of substitution t = v/u, we get: