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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1988) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVIII (1988)

Lu c y n a Re m p u l s k a (Poznan)

On the summability of numerical and orthogonal series by the (B, n)-method

A bstract. In this paper we compare the summability methods (C , 1) and (B, n) for numerical and othogonal series. We study the summability, the absolute summability and the strong summability of series by these methods.

The methods (B, n) are considered in [1] (n = 0) and [7] (n = 1, 2, ...) in connection with some boundary problem for the Helmholtz equation.

1. The summability of numerical series. Write, as in [2], D°(h{r)) = h(r),

^ ^ ^ d

D"(h(r)) = D" -1 ( h ( r ) ) + ~ - D"~1 (h(r)),

n = 1 , 2 , . . . , for every function h defined in <0, 1 ) and having the deriva

tives of all orders in <0, 1).

Denote, further, by N the set of all non-negative integers and write

Л(г) = 7777 ( k e N , r e <0, 1», where I k is the Bessel function defined by the formula

h(r)

00 / r \ 2 p + k !

h \ y P'-(p + k)\

(cf. [5]).

In [2] and [7] there are given the following definitions:

The series

(1-2)

Y.

k= 0

(uk = real number) is (Я, n)-summable, n e N , convergent in <0, 1) and if the function H n,

00

to S if the series £ ukrл is k= 0 00

н „ ( г ) = Y k= 0

(re <0, 1)), (1.3)

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satisfies the condition Km H n(r) = S.

r-> 1 -

The series (1.2) is (В , w)-summable, n e N , to S if the series £ мкг* is fc= о convergent in <0, 1) and if the function U„,

00

(1.4) U„(r) =

I

D " (M r)K (re <0, 1)).

k = 0

00

satisfies the condition lim Un(r) = S.

r~* 1 —

The series (1.2) is absolutely (H, n)-summable or, shortly, \H, n\- summable, n e N , if the integral

- f H.M

dr dr

is convergent ([3], [8]).

We shall say that the series (1.2) is absolutely (В , n)-summable or, shortly, \B, nj-summable, n e N , if the integral

dr Un(r)^dr

is convergent.

The series (1.2) is absolutely (C, l)-summable, (|C, l|-summable) if the

00 П

series £* |оп- о п- х\, an = ^ (1 - k / ( n + l))uk, is convergent ([3]).

n — 1 k = 0

We shall call the series (1.2) (C, l)9-summable, q > 0, to S if (1.2) is strongly summable with the exponent q to S by the (C, l)-method, i.e.,

(1.5) lim t~ 7 S IS,—S|* = 0,

k~*oo К + 1 p= о

Sk = u0+ ••• +Uk ([9]).

We shall say that the series (1.2) is strongly summable with an exponent q > 0 to S by the method (В , n), n e N , or (B, n)„-summable to S if

oo 4

(1.6) lim £ ID" (yt (r)—л +i (r))| |S, - S|« = 0.

r - 1 - k = 0

The methods (H , n) and (В , n), n e N , are regular and linear ([2], [7]).

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Summability o f numerical and orthogonal series 137

In the sequel of the paper by M k(a,b), к = 1, 2, we shall denote some positive constants depending on a, b only.

The purpose of the present paper is the comparison of the methods (B , ri), (C, 1) and (H, n).

In [2] and [8] there are proved the following lemmas.

Le m m a 1. The series (1.2) is (H, ri)-summable, 1 ^ n e N , to S if and only if

lim (1 - rf — H 0(r) =

*1 - drk

if к = 0, if k = 1, , n.

Lemma 2. I f the series (1.2) is summable to S by the Cesàro (C, 1)- method, then it is (H , n)-summable to S for every n e N .

Lemma 3. The series (1.2) is \H, n\-summable if and only if the integral

J(l-r)"

0

dn+l

drn+1H 0(r) dr

is convergent.

I f {1.2) is IH, n+l\-summable, then it is \H, n\-summable.

Le m m a 4. I f the series (1.2) is |C, l\-summable, then it is \H, n\-summable for every n e N .

Arguing as in [2] (Corollary 1 and Lemma 3) and [8] (Lemmas 4 and 5), we obtain

Le m m a 5. The series (1.2) is (B, n)-summable, l ^ n e N , to S if and only if

lim (1

r - 1 -

if к = 0, if к = 1, •••, n.

Lemma 6. The series (1.2) is |JB, n\-summable if and only if the integral

(1-r)"

6

dn+l

drn+l U0{r) dr

is convergent.

I f {1.2) is IB, n+ Ц-summable, then it is \B, n\-summable.

Now we shall prove the following properties of the functions H 0 and U0, defined by (1.3) and (1.4).

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Lemma 7. I f uk = o(k), then

Urn ( l - r ) " ^ ( H o(r)-l/„(r)) = 0

r - i - a r

for every n e N .

P roof. By (1.3) and (1.4),

for re <0, 1) and

H 0(r) — U0(r) = £ (r*-y*(r))wk

k = о

j oo 2 p — 1 j

2‘ (r) = 7 j l ) ^ 2 ^ V ( P + ï ) ! for re< 0, 1) and k e N .

Let us remark that

M r)| ^ 1 1 1

and

d?_

drq

2/, (1) ,^ 0 22f+‘ p ! (p+ fc+ 1) ! 2 (k + 1)

2«+1 °°

AW p > ( q + l ) l 2 2 2p+kpl(p + k)l for k e N , 4 = 1 , 2 , . . . and r e <0, 1 ).

Hence, dP_

dr*(f*zk(r)) ^ M 2(p, r0)rk(k+ l)p 1 for k, p e N and re <r0, 1>, r0 > 0.

Consequently,

|H⁰(r)-l/oWI < ( 1 —r) I ^fc=0 2(k + 1)⁷⁷r*—rrkl (re<0, 1))

and

(1-r)" d*

dr-(H „ (r)-lM r))

< M 3(n ,r0){ ( i- r ) " +1 X ( t + i r ^ k l + d - r ) " I № + i) " - V |Mi|}

k = 0 fc = 0

for 1 ^ n e N and r e (r0, 1), r0 > 0.

Now, by the condition uk = o(k), we easily obtain our thesis.

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Summability o f numerical and orthogonal series 139

Arguing as in the proof of Lemma 7 and applying the equality

(1-r)"

0

r dr =

{k 4- 1)... (k + n + 1) (k, neN ),

we obtain the lemma.

00

Le m m a 8. I f the series £ \uk\/k2 is convergent, then the integral

k = 1

(1-r)" ( H 0 ( r ) - U 0 (r)) dr,

with H 0 and U0 defined by (1.3) and (1.4), is convergent for every n e N . It is easily verified that the following lemma holds.

00

Le m m a 9 . I f the series (1 .2 ) is |C, 1 |-summable, then the series £ k ~ 1 \uk\

k= 1 is convergent.

From Lemmas 1, 5 and 7, we obtain the following:

Co r o l l a r y 1. I f uk = o(k), then (1.2) is (B , n)-summable, ne N, to S if and only if it is (H , n)-summable to S.

Lemma 2 and Corollary 1 imply

Co r o l l a r y 2. I f the series (1.2) is (С, 1 )-summable to S, then it is (В , n)- summable to S for every ne N.

Applying Lemma 3, Lemma 6 and Lemma 8, we obtain

Co r o l l a r y 3. I f the series k~2 |uk| is convergent, then (1.2) is |B, n\-

k = 1

summable, n e N , if and only if it is |H, n\-summable.

Lemma 4, Lemma 9 and Corollary 3 imply

Co r o l l a r y 4. I f the series (1.2) is \C, \\-summable, then it is |B, n\- summable for every n e N .

In paper [7] the following lemma is proved.

Le m m a 10. I f k, n e N and r e (r0, 1) (r0 > 0), then

\Dn(y* (r)- yk +1 (r))| ^ M 6 (n, r0) (1 - r)n+1 (к + 1)”rk.

Applying Lemma 10, we shall prove

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Lemma 11. I f the series (1.2) is (C, \)q-summable, q > 0, to S, then it is (B, n)q-summable to S for every n e N .

P roof. By Lemma 10,

£ |ü" (yk (r) - y, +1 (r))| |Sj - S|4

k = 0

< M 6(n,r0) ( l - r ) ”+1 £ (* + l ) V |S t -S |«

k= 0

oo к

^ M7(«, r0)( l- r ) " +2 £ ( * + l ) V X |SP-S |*

k = 0 p = 0

for re <r0, 1) and ne/V.

Hence condition (1.5) implies (1.6).

2. The summability of orthonormal series. Let (<p*(x)} be an orthonor

mal system on the interval <0, 1) and let q>k, k e 0, 1, ..., be real functions.

We shall consider the orthonormal series

00

(2.1) £ ck<Pk(x) ( xe <0, 1 » k= 0

with real coefficients such that

£ cî < °° •

k= 0

The following property can easily be obtained Lemma 12. The series

£ k*<pk(x)|fc 1

к = 1

is convergent almost everywhere in <0, 1) for every series (2.1).

In this part we shall examine the relations between the methods (В , n), (C, 1) and (H , n) of summability of the series (2.1).

We shall say that two methods of summability are equivalent almost everywhere if the summability of (2.1) in a set E of positive measure by one of these methods implies the summability of (2.1) almost everywhere in E by the other method and to the same sum.

Similarly, for two methods we define the equivalence almost everywhere in the sense of absolute summability.

It is known ([4]) that the method (C, 1) and the Abel method (i.e., the (Я, 0)-method) are equivalent almost everywhere.

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Summability o f numerical and orthogonal series 141

In [2] and [6] there was proved that the (C, I)-method and the (H , n)- method, n s N , are equivalent almost everywhere.

Now we present four theorems on summability of the series (2.1).

Th e o r e m 1. The methods (C, 1) and (В , n), n e N , are equivalent almost everywhere.

Proof. The theorem follows by Lemma 12, Corollary 1 and the equiv

alence of the methods (C, 1) and (Я, n), n e N .

Th e o r e m 2. The methods (B , ri) and (Я, n), for a fixed n e N , are equiv

alent almost everywhere in the sense of absolute summability.

Proof. Applying Lemma 12 and Corollary 3, we obtain the theorem.

Th e o r e m 3. I f the coefficients o f the series (2.1) satisfy the condition 00 2”+1-2

(2.2) К I cl)m < oo,

n = 0 k = 2 " - l

then (2.1) is |B, n\-summable almost everywhere in <0, 1) and for every n e N . Proof. It is known ([10]) that condition (2.2) implies the |C, 1|- summability of (2.1) almost everywhere in <0, 1). Applying Corollary 4, we obtain the desired thesis.

Th e o r e m 4. I f the series (2.1) is (B, 0}-summable to S(x) almost every

where in <0, 1 ), then it is (В , n)q-summable to the same sum almost everywhere in ^0, 1) and for every n e N and q > 0.

Proof. By Theorem 1, the (B, 0)-summability of (2.1) to S(x) almost everywhere in <0, 1) implies the (C, l)-summability of (2.1) to S(x) almost everywhere in <0, 1).

In [9] it is proved that if (2.1) is (C, l)-summable almost everywhere in

<0, 1 ) to S{x), then it is (C, l)e-summable, q > 0, almost everywhere in

<0, 1> to S(x).

These results and Lemma 11 complete the proof.

References

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[2] Z. D o p ie r a ia , L. R e m p u ls k a , On the summability o f series by harmonic methods, Comment. Math. 23 (1983), 199-213.

[3] T. M. F le tt , On an extension o f absolute summability and some theorems o f Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113-141.

[4] S. K a c z m a r z , H. S t e in h a u s , Theory o f orthogonal series (in Russian), Moscow 1958.

[5] N. N. L e b ie d ie v , The special functions and their applications (in Russian), Moscow 1963:

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[6] L. R e m p u ls k a , On the (A, p)-summability o f orthonormal series, Demonstration Math. 8 (1980), 919-925.

[7] —, The summability o f the Fourier series and the Dirichlet problem, Bull. Acad. Polon. Sci.

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