Comparison of speeds of convergence in Riesz-type families of summability methods/Summeeruvuskiiruste vordlemine Rieszi-tuupi peredes.

1. INTRODUCTION AND PRELIMINARIES

Let us consider the functions x = x(u) defined for u [less than or equal to] 0, bounded and measurable in the sense of Lebesgue on every finite interval [0,[u.sub.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.sub.n])) into functions Ax = y = y(u) [member of] X. If the limit [lim.sub.u[right arrow][infinity]]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)[right arrow]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 [omega]A 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever x [member of] X.

The most common summability method for functions x is an integral method A, defined with the help of the transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where a(u,v) is a certain function of two variables u [less than or equal to] 0 and v [less than or equal to] 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 Norlund method (N,P(u),Q(u)), defined with the help of the transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P = P(u) and Q = Q(u) are nonnegative functions from X such that R(u) = [[integral].sup.u.sub.0] P(u - v)Q(v)dv [not equal to] 0 for u > 0.

For sequences x = ([x.sub.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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.n](u) (n = 0,1,2,...) are some functions from X.

As examples on semi-continuous methods the Abel-type methods [A.sub.[alpha]] = (A, [alpha]) with [alpha] > - 1 (see [1]) and the Borel-type methods [A.sub.[alpha]] = (B, [alpha]) with [alpha] > [[alpha].sub.0] (where [[alpha].sub.0] is some fixed number) can be considered (see [2,3]). The Abel-type methods (A, [alpha]) are defined by the transformation of x = (xn) into y [alpha] = y [alpha](u) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [A.sup.a.sub.n] are the Cesaro numbers. In particular, if [alpha] = 0, we have the Abel method A = (A,0).

The Borel-type methods (B, [alpha]) are defined by the transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where [GAMMA](*) is the Gamma-function and N is the smallest integer satisfying the inequality N > max {-[alpha].sub.0] 1/2]}. In particular, if [alpha] = 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 [lambda] = [lambda](u) be a positive function from X such that [lambda](u) [right arrow] [infinity] as u [right arrow] [infinity]. We say that a function x = x(u) is convergent to s with speed [lambda] if the finite limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exists. Note that the limit can be zero. If we have

[lambda](u) [x(u)- s] = O (1)

as u [right arrow] [infinity], then x is said to be bounded with speed [lambda]. We use the notations [c.sup.[lambda]] and [m.sup.[lambda]] for the sets of all these functions x = x(u) which are convergent to some s with speed l and bounded with speed l, 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 l with respect to the summability method A if Ax [member of] [c.sup.lambda] or Ax [lambda] [m.sup.lambda], respectively.

2. 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 Norlund methods [A.sub.[alpha]] = (N,[u.sup.[alpha]-1], q(u)), where [alpha] > 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.sub.[alpha]x = [y.sub.[alpha]] = [y.sub.[alpha]](u) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [r.sub.[alpha]] = [r.sub.[alpha]](u) = [[integral].sup.u.sub.0] [(u - v).sup.[[alpha]-1] q(u) dv.

It can be easily shown that any two methods [A.sub.[gamma]] and [A.sub.[beta]] with [beta] > [gamma] > 0 are connected through the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Let us prove first relation (2.2), starting from its right side and using the substitutions v' =v - t and v" = v'/u-t.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where B(., .) denotes the Beta-function. The verification of (2.1) follows along the same lines; we just have to replace [r.sub.[gamma]](u) by [r.sub.[gamma]](u) [y.sub.[gamma]] (u) and [r.sub.[beta]] (u) by [r.sub.[beta]] (u) [y.sub.[beta]] (u).

In particular, if q(u) = 1 (u [less than or equal to] 0), we have that [r.sub.[alpha]](u) = [u.sup.[alpha]]/[alpha] and methods (N, [u.sup.[alpha]-1]],q(u)) turn into Riesz methods (R, [alpha]) (see [9]), and (2.1) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Note that the same connection formula (2.4) appears for Abel methods (A, b) and (A, g) defined by (1.1). We have only to exchange places of y g(u) and y b(u) in it. More precisely, we have the relation (see [1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

where [M.sub.[gamma][beta]] is defined by (2.5).

Example 2. Connection formula (2.6), together with (2.5), appears also if we consider the methods [A.sub.[alpha]] = (D, [alpha]) ([alpha] > -1), defined with the help of the integral transformation (see [10])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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.sub.[alpha]]} be a family of summability methods [A.sub.[alpha] where (1) [alpha].sup.>.sub. (-)] [[alpha].sub.1] and which are defined by transformations of functions x = x(u) [member of] [omega] [A.sub.[alpha]] [subset] X into functions [A.sub.[alpha]]x = [y.sub.[alpha]] = [y.sub.[alpha]](u) [member of] X. Suppose that for any [beta] > [gamma].sup.>.sub.(-)] [[alpha].sub.1] we have the relation

[omega] [A.sub.[gamma] [subset] [[omega]A.sub.[beta]] (2.8)

or

[omega]A[beta] [subset] [omega]A [gamma]. (2.9)

Definition 1. A family {A[alpha]} ([[alpha].sup.>.sub.(-)] [[alpha].sub.1]) is said to be a Riesz-type family if for every [beta] > [gamma].sup.>.sub.(-)] [[alpha].sub.1]

A ) relation (2.8) holds and the methods [A.sub.[gamma]] and [A.sub.[beta]] are connected through (2.1) or

B) relation (2.9) holds and the methods [A.sub.[gamma]] and [A.sub.[beta]] are connected through the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

where [r.sub.[gamma]] = [r.sub.[gamma]] (u) and [r.sub.[beta]] = [r.sub.[beta]] (u) are some positive functions from X related through (2.2) and [M.sub.[gamma],[beta] is a constant depending on [gamma] and [beta].

In other words, a Riesz-type family is a family where every two methods are connected through the connection formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case A ), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in case B), where [C.sub.[gamma],[beta]] is the integral method defined with the help of the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that Definition 1 in case A ) was given in [6,8]. We see that the methods (N, [u.sup.[alpha]-1], q(u)) ([alpha] > 0) and (A, [alpha]) and (D, [alpha]) (a > -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.sub.[alpha]} be the family of generalized Norlund methods (N,[p.sub.[alpha]](u),q(u)) ([alpha] > [alpha].sub.0]), defined with the help of positive functions p = p(u) [member of] X and q = q(u) [member of] X and number [[alpha].sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [p.sub.[alpha]] (u) = [[integral].sup.u.sub.0] [(u - v).sup.[alpha]-1] p(v)dv. It is known that relation (2.1), together with (2.2) and (2.3), holds for any [beta] > [gamma] > [[alpha].sub.0] (see [11]), and thus this family is a Riesz-type family of case A ).

Example 4. Consider the family {A.sub.[alpha]} of Borel-type methods [A.sub.[alpha]] = (B, [alpha],[q.sub.n]) defined in [8]. Let ([q.sub.n]) be a nonnegative sequence with [q.sub.0] > 0 such that the power series [SIGMA] [q.sub.n] [u.sup.n] has the radius of convergence R = [infinity] and [q.sub.n] > 0 at least for one n [member of] IN. Denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and define the methods (B,[alpha],[q.sub.n]) ([alpha] > - 1/2) for converging sequences x = ([x.sub.n]) with the help of the transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The methods (B,[alpha],[q.sub.n]) satisfy relations (2.1) and (2.2) with [M.sub.[gamma],[beta]] = 1/[GAMMA] ([beta]-[gamma]) (see [8]). Thus {[A.sub.[alpha]} is a Riesz-type family of case A). In particular, if [q.sub.n] = 1/n! , we get the Borel-type methods (B,[alpha]) = (B,[alpha],1/n!) (see (1.2)) because in this case r[alpha](u)~ [e.sup.u] as u [right arrow] [infinity].

C. We discuss here the property of monotony of a Riesz-type family.

Lemma 1. Let {[A.sub.[alpha]]} ([alpha].sup.>.sub.(-)] [[alpha].sub.1]) be a Riesz-type family. The methods [C.sub.[gamma],[beta]] are regular for all [beta]>[gamma]> [[alpha].sub.1].

These methods are regular also for all [beta] > [gamma] = [[alpha].sub.1], provided that the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

holds.

Proof. For the case [beta]>[gamma]> [[alpha].sub.1], this result was proved in [1] as Proposition 1. It remains to prove our statement if [beta] > [gamma] = [[alpha].sub.1]. Because of the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([beta]>[delta]> [alpha].sub.1])

(which follows from (2.1)) in case A) and the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([beta]> [delta]> [[alpha].sub.1])

(which follows from (2.10)) in case B), it suffices to verify our statement only for [[alpha].sub.1] < [beta] < [[alpha].sub.1]+1. We use Theorem 6 from [9], which gives the sufficient conditions for the regularity of integral methods. Since the methods [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are defined by positive functions and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](u,v) dv = 1 by (2.2), it remains to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

for every finite [v.sub.0] > 0. Supposing that v [less than or equal to] [v.sub.0] < u, we get with the help of (2.11) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly for 0 < v [less than or equal to] [v.sub.0] as u[right arrow][infinity]. Hence condition (2.12) is satisfied for every [v.sub.0] > 0.

Remark 1. As we can see from the previous proof, the transformations [C.sub.[gamma],[beta]] ([beta] >[gamma].sup.>.sub.(-)] [[alpha].sub.1]) transform all bounded functions of X into bounded functions of X again.

Proposition 1. Let {[A.sub.[alpha]]} ([[alpha].sup.>.sub.(-)] [[alpha].sub.1]) be a Riesz-type family. Then we have for functions x = x(u) and numbers s and [beta] > [[gamma].sup.>.sub.(-)] [[alpha].sub.1] in case A) that

x(u) = O([A.sub.[gamma]]) [??] x(u) = O([A.sub.[beta]]) and x(u) [right arrow] s ([A.sub.[gamma]) [??] x(u) [right arrow] s ([A.sub.[beta]),

and in case B) that

x(u) = O([A.sub.[beta]) [??] x(u) = O([A.sub.[gamma]) and x(u)[right arrow]s ([A.sub.[beta]) [right arrow] x(u) [right arrow] s ([A.sub.[gamma]),

provided in both cases that (2.11) is satisfied if [gamma] = [[alpha].sub.1] is included.

Proof. This result follows directly from Definition 1 because the methods [C.sub.[gamma],[beta]] are regular by Lemma 1.

3. 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.sub.[alpha]]} ([alpha] > [[alpha].sub.0]) be a Riesz-type family. Let there be given some positive function [lambda]= [lambda](u)[right arrow][infinity] from X and some number [gamma] > [[alpha].sub.0] such that [r.sub.[gamma]](u) / [lambda](u) [member of] X.

(i) Then we have for functions x = x(u) and numbers s and [beta][greater than or equal to] [gamma] in case A) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and in case B) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the speeds are related through the formulas

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

(ii) Moreover, we have

in case A) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and in case B) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided in both cases that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Proof.

Case A). Set [a.sub.1] = [gamma] and consider another family of summability methods B[alpha] ([alpha][less than or equal to] [gamma]), defined by the transformations of x into [[eta].sub.[alpha]] = [[eta].sub.[alpha]] (u) with

[[eta].sub.[alpha]] (u) = [[lambda].sub.[alpha]] (u) [y.sub.[alpha]](u),

where [[lambda].sub.[alpha]] = [[lambda].sub.[alpha]] (u) is given according to (3.1). The methods [B.sub.[alpha]] obey the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

and form therefore a Riesz-type family. Notice that we have for [alpha][greater than or equal to] [gamma].

[[lambda].sub.[alpha]](u) [[y.sub.[alpha]](u) - s] = O(1)[??] x (u) - s = O([B.sub.[alpha]]), (3.4)

[[lambda].sub.[alpha]](u) [[y.sub.[alpha]](u) - s] [right arrow] t [??] x (u) - s [right arrow] [omicron]([B.sub.[alpha]]), (3.5)

where [[lambda].sub.[gamma]](u) = [lambda](u). Now Proposition 1 in case A) (apply it to [B.sub.[alpha]] and x(u) - s instead of [A.sub.[alpha]] and x(u)) yields the desired result. Notice that relation (3.3) defines the connection methods [C.sup.*.sub.[gamma],[beta]] such that [B.sub.[beta]] = [C.sup.*.sub.[gamma]], [omicron] [B.sub.[gamma]].

Case B). Define the methods [B.sub.[beta]] and [B.sub.[gamma]] by transformations of x into the functions [[eta].sub.[beta]] and [[eta].sub.[gamma]], respectively, where [[eta].sub.[beta]] (u) = [lambda](u) [y.sub.[beta]] (u) and [[eta].sub.[gamma]] (u) = [[lambda].sub.[beta]] (u) [y.sub.[gamma]] (u). Now we have the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

which yields the desired result due to the regularity of connection methods [C.sup.*.sub.[gamma],[beta]] which have in case of (3.6) the same shape as in case of (3.3).

Remark 2. Under restriction (3.2) the condition [lambda](u)[right arrow][infinity] implies [[lambda].sub.[beta]](u)[right arrow][infinity] in Theorem 1. This follows from the regularity of methods [C.sub.[gamma],[beta]] and [C.sup.*.sub.[gamma],[beta]] (apply [C.sup.*.sub.[gamma],[beta]] to the function [lambda](u)[x(u) - s][right arrow]t, where t [not equal to] 0 and [C.sub.[gamma],[beta]] to the function x(u)).

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 Norlund methods in Banach spaces were proved in a recent paper [12].

Next we will compare the speeds [lambda] = [lambda](u) and [[lambda].sub.[beta]] = [[lambda].sub.[beta]] (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.sub.1], [c.sub.2], and u0 such that the condition

c1 b(u) [less than or equal to] a(u) [less than or equal to] [c.sub.2] b(u) (3.7)

holds for every u > [u.sub.0], we write

a(u) [approximately equal to] b(u).

If the function b = b(u) is nondecreasing and condition (3.7) is satisfied with some positive numbers [c.sub.1] and [c.sub.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.sub.[alpha]]}([alpha] > [[alpha].sub.0]) and an almost nondecreasing function [lambda] = [lambda](u). Suppose that [[lambda].sub.[beta]] = [[lambda].sub.[beta]] (u) ([beta]> [gamma] > [[alpha].sub.0]) is defined through (3.1). Then for [beta] > [gamma] > [[alpha].sub.0] we have

[[lambda].sub.[beta]] (u) [less than or equal to] M [lambda](u) (u > 0),

where M is some positive constant independent of u.

Proof. By the relation [r.sub.[gamma]](u) = [b.sub.[gamma](u) [lambda](u) and the other formulas (3.1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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.sub.[alpha]]}([alpha] > [[alpha].sub.0]) and a positive function [lambda] = [lambda](u).

Suppose that [[lambda].sub.[beta]] = [[lambda].sub.[beta]](u) ([beta]> [gamma]> [[alpha].sub.0]) is defined through (3.1). If [b.sub.[gamma]](u) = [r.sub.[gamma]](u)/[lambda](u) is almost nondecreasing, then for [beta] > [gamma] > [[alpha].sub.0] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where K is some constant independent of u.

Proof. With the help of formulas (3.1) we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the coefficients [M.sub.[gamma],[beta]] are determined by the given Riesz-type family, and N and K depend on [gamma] and [beta] but not on u.

Remark 3. If both [lambda](u) and [b.sub.[gamma]](u) are almost nondecreasing, then for [beta]>[gamma]> [[alpha].sub.0] we have by Propositions 2 and 3

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where K and M are positive constants independent of u.

4. 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.

Example 5. We consider the family of Riesz methods [A.sub.[alpha]] = (R,[alpha]) = ([N,[u.sup.a-1],1) ([alpha] > 0). Let us choose the speed of convergence [lambda](u) = ([u+1).sup.[rho]] ([rho]> 0) and some number [gamma] > 0.

Suppose that x = x(u) is a function having the given speed of convergence [lambda](u) with respect to the method [A.sub.[gamma]] = (R,[gamma]). Determine with the help of Theorem 1 the speed of convergence [[lambda].sub.[beta]](u) of x = x(u) with respect to the methods [A.sub.[beta]] = (R,[beta]) for [beta] > [gamma].

Using formulas (3.1), we have [[lambda].sub.[beta]](u) = [r.sub.[beta]](u)= [b.sub.[beta]](u) with [r.sub.[beta]](u) = [u.sup.[beta]] = [beta] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

where [M.sub.[gamma],[beta]] is the constant defined by (2.3).

It follows directly from (4.1) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

for every u > 2.

a) If [rho] < [gamma] + 1, then (4.1) yields due to Theorem 42 of [9] the equivalence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as x[right arrow][infinity]. Calculating the last integral with the help of substitution t = v=u, we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus we have in this case that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

as u[right arrow][infinity].

Evaluating the functions [b.sub.[beta]](u) and [[lambda].sub.[beta]](u), we do not calculate in further estimations (as we did in (4.2) and (4.3)) the exact values of numerical coefficients any more. Moreover, in order to shorten our writings, we do not emphasize further the dependence of these coefficients on parameters [gamma],[beta], and [rho] with the help of indices in these coefficients.

b) If [rho] = [gamma] + 1, it follows from (4.1) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every u > [u.sub.0], where [u.sub.0] is some positive number and [L.sub.1], [L.sub.2],..., [L.sub.6] are constants independent of u.

On the other hand, inequality (4.2) gives us that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every u > [u.sub.1], where u1 is some positive number bigger than 2, and [M.sub.1], [M.sub.2], and [M.sub.3] are constants independent of u. Thus we have shown that in this case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

c) If [rho] > [gamma] + 1, then starting from (4.1) and (4.2) and discussing analogously to case b), we come to the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

As a result we have proved by (4.3)-(4.5) the following estimations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that case A) of statement (i) of Theorem 1 holds here for any [beta] > [gamma]> 0 and [rho] > 0. Moreover, if [rho] < [gamma]+1 or [rho] = [gamma]+1, then condition (3.2) is satisfied and also case A) of statement (ii) of Theorem 1 works here.

Example 6. Consider the family of Abel-type methods [A.sub.[alpha]] = (A,[alpha]) ([alpha]> -1). Suppose that [lambda](u) is the same as in the previous example. Fix some number [gamma] > -1 and pose the same task as in the previous example to find [[lambda].sub.[beta]](u) for [beta]> [gamma]> -1.

Here the connection method [C.sub.[gamma],[beta]] is the same as for the Riesz methods (compare relations (2.6) and (2.4). So the same situation as in the previous example appears here and we get the same speed [[lambda].sub.[beta]](u). Thus, case B) of statement (i) of Theorem 1 holds here. Moreover, if [rho] < [gamma]+1 or [rho]= [gamma] +1, then condition

(3.2) is satisfied and also case B) of statement (ii) of Theorem 1 works.

The same situation appears if we consider the family of methods [A.sub.a] = (D, [alpha]) ([alpha] > -1) (see Example 2).

Example 7. Let us consider the Borel-type methods [A.sub.[alpha]] = (B,[alpha],1/n!) = (B, [alpha]) ([alpha] > - 1/2). Here we have [r.sub.[alpha]](u) ~ [e.sup.u] (see Example 4).

Suppose that [lambda](u) = [(u+1).sup.[rho]] [e.sup.u], fix some [gamma] > - 1/2], and find [[lambda].sub.[beta]](u) for [beta] > [gamma] again. Now we get for [beta] > [gamma] with the help of relations (3.1) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evaluating the last integral in the same way as in Example 5, we get the following results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 8. Consider here the methods [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where 0 < [phi] < 1 is some fixed number. Suppose that [lambda](u) = [(u + 1).sup.[rho]] (r > 0). We have for [beta] [greater than or equal to] [gamma] that (see [7, p. 236])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we get with the help of (3.1) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[[lambda].sub.[beta]](u) [approximately equal to] [u.sup.[rho]] [lambda](u)

if [beta] > [gamma]. So, both statements (i) and (ii) in case A) of Theorem 1 apply here with

[[lambda].sub.[beta]](u) [approximately equal to] [lambda](u).

Example 9. Let [A.sub.[alpha]] be the same methods as in the previous example, but suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then we have by (3.1) that [b.sub.[gamma]](u) [approximately equal to] [[u.sup.(1-[phi])[gamma]] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [beta] > [gamma]. Therefore statements (i) and (ii) of Theorem 1 in case A) are true with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This series of examples could be continued.

ACKNOWLEDGEMENT

This research was supported by the Estonian Science Foundation (grant No. 7033).

Received 25 January 2008, in revised form 22 April 2008

REFERENCES

[1.] Borwein, D. On a scale of Abel-type summability methods. Proc. Cambridge Phil. Soc., 1957, 53, 318-322.

[2.] Borwein, D. On Borel-type methods of summability. Mathematica, 1958, 5, 128-133.

[3.] Borwein, D. and Shawyer, B. L. R. On Borel-methods. Tohoku Math. J., 1966, 18, 283-298.

[4.] Kangro, G. On the summability factors of the Bohr-Hardy for a given rapidity. I. Eesti NSV Tead. Akad. Toim. Fuus. Mat., 1969, 18, 137-146 (in Russian).

[5.] Kangro, G. Summability factors for the series l-bounded by the methods of Riesz and Cesaro. Tartu Ulik. Toimetised, 1971, 277, 136-154 (in Russian).

[6.] Pavlova, V. and Tali, A. On the convexity theorem of M. Riesz. Proc. Estonian Acad. Sci. Phys. Math., 2002, 51, 18-34.

[7.] Stadtmuller, U. and Tali, A. Comparison of certain summability methods by speeds of convergence. Anal. Math., 2003, 29, 227-242.

[8.] Stadtmuller, U. and Tali, A. Strong summability in certain families of summability methods. Acta Sci. Math. (Szeged), 2004, 70, 639-657.

[9.] Hardy, G. H. Divergent Series. Oxford Press, 1949.

[10.] Kuttner, B. On "translated quasi-Cesaro" summability. Proc. Cambridge Phil. Soc., 1966, 62, 705-712.

[11.] Tali, A. Zero-convex families of summability methods. Tartu Ulik. Toimetised, 1981, 504, 48-57 (in Russian).

[12.] Meronen, O. and Tammeraid, I. Generalized Norlund method and convergence acceleration. Math. Model. Anal., 2007, 12, 195-204.

(1) The notation [[alpha].sup.>.sub.(-)] [[alpha].sub.1] means that we consider parameter values [alpha] > [[alpha].sub.1] or a, a1 with some fixed number [[alpha].sub.1].

Anna Seletski and Anne Tali *

Department of Mathematics, Tallinn University, Narva mnt. 25, 10120 Tallinn, Estonia

* Corresponding author, [e-mail]