On the absolute and strong summability of numerical and orthogonal series by the methods connected with the Dirichlet problem for an elliptic domain

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVTII (1989)

Bar ba r a Firlej and Lu c y n a Re m p u l s k a (Poznan)

On the absolute and strong summability of numerical and orthogonal series by the methods connected with the Dirichlet problem for an elliptic domain

In this paper we shall investigate the absolute and strong summability of numerical and orthogonal series by the methods introduced by Gôrowski in [2] and [3] for the construction of a solution of the Dirichlet problem for an elliptic domain. In Section 1 we shall give relations between the methods (Л,) of the Gôrowski type and the Abel method for absolute and strong sum­

mability of numerical series. The similar problem for orthogonal series we shall consider in Section 2.

Some properties of the methods (Л,) are given in [5].

1. The absolute and strong summability of numerical series.

1.1. Write, as in [5],

(1) 4 » = U

(2) Я‘„2’(г)= 1,

4 " «

cosh/a cosh к T ’ 4 2, (t) = sinh/a

sinh kT for re < 0, T} and k — 1 , 2 ,..., where 0 < T < oo.

We say ([5]) that the numerical series

(3 ) f Щ

k= 0

(uk are real numbers) is (A^-summable, i = 1 or i = 2, to S if the series

(4) V M = t W (')»»

k= 0 is convergent in <0, T) and if lim (/,• (t) = S.

t ->t —

In [5] the following results were proved.

Lemma A. The function Ub i = 1 or i — 2 (defined by (4)) exists in <0, T) if and only if the series

00

(5) V(t)= £ e‘(- T4

k = 0

is convergent in <0, T).

Theorem A. The method (At), г = 1 , 2 , and the classical Abel method are equivalent.

Lemma B. There exists the positive constant (t0), depending on t0 e(0, T) only, such that

W ° (0 -4 ° +i (01 < (t0)( 1 - е * - т)ек« - Т) for t e <f0, Г), к = 0, 1, ... and i = 1,2.

1.2. Now we shall give some auxiliary results. Let N be the set of all non-negative integers.

It is easily verified that the following results are true.

Lemma 1. I f series (5) is convergent in <0, T), then for every n e N the

00

series £ kn ek(t~T) uk is convergent in <0, T) also. Moreover, this series is

k = о

absolutely and uniformly convergent in every interval <0, by, b < T.

Lemma 2. There exists the positive constant M 2(T), depending only on T, such that

° ^ S ' ^ M 2 ^ k neHt~T) for t e ( 0, T), / = 1 ,2 and all k, ne N.

Lemma 3. I f series (5) is convergent in (0, T), then the series 00 dn

(6) I

k= 1 Ш

for every / = 1 ,2 and n e N, is convergent in (0, T) also.

Le m m a 4. If, for a fixed i and n, series (6) is convergent at t = p, p e (0, T), then it is absolutely and uniformly convergent in every interval (0, by, b < p.

Applying Lemma A, Lemma 3 and Lemma 4, we immediately obtain:

Le m m a 5. Suppose that (3) is such that series (6) with i = 1 and n = 0 is convergent in <0, T). Then the functions U1 and U2, 4efined by (4) in <0, T),

have the derivatives of all orders in <0, T). Moreover, d" 00 dn

dt _{k= i} dtr_Ui

for re<0, T), i = l , 2 and every neN.

Lemma 6. I f series (6), with i — 1 or i = 2 and n = 0, is convergent in

<0, T), £/ien

00

X /с”е- *7> к| < 00, n = 0, 1, ...

л= 1

Now we shall prove the following lemma:

00

Lemma 7. / / f/ie series ]Г /се~кГ|мк| is convergent, then the series k= 1

Z 4: (4° (0 “ ^ ~ T)) uk (i = 1, 2) k=o“t

is absolutely and uniformly convergent in <0, T>.

Proof. Let i = 1. By (1),

dt(4 1>(0-е«*-г>)цк /с|мк| _*(* - Г)'

t f T + e~kT

e ~ kt e k(t~ 2T)

= к \uk\ -^f-f e-k:r < 2/c Ы e_fcT

for re < 0, T ) and к ^ 1.

From this and our assumption we obtain the desired result.

The proof for i = 2 is similar.

1.3. In this section we shall give some remarks on the absolute sum­

mability of numerical series.

Definition 1. Series (3) is absolutely summable by the method (Л,), i = 1 or i = 2 (shortly is \Ai\-summable) if series (4) is convergent in <0, T) and if the integral

(7)

T

s dt

is convergent.

The absolute summability by the classical Abel method was considered in [1]. The following lemma is easily proved:

Lemma 8. Series (3) is absolutely summable by the classical Abel method

(is \A\-summable) if and only if there exists V(t) (defined by formula (5)) in Л), T) and if the integral

(8)

dV dt dt is convergent.

From Definition 1 we immediately obtain the following theorem:

Theorem 1. I f series (3) is |A^-summable, i = 1 or i = 2, then it is (Л,)- summable.

Applying Lemmas 5-8, we shall prove the following theorem:

Th e o r e m 2. Series (3) is absolutely summable by the method (Л,), / = 1 or i = 2, if and only if it is absolutely summable by the classical Abel method.

Proof. Suppose that (3) is |Л|-8иттаЬ1е. Then, by Lemma 8, the integral (8) is convergent.

By Lemma A there exist U Y(t) and U2(t) in <0, T), defined by (4).

Applying Lemma 1 and Lemma 5, we get

for t e <0, T) and / = 1,2. Moreover, by Lemma 6 and Lemma 7, the functions — ( L/f (t) — K(t)), i = 1 and i = 2, are bounded in <0, T).

Hence,

T d ^T dV ^{T d}

bf 7 , u ' « dt ^ \

b dt dt+ \

b

C! . 1

for / = 1,2.

Similarly we can prove that the ^,|-summability (/ = 1 or i = 2) of (3) implies the ^|-summability of (3).

From Theorem 2 it follows the following corollary:

Co r o l l a r y 1. Then methods |A t \, \A2\ and \A\ of the summability of numerical series (3) are equivalent.

It is known ([1]) that the absolute summability of series (3) by the Cesàro (C, l)~method implies the absolute summability of (3) by the Abel method. This fact and Theorem 2 imply the following:

Co r o l l a r y 2. I f series (3) is absolutely (С, 1 )-summable, then it is |Л,|- summable, / = 1,2.

1.4. Let S„ be the и-th partial sum of series (3).

Definition 2. Series (3) is strongly (Af-summable, i = 1 or i = 2, to S and with an exponent q > 0 (shortly (A^-summable) if the series

00

(9) H ,( f )= I Wi,(0 - 4 iV1(l)l|St -S |«

k = 0

is convergent in <0, T) and if lim H,-(t) = 0.

t ->т —

We shall say that series (3) is (A)q-summable, q > 0, to S if (3) is strongly Abel-summable to S and with exponent q > 0 ([4]), i-e.,

00

lim (1 - r) X rk \Sk — S\q = 0.

r - » l - k = 0

We observe that the following lemma is true.

Lemma 9. Series (3) is (A)q-summable, q > 0, to S if and only if

lim (l—e‘~ T) f e*(f- r ) |Sk- S | 9 = 0.

t - > T - k = 0

Applying (9), Lemma В and Lemma 9, we obtain the following theorem:

Theorem 3. I f series (3) is (A)q-summable, with an exponent q > 0, to S, then it is (A^-summable, / = 1,2, to S.

2. The absolute and strong summability of orthogonal series.

2.1. Let ((p„{x)} be an orthonormal system on the interval <0, 1). We shall consider real orthonormal series

00 00

(10) X c k < P k ( x ) with X cfc2 < 0 °-

k =0 k = 0

Let as usual S„(x) be the n-th partial sum of (10).

Since, for series (10), ck (pk{x) = ox(l) holds almost everywhere in <0, 1), there exists the functions t/f, / = 1,2, defined by the formula

00

(11) [/,.((, x )= I 4 " (t)<•„%(*)

k = 0

for t e <0, T) and almost every x e <0, 1 >.

In [7] there is proved that if the coefficients of (10) satisfy the condition

00 2n + 1 - l

(12) Z C I c*2r < 00,

n = 0 k = 2"

then (10) is absolutely (C, l)-summable a.e. in <0, 1). This result and Corollary 2 imply the following theorem:

Th e o r e m 4. I f the coefficients of series (10) satisfy condition (12), then (10)

is \Ai\-summable, i = 1, 2, almost everywhere in <0, 1), i.e., the integral

is convergent a.e. in <0, 1 ).

0i = U 2)

2.2. Applying Theorem A and Theorem 3, we shall prove the following theorem:

Theorem 5. I f series (10) is (A^-summable, i = 1 or i = 2, to S(x) almost everywhere in <0, 1), then it is (A^-summable to the same sum almost everywhere in <0, 1 ) and for every q > 0, i.e.,

lim £ 14° (0 - 4%. (01 |S* M - s (x)|« = 0 t -*T — <£=0

for / = 1,2, q > 0 and a.e. in <0, 1 ).

P roof. Let i = 1. The (yl,)-summability of (10) a.e. in < 0,1) and Theorem A imply the Abel-summability of (10) a.e. in <0, 1) and to the same sum.

In [4] there is proved that if (10) is Abel-summable to S(x) at almost every xe<0, 1), then it is (^-sum m able to the same sum a.e. in <0, 1) and for every q > 0.

Now applying Theorem 3, we obtain our theorem.

The proof for i = 2 is analogous.

2.3. Finally, we shall give an approximation theorem of the Leindler type for series (10).

By the Riesz-Fischer theorem series (10) is convergent in the mean to a square integrable function /. Hence, by (1), (2), (10) and (11), we get,

U,(t, x ) - / ( x ) = £ (4 ° ( t)-4 '!n W)(St (x )-/(x ))

k = 0

for t ^e <0, T), i = l , 2 and a.e. in <0, 1 ).

Using Lemma В and arguing as in paper [6], we obtain the following theorem:

Theorem 6. I f the coefficients of series (10) satisfy the condition

00

£ cl k2a <oo (a > 0),

k = 1

then

TY) if 0 < a < 1, Vi(t, x ) - f ( x ) = <ox( ( l - e , " T)|lo g (l—el - r )|) if a = 1,

(Ox( l - e ‘~T) if a > 1

T —, i = l , 2 and almost everywhere in <0, 1).

References 1

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