On the summability methods connected with the Dirichlet problem for an elliptic domain

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

Ter esa Ma r k ie w ic z and Lu c y n a Re m p u l s k a (Poznan) On the summability methods connected with the Dirichlet problem

for an elliptic domain

The present paper was motivated by two articles ([2], [3]) by J. Go- rowski. In [2] there was constructed the solution of the Dirichlet problem for the Laplace equation by the application of some new summability method of series. In [3] there was compared this new summability method with the Abel method in application to Fourier series.

In this paper, we shall investigate some properties of the summability methods of the Gôrowski type for a numerical series and the Fourier series.

1. Summability of numerical series.

1.1. Let

(1 ) I Ut

k = 0

be a numerical series and let uk be real numbers.

Write

(2⁾ A t f ' W 3 1 , = cosh kt

cosh k T ’ (3)

for t e <0, T } and

A'o?>(f) s 1, A H ') sinh kt sinh kT к = 1, 2, ..., where 0 < T < oo.

De f in i t i o n. Series (1) is summable to S by the method (Л,), i = 1 or i = 2 if the series

(4) t/,(f)= X

fc= 0 is convergent in <0, T) and lim L, (r) = S.

t -*t—

The (zhl-summability of the Fourier series was considered in [3].

In this section we shall prove the equivalence of the method (Л,) (i

= 1,2) and the classical Abel method.

Below by Mk(a, b), к = 1 ,2 ,..., we shall denote some positive con­

stants depending only on the parameters a, b.

1.2. First we shall prove some auxiliary results. It is easily verified that the following lemmas hold.

Lemma 1. Series (1) is summable to S by the classical Abel method if and only if lim V (t) = S, where

t-*T-

(5) K (t)= £ e‘,,- T4 (te<0, T)).

fc= 0 Lemma 2.

i £k(t - T) ^ Д1 > ^ ^ 2ek{t ~ T), 0 ^ Л*2)(0 ^ ef(t~T){\ — e~2T)~1 for к = 1, 2, ... and £e<О, Г).

Lemma 3. There exists M 1(T) such that

00

k= 0 for р е ф , T } and i — 1, 2.

P ro o f. Let i = 1. By (2),

0 0 00

I

^ £ е-«Г+» ^

k= 1 - 1

for £G<0, T).

The proof for i = 2 is similar.

Lemma 4. I f t e ( t 0, T), f0 > 0 , then there exists M 2(t0) such that

\Ak)(t) — ek(t~T)\ ^ M 2{t0)eki,- T)( l - e t ~T) for к = 0, 1, ... and i = 1, 2.

P ro o f. We shall consider the case i = 1, because the proof for i = 2 is

similar. For t e <0, T) and к ^ 1, we get

2 k T _ 2kt

w e - * * - * i- * - V ë»t+1)

^ jM,-T>2*(gT- e')e<2t~lir

^ * e 2 k ( T + t)

< 2e*(t-T)(l — е*~т)ке~2кх.

But, if t e ( t 0, T ) , t 0 > 0, and к ^ 1, then ke~2k,^ k e 2kt° ^ M 3(t0).

Hence the proof is completed.

Lemma 5. If t e ( t 0, T), t0 > 0, then there exists M4 (t0) such that (6) 14° ( O - 4 + i (01 < M4(t0) ( i - ^ " r) ^ (," r)

for к — 0, 1, 2, ... and i — 1,2.

P ro o f. For к = 0, 1, 2, ..., t e <0, T) and i = 1, 2 we get (7) |4 “ (0 - Я а ,(г ) |

< 14° (0 — e*(' ” T)| + ( 1 — é ~ T) ek(t ~T) + |4 + ! (0 — e(k+1 )(t ~ r)| .

Now applying Lemma 3, we obtain (6) immediately.

Lemma 6. Suppose that series (1) is such that uk = o(ekT), where T is the number as in (2) and (3). Then the series

00

(8) Щ (0 = £ (Д<‘> (t) - e«> - " K (i = l. 2) fc = 1

is convergent in (0, T) and lim Wt (0 = 0.

t ->T —

P roof. Let i = 1. Since

\ u M l)(t)~ek^ \ = \uk\--- i f f --- ^ \uk\e~kTe~kt, eZK + 1

for t e(0, T) and к = 1, 2, ..., and the condition uk = o(e*T) holds, we get the convergence of series (8) for i = 1 and t e(0, T). Moreover,

со e k ( T - t ) _ J i ( t - T ) oo

(9) W^(0 = £ ---злёГГк--- uk = Z ^ (Om^ - *7,

k=l e +L k= 1

for re (0, T), where

& (0 =

, k ( T - t ) _ e k ( t - T ) Л Т

e2kT + l

We observe that

lim gk (0 = 0 for any fixed к

t -*т —

and

00 00 00

I IMOK I

k = 1 f c = l k = 1

for t e ( t 0, T), where t0 > 0. Hence, by uk =o(ekT), (9) and the Toeplitz theorem, we obtain lim Wx (t) = 0.

t - *t —

The proof for i = 2 is analogous.

From (2)-(4) we immediately obtain

Lemma 7. I f for series (1) there exists Ux(t) or U2{t) in the interval

<Ю, T), then uk = o(efcT) as к ->oo.

Applying Lemma 6 and Lemma 7, we shall prove the following

Lemma 8. Suppose that for series (1) there exists one of the functions Ux, U2 and V, defined by (4) and (5), in the interval <0, T). Then there exist also the other two of these functions in <0, T).

P ro o f. Suppose that there exists Ux (t) for t e <0, T). Then, by Lemma 7, uk = o(ekT) holds. Hence, applying Lemma 6, we get the convergence of (8) for te(0, T) and i — 1, 2,

V(t) = Ux (t) - Щ (t) and U 2 (r) = W2 (t) + V(t) for t e(0, T).

Similarly we can prove the other cases.

1.3. Applying Lemmas 1-8, we shall prove the main properties of the methods (Л,), i = 1, 2, for numerical series.

Theorem 1. The methods (Лх) and (Л2) are regular.

P roof. From (7) and Lemma 3 it follows that sup £ 14°( t ) - 4 + i {t)\ < °o

O ^ t < T k = 0

for i = 1 and i = 2. These inequalities and (2) and (3) imply the Toeplitz conditions for the methods (Лх) and (Л2). Hence the proof is completed.

Theorem 2. The method (Л,), i = 1 ,2 , and the classical Abel method are equivalent.

P roof. We assume that series (1) is ^^-sum m able to S, i.e., Ux(t) — S

= o(l) as t —*■ T —. Then, by Lemma 8, there exists V{t) for t e ( 0 , T).

Moreover, by Lemma 7 and Lemma 6, we have W1(t) = o(l) as t ->T—.

Since, by (8), V(t) = Ul ( t) - W l (t) for te(0, T), we get F (r)-S = o(l) as t - + T ~ .

Now applying Lemma 1, we obtain the summability of (1) to S by the classical Abel method.

Arguing similarly as above, we can prove that if (1) is summable to S by the classical Abel method, then it is (Л ^-summable to S.

The proof of the theorem for i = 2 is analogous.

From Theorem 2 we obtain the following

Co r o l l a r y 1. The methods (A 2) and (A 2) of summability of series (1) are equivalent.

2. Summability of the Fourier series.

2.1. Let LP2n, 1 < p ^ oo (L2n = L ln) be the class of all real functions 2n- periodic and Lebesgue integrable with p-th power in ( — n , n \ i f l ^ p < o o , and continuous everywhere if p = oo. Let f eL 2„ and let

00 00

(10) %a0+ £ (ak cos foe+ bk sin be) = £ Ak(x;f)

k=l fc= о

be its Fourier series.

Clearly, if / eL 2„, then there exists

(11) V(t, x ; f ) = X ek(t~T)Ak{x;f) k= о

for t e <0, T) and every x. Moreover, by Lemma 2 or Lemma 8, the series (12) V,(t, x-,f) = £ XP(t)Ak(x ;f) (i = 1, 2)

k = 0

is convergent for t e ( 0 , T) and x e ( — oo, +oo).

Applying the functions U, defined by (12), we obtain the Gôrowski means of the Fourier series o f /gL2„:

00

(13) U(t, x ; f ) = ^ a 0+ Y, (^кЧ1) ^ coskx + À{k2)(t)bksinkx),

k = 1

t s (0, T) and x e( —oo, +oo), considered in [2].

For / gL2„ we have

(14) U(t, x ; f ) = Uiit, x; F l)+ U 2(t, x; F2) , (fe<0, T), xg( —oo, +oo)), where

F i (x) = f( x ) + f ( - x )

and F 2 (x) = f ( x ) - f ( - x )

(15) 2 2

Applying (11), (13) and Lemma 6, we obtain

Th e o r e m 3. I f f e L 2n, then

m ax|U(t, x ; f ) — V{t, x ; f )| = o(l) as t -* T — .

X

Applying the well-known theorem on the Abel summability of the Fourier series ([5]) and Theorem 2, we obtain

Co r o l l a r y 2. The Fourier series of the function f e L ln is (Aj-summable, i

= 1, 2, to f almost everywhere.

Lemma 1 and Theorem 3 imply the following corollary.

Co r o l l a r y 3 (see [2]). I f f e L 2n, then

(a) lim U(t, x \ f ) = f( x ) if and only if (10) is summable in x to f( x ) t ->t—

by the classical Abel method.

(b) lim U(t, x ; f ) = f ( x ) almost everywhere,

t -*t —

2.2. In this section we shall give an approximation theorem for the function f e lf2n, 1 < p < oo. Let

II/(-)IL p

( J‘ \f(x)\pdx) 1,p if 1 < p < 00,

— n

max I / (x)| if p = oo

X

for / e l p2n. Let E„{f)LP be the best approximation of /eL^„, 1 < p < oo, by the trigonometric polynomials of the order ^ n in the metrics of U2n.

Considering the function Ut, defined by (12) for / eL2„ we remark that, by (7), Lemma 3 and Lemma 2, the equality

QO

I (4nW-4'iiW) = i

fc= 0

holds for t e <0, T) and i = 1,2. Moreover, we can apply the Abel trans­

formation in the series (12). Hence, i f / e L 2n and Sk(x ;f) = A 0(x ;f)+ ...

... -M *(* ;/), then

(16) * ; / ) - / ( * ) = f (4° M -A & , m s k(,x;f)-f{x)) k= 0

for t e ( 0 , T), x e ( — oo, -f oo) and / = 1,2.

Using (16) and Lemma 5, and arguing as in [4] (p. 48) and as in the proof of Theorem 3 in [1] (p. 207), we obtain the following theorem:

Th e o r e m 4. I f f eLP2n, 1 ^ p ^ oo, then there exists a positive constant M6 (p, t0) such that

IIu,(t, M 6(p, t0)(l —e'~T) £ E„(/)lp

k= О

for t e ( t 0, T) and / = 1,2, where t0 > 0 and N is the integral part of (1 — e*~T)~l .

2.3. Now we shall give a theorem on approximation of / e lf2n by the Gôrowski means (13).

If / e L2„, then by (14) and (15) we get

(17) U(t, x ; f ) - f ( x ) = Ux{t, x; F i) - F 1(x)+U2(t, x ; F2) ~ F 2(x) for t e ( 0, T) and every x.

From (15) it follows that

(18) EAFÙlp ^ En{f)LP

for / eLP2n, 1 ^ p ^ oo, n = 0, 1, 2, ... and / = 1,2.

Applying (17), (18) and Theorem 4, we obtain the following theorem:

Th e o r e m 5. I f f eLp2n, 1 ^ p < oo, then there exists M 7(p, t0) such that

\W(t, ■ ;/)—/(-)lltJ> « MAP, t o W - e 1- 7) I fia /liP k= 0

/or t e ( t 0, T), w/iere t0 > 0 and N is the integral part of (1 — e, - r )-1 .

References

[1] Z. D o p ie r a la and L. R e m p u ls k a , O n th e s u m m a b il ity s e r ie s b y t h e h a r m o n ic m e th o d s ,

Comment. Math. 23 (1983), 199-213.

[2] J. G ô r o w s k i, O n th e D ir ic h le t p r o b le m f o r in te r io r a n d f o r e x t e r io r o f a n e llip tic d o m a in ,

ibidem 20 (1978), 337-340.

[3] —, T h e D ir ic h le t p r o b le m f o r th e e llip tic d o m a in a n d s u m m a b il ity o f th e F o u r ie r se rie s,

ibidem 24 (1984), 257-261.

[4] L. L e in d 1er, S tr o n g a p p r o x im a ti o n b y F o u r ie r s e r ie s , Budapest 1985.

[5] A. Z y g m u n d , T r ig o n o m e tr ic s e r ie s (in Russian), Moscow 1965.

— Commentationes Math. 28.2