On the summability of series by harmonie methods

ANNALES SOCIFTATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIII (1983) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEOO

Séria I: PRACE MATFMATYCZNE XXIII (1983)

Zenon Dopierala and ^{Lucyna Rempulska} (Poznan)

On the summability of series by harmonie methods

Abstract. In this paper we shall define the n-harmonic method of summability of the series.

We shall prove that the n-harmonic method is equivalent to the Abel method (A , n). In Section 2 we shall investigate the order of the approximation of the function by the means of the trigonometric Fourier series. In Section 3 we shall give the example of the application of the harmonic mean for the Laplace equation.

1. Summability of the numerical series. Let Cx <0, 1) be the class of all real-valued functions /having the derivatives p =

0

,

1

, ..., continuous in

<0, 1). Let Dn and Dn be the differential operators defined for the functions of the class C * <0, 1) by the formulae

(1.1) D °f (r) = f{r ), D"f(r) = D"~ 'f(r)+ ^~2n Tr D"~1/(r>’

(

1

.

2

) 5 ° f(r ) = f(r ), D”/('-) = Ô”-

1

/ ( r ) + y ^ i î ; l f i " - i / ( r)

n dr

{n =

1

,

2

, ...; re<

0

,

1

» . We shall say that the numerical series

00

(1.3) I Uk

k= 0

(uk are the real numbers) is summable to s by the n-harmonic method,

00

w = 0, 1 ,... (is (H, n)-summable to s) if the series £ rkuk is convergent fc= о

in <

0

,

1

) and if

(1.4) H (r;n)^>s if r —► 1 —,

where

H (r; и)= I

k=0 (re<

0

,

1

)).

(1.5)

200 Z. D o p i e r a l a and L. R e m p u lsk a

We shall say that series (1.3) is summable to s by the Abel method of

00

the order n, n = 0, 1, ... (is (A, n)-summable to s) if the series £ rkuk is con­

vergent in <

0

,

1

) and if k~°

(1.6) H (r;n )-+ s if r —► 1 —,

where

(1-7) H (r ;n )= f Dn(rk)uk (re<0,1)).

*=o

The function H (r; ri) (H (r; «)) will be called the (Я, ri) mean ((A, n) mean) of series (1.3). The definition of the (A, ri) method is equivalent to a definition given in [5].

From (1.1), (1.2), (1.5), (1.7) follows

(1.8) H (r; ri) = H {r; и -1 ) + ^ ^ — y- H (r; n - 1),

2

n dr

(1.9) Й(г;п) = Й ( г ; п - 1 ) + У ^ ^ ^ - Я ( г ; п - 1 ) n dr

(n = 1 ,2 ,..., re < 0, 1)) and H (r; 0) = H {r; 0).

The two methods of summability are called the equivalent methods if the summability of series (1.3) by one of those methods implies the summability of (1.3) to the same sum by the other method (cf. [2]). We shall prove that the methods (Я, n) and (A, ri) are equivalent.

First, we shall give the some auxiliary results.

Lemma 1. I f f eC°° <0, 1) and n ⁼ 1 , 2 ,. .. , then

(1.10) Dnf (r) = f( r ) + £

[(1

- r

2

)r]‘ ^ 7 Z Bp(k, n) (1 - r 2y,

fc=i a r p=0

(1.11) D " f(r )= f(r )+ £ [(1 - r ) r ] ‘ "Z Bp(k, n) (1 - r f

. * = 1 a r p = 0

(re<0, 1)), where Bp(k, ri), Bp(k, n) are some real numbers and B0(k, n), B0(k, n) > 0 for 1 ^ к < n.

Proof. We shall prove (1.10). We use induction. Relation (1.10) for n

= 1 is obvious. Suppose that (1.10) is true for n = 1, 2, ..., N. We shall prove (1.10) for n = jV+1. By (1.1) and (1.10) for n — N, we have

D"+1f(r ) = f( r ) + £ X ( l - r

2

)r]* £ £ Y Щ 1 - г 2Г +

k = 1 a r p = 0

Summability o f series by harmonic methods 201

+ < ! у

+ 2(N + \)dr ^{^ k%} 2 (N + 1 ) i t r%

JV Г(\—r 2)r~\k+ l dk+l f N~k

y LU n r J---O J BJ k , N) (1 —r2f — it",

2

(JV+

1

) Л *+ , Д P

N r 2 г/ 1 — „2\ „п Ь Jft f N — к

АН- 1

/М + Z И

4

-(г; Л

0

^

fc =

1

^drk

where

Wi(r; JV) = ( l - r 2)r

1

^{N — 1}

2{N + ¹⁾ + У Bp(l, N) (l — r2y +^{p= о}

N - 1

+

1

1 — 3r: /v-l pr2

2(N +1)

WN+l(r; N) =

Bp( l , N ) ( l - r 2Y - £ J L ^ B p( l , N ) ( \ - r 2f } ,

p= i * + i

[ ( l - r

2

)r

] iV+1

2

N+1

(JV + 1)! ’

^ ( r ; JV) = [ ( l - r

2

)r]fc l B 0(k, N) + k} \ T ЗГ' } Б

0

(/с, N)+ - °}k„ U N ) + 2(JV+1)

N - k

+ Z t1 - ' - 2)'

p= 1

^ N ) H (

1

:

3

:

2

) -

2

pr

2

^ N ) + ^

2(JV+1) Bp(/c-l,iV)

2

(N +

1

)

2

(N +

1

) +

I Вы+1-к(к ^ ^ _ г

2

^лг+ l -л 2(JV + 1)

= [ ( l - r

2

2, J 2 (JV + 1 )r] - k) B0(k,N ) + B0( k - 1, AT) 3/сЯ0 (/c - 1, JV) /1 2ч

j

2(N +

1) +

2(N+

1)

( l - r 2) +

N - k

+ z

(1

- r 2f

p= 1

2 ( N + l—k — p )B Jk , JV) + fip (/ c -l, JV)

+ 3<: + 2L B (k, N )(l—r2)

2(N + 1) p +

2(JV+1)

Ву+

1

- к ( к -

1

,Л

0

2(JV+1)

+

(

1

- r 2)2\ЛГ+ 1 - к

We remark that, for 1 < к ^ JV,

N + 1 - k

И4(г; JV) = [(1 - г 2)г]‘ J B,(fc, N + l ) ( l - r 2y,

p = 0

202 Z. D o p i e r a l a and L. R e m p u lsk a

where

B0{ 1, N + l) =

2

jVB

0

(l, N )+ l 2(N +1)

N — p 2p+\

в р(

1

, N + l) = B ,(l, Л

0

+ ^ T T T T ( ! ’ * ) ,

Д0( * ,Л +

1

) =

ЛГ

+1 7 2

(Af+l)

2 ( N + l - k ) B0(k, N) + B0( k - 1, AT)

2

(iV+l)

D/,

1

, Bp(/f-l,N ) + 2 ( N + l - * - p ) f i p(fc,A0 + (3^ + 2 p - 2 ) B ^

1

(fc,N)

* ' (* ’ N + 1 ) e ---

2

(ïvTT)---

BN+1- k(k ,N + l) = Consequently,

(1

< p ^ N-fc), gjv+i - fc( ^ - l , N) + (ЗА: + 2N — 2k) BN_k(k, N)

2 (N + l)

Di,+

1

/ W = / W +

T

* В Р№,ЛГ-И)(

1

- Г2г

k= 1 _{P = 0}

and B0(k, N + l) > 0. Hence, formula (1.10) is true for n = 1, 2, ... Similarly we can prove (

1

.

11

).

From Lemma 1 the following corollaries can be immediately obtained.

Corollary 1. I f the series X ifuk is convergent in <0, 1), then

k = 0

(1.12) H(r ;n) = H(r ; 0)+ £ [(1 - r 2) r f —^ X 5p(/c, w) (1- r

2

f,

*=i “r p= о

(U 3) Й(г;п) = Я(г;0) + f [ ( 1 - » ] ‘ ‘'*Я,1' ;01

" f B„(k, n ) ( l - r Y

k = 1 ' ilr p= 0

(re<0, 1)), where Bp{k, n), Bp(k, n) are some real numbers and B0(k, n), B0(k, n) > 0 fo r 1 ^ к ^ n.

Corollary 2. For any fixed n, n = 0, 1, ..., and re< 0, 1) we have

00

(1.14) L > V )-> 0 if к -+ oo; X # V - r k + 1) = 1.

t=о Using the induction we can prove

Summability o f series by harmonic methods 203

Lfmma 2. I f к, n = 0, 1, ... and re < 0, 1), then

Dn(rk) - D n(rk + l ) = ( l - r

)"+1

^ + rfc,

П

Dn(rk) — Dn(rk+i) = (1 — r)”+

1

rk X Lp(r;n)/fp,

p= о

where Vp(r ; и) are some algebraic polynomials o f the variable r o f order ^ n, whose coefficients are dependent on n only.

Hence, for к = 0, 1, . . r e <0, 1) and a fixed n, (1.15) \Dn(rk — rk+l)\ ^ M{n) (1 —r

)”+1

rk(k + l)n, where M(n) is some positive constant depending on n only.

Now, we prove the relation between the methods of order n and n+1.

Lemma 3. I f series (1.3) is (H , n+ 1 )-summable to s ((A, n + 1 fsummable to s), then it is (H , n)-summable to s ((A, n)-summable to s).

Proof. Lemma 3 for the Abel methods is proved in [5]. Let (1.3) be (H, n+ l)-summable to s. By (1.4) and (1.5).

H (r; n + 1) = s + h(r) (re<0, 1)),

where h is a function continuous in <

0

,

1

) and /г(г) —>

0

if r -> l —.

Hence, by (1.8), we get the equation

(1.16) h(r) + s = H {r ;n )+ ^ rJ u ^ r H (r ’ n)

2

(n+

1

) dr (г е <°Л)).

Let roe(0, 1) and H0 = H(r0; n). The function

H{r ; n) =

l _ r

2

J + l

2(n+

1

) (h(t) + s)t2n+1

(1

- t 2)n+2 dt “Ь r²0

1

— r₀²

is the solution of the linear differential equation (1.16) in <r0, 1). Since

.2 n+ 1 n

(1

- n^2\n+2dt

( - 1 ) *

i Z — r 1

=

0

n

+1

-{(1—r2) * - " - 1—(1 —rgy„2\ k -n - 1 \

r 0) i

204 Z. D o p i e r a l a and L. R e m p u ls k a

and

ro

then H {r\ n )-*s if r - + l —. The proof is completed.

Now we shall prove the theorem on equivalence of methods.

Theorem 1. The methods ^(Я, n⁾ and (A, n^{) (}n is a fixed number⁾ o f summability o f series (1.3) are equivalent.

Proof. The equivalence of the methods (H , 0) and (A, 0) is obvious.

Let us suppose that series (1.3) is (Я , n)-summable to s, i.e. H(r;n)-^>s if r —► 1 — (n ^ 1). By Lemma 3 we get

(1.17) H ( r ;p ) ^ s if r -► 1 - (p = 0, 1, ..., n).

From (1.12) and (1.17) for p — 1 it follows

(1.18) (1 — г) Я (r ; 0) -> 0 if r -> 1 — . dr

Further, by (1.12), (1.17) for p — 2 and by (1.18), if r —>

1

—.

if r -»

1

— {p = 0, 1, ..., n). From (1.17), (1.19) and (1.13) follows

H (r; n) -> s if r -*■

1

—, which proves that (1.3) is (A, n)-summable to s.t

Similarly, using Lemma 3 and (1.12), (1.13) we can prove that the (A, n)- summability of series (1.3) to s implies the (Я, n)-summability to s.

Theorem 2. I f series (1.3) is summabte to s by the Cesàro method (C, 1), then it is (H, n)-summable to s ((A, ri)-summable to s) for any n = 0, 1, ...

Proof. The theorem for the method (T; 0) is proved in [2], p. 545, and for methods (A, n) in [5]. Hence, by Theorem 1 we obtain the (Я, n)- summability.

2. Summability of the trigonometric Fourier series. Let 1 ^ p <

00

(l}2n = L2n) be the class of all real functions 27u-periodic and Lebesgue

(

1

- г )

2

^ Я ( г ;

0 ) - 0

Reasoning as above, we obtain (1.19) ( \ - r f ~ Я ( г ; 0 ) - 0

Sumrmbility o f series by harmonic methods 205

integrable with p-th power over the interval ( — n, к ) if

1

^ p < oo and continuous everywhere if p = oo. We define the norm as usual

if 1 ^ p < GO, if p — oo.

Let Sk( x ;f) be the /с-th partial sum of the Fourier series

00 00

(

2

.

1

) i ^ o + Y (a k c o skx + bksin kx) = Y, Tk( x ; f )

k = 1 k= 0

of the function / e L 2jt. We shall say that the series (2.1) is (H, n)-

00

summable ((A, n)-summable) in x0 to s = s(x0) if the series Y, Tk(x0; f ) is

k = 0

(Я , «)-summable ((Л, n)-summable) to s. The ^{( i f ,} n) mean ({A, n) mean) of series (2.1) we shall denote by H(r, x; n ,f) (H(r, x; n ,f)). Clearly, the functions H(r, x; n ,f), H(r, x ; n ,f) are defined for any / e L 2n and n =‘

0

,

1

, ... (xe( — oo, + x ), r e <

0

,

1

)).

From Theorem 2 and the theorems on the (C, l)-summability of (2.1) ([10], t. I, p. 149-151) follow the corollaries on the (Я, n)-summability of

(2.1) .

In this part we shall give the theorems on the order of approximation of a function /eLp2k by (Я, n) and (A, n) means. We shall denote by the p-th modulus of smoothness of/ eLPln and by £„(/

1

^ the best approximation of / eLP2n by the trigonometric polynomials of order ^ tt ([9], p. 41, 116).

Below, by Mf, M,(a, /1) (i = 1, 2, ...) we shall denote the suitable posi­

tive constants — absolute or depending on the indicated parameters only.

By definition (1.5), for series (2.1), we get

0

° i f It’

(2.2) H(r, x; n, f ) = £ Dn(rk) Tk(x; f ) = - f ( x + t) K (r, t; n)dt,

k = 0 K J

— П

where

X

K(r, t; ri) = j + Y, Dn(rk) cos kt ( r e <0, 1), n = 0, 1, ...).

k = 1

Since

П

- I K (r, t; n)dt =

1

(re<

0

,

1

), n =

0

,

1

,...) , j J j If{ x r d x } 4 *

j шах I / (x)|

n

— It

206 Z. D o p i e r a l a and L. R e m p u lsk a

then

(2.3) H{r, x; n , f ) - f ( x) = - f x(t) K (r, t; n)dt к

( r e <

0

,

1

), x e ( - o c , + x ) , n = 0,

1

,...) , where f x(t) = f ( x + t ) + f { x - t ) '—2f(x ). Similarly we obtain

(2.4)

where

Й(г, x ; n , f ) - f ( x ) = - | f x(t) K (r, r; n)</f,

71

K(r, r; n) = ? + £ Dn(rk) cos /сГ. , k= 1

Clearly

(2.5)

K (r, t ; n + 1) = K (r, t ; n) +

X(r, r; n + /) = £ (r , r; w) +

( 1 — r 2) г dK(r, t; n⁾

2 ( и + 1 ) d r ’

( l - r ) r 5K(r, r; n)

И + 1 dr

(re<

0

,

1

), re<

0

, tü>, n — 0,

1

, ...).

As is well known

K ( r ,t ;

0

) = ^{1 - r 2}

2(1

—

2

r cos f+ r2) (re<0, 1), re<0, ^{7 i » .} By (2.5),

(2.6)

K ( r , * ; l )

(1

— r

2)2 (1

— r cos t)

2(1

—

2

r cos r-f r

2)2

’

^ ( r , f ; l ) =

(

1

—r

)2 (1

— r

2

+

2

r(l — r cos t)}

2(1

—

2

r cos t + r2)2

(re<

0

,

1

), re<

0

, я » . The mean H(r, x; n ,f) (similarly H(r, x; n, /)) can be written in an other form. By relation (1.10) with / (r) = rk, we can apply the Abel trans­

formation to the sum in H(r, x; n\f).

Summability o f series by harmonic methods 207

Hence

H(r, x; n , f ) = I

k = 0

{0"(г*)-В "(г*+

1

)}5 ,(х ;/ ) and, by (1.14),

00

H(r, x ; n,/ )—/(*) = X D"(r‘ - r ‘ +

1

){S » (x ;/ )-/ (x )}

(2.7) *"°

(re<

0

,

1

), x e( — oo, + oo), n =

0

,

1

, ...).

Difference (2.3) ((2.7)) in the case n = 0 is estimated in [4], [7] and in the case n — 1, / eL®„ in [1]. We shall prove the similar theorems for f e U 2n and n =

0

,

1

, ...

Theorem 3. I f f eLp2„ (1 ^ p ^ oo) and N = [1/(1 — r)], then

(2.8) НЩг; n,/) —/||^

||H (r;n,/)-/llLP

^ M(n, p) ( l - r

)"+1

X ( * + ! ) " £ * ( / ) „ , .

к =

0

(re<

0

,

1

), n =

0

,

1

, ...).

Proof. Estimation (2.8) for the Abel mean is the corollary from Theorem 1 given in [

6

]. We shall apply the estimation

2m

|| I 0 "(r* -ri + 1){S l (/)-y}||

k = m ^LP

2m

< M , (?)£„(/ )„ !(m +l) X [ D V - r

^ 1)]2}1'2

k = m

(m, n = 1, 2, ..., r g <0, 1); see [3], [

6

]) and (1.15). Setting

?' =

2

« - l and JV, = [l° g

2

([

1

/(

1

—'•)] + !)], we have

||Я(г; « ,/ ) - / !! „ < X || X D"(r‘ - r ‘ +

1

)-{St (/)-/}||„,

4 = 0 к = 4’

".

^ M 2(n, p) (1 — r ) " +1 { X 2 « “+ 11 £ , < / ) „ +

4=0

oo 2<j'

+ x ( I *

2

"+ , / T

2

£,.(/ )„}

4 = Nx + 1 k = q'

= M

2

(n, p) ( l - r

)n+1

{Z j(r; n) + Z

2

(r; n)}.

208 Z. D o p i e r a t a and L. R e m p u ls k a

Further, denoting by N = [1/(1— r)], we have

ос 2 q' oo

Z

2

(r; n) « :£ „ (/ )„ { £ I (/c + l

)2"+2

г

*}1'2

{ £ 2

- ’}1'2

, = N, + 1

<j =

0

k = q'

« M 3 ( « ) £ „ ( / ) „ } j _2n+2

j

2

/i+

2

oo

1/2 ( Z ^ f

k = 0 n - 1

= м 4 («)

Moreover, the following inequality holds

Z ,( r ; » K M s (») £ (A: + !)” £»(/)„,.

k= 0

(2.9)

Collecting the results we obtain (2.8).

From (2.8) follow the inequalities

\ l|W(r;n,/)-/||^

1 | Я ( г ; « , / ) - / | | 1/ < м 6 (я , p ) ( « „ ( i - ^ * ; / ) ^

(re<0, 1)) for / e/f2„ and « = 1 , 2 ,. .. Estimation (2.9) in the case n = 1 can be improved. First, we shall give a lemma.

Lemma 4. J/~g = 0, 1 , 2 , then

(2.10)

(l-r )-« Jf« | K (r ,t; l)\ dtsiM 7(q),

0

( 1 - r ) - ’ p ’ |K(r, t; l)\dt « Afs (<j) ( r e <0, 1)).

0

Proof. We shall prove (2.10). By (2.6) we have K(r, r; 1) > 0 if re<0, 1), fe < 0,

7

i). Hence, by the definition of K (r, t; 1) ((2.4)), we have

Я

J K(r, t;

1

) dt = n/2, i.e., (

2

.

10

) for q = 0.

о

Clearly, for re < 0, 1),

J t9 K (r, t; l)dt = { J + { }tq K (r, t; l)dt = Ii(r, q) + I 2(r, q). -

0 ~ 0 1 - r

Using the inequalities

1

— r cos r ^ l — r + r

2

^

2(1

- r )

5 1

if re<

0

,

1

— r), if t e {

1

—r, n), 1 — 2r cos t + r2 ^

(1

~ r)2

t2/n2

if t e ( 0 , 1 — r), if re (l — r, n)

Summability o f series by harmonic methods 209

(re<

0

,

1

)) we get

1 - r

r . . . f*(l — r)2

{1

- r

2

+

2

r(l —r cos t)}

I i ( r , q ) = ---Z

2(1 7

j— =--- —

2

r cos t + rzy

— 272

---~ dt

3(1 ~ r f

0

1 - r

^ 3 (1 —r

)~1

J tqdt

о 4 + 1

I 2(r, Ч) ^ 27i4(l — r)2 J tq

4

(1— r + t2)dt

1 - r

t Юл

(1

— r)2 J t~2 dt

< ^{1 - r}

27u4(1 — r

)2

{ f (

1

— r)t 2dt + J dt]

^

10

я

4(1

—r)q.

Hence, inequality (2.10) is satisfied. The lemma is proved.

Applying Lemma 4 we shall prove Theorem 4. I f f LP2n, 1 ^ p ^ oo, then

if q =

1

if q =

2

(2^.11⁾ IIH ( n l , / ) - / ! ! „

II й (г;

1

J ) - f \ \ lF < M9 -w2( l - r ; f ) LP (r e <

0

,

1

)).

Proof. This theorem for H (r; 1,/) and p = ^oo is given in [1]. We prove (2.11) for the Abel mean. Using formula (2.4) and the inequality

®

2

( U / ) ^ < ( l+ w

2

( i - r ; / ) tf ([9], p. 116) we get

ll#(r; l,/ )-/ ll^ ^ J c o A r J ^ K i r , t;

1

)|dt

о

J \K(r,t;l)\dt.

0

< ш

2

(

1

- г ;/ ) ^ j

^1

+ pZ

Now, applying (2.10) we obtain (2.11). The proof for the harmonic mean is similar.

2 — Prace Matematyczne 23.2

210 Z. D o p i e r a l a and L. R e m p u ls k a

3. Application. Let A = A1 be the Laplace operator and let An — A A

n_1

(n = 2, 3, ...). We shall consider the real functions и of two variables r, q>, v/here r, (p are polar coordinates. The class of all functions и with the derivatives of order ^ m continuous in the set P shall be denoted by Cm(P).

Below, we shall apply some properties of the operator _ d2 I d I d 2

^ dr2 + r dr + r

2

d(p2'

If u, veC°°{\rel<p\ ^ 1), then for n = 1, 2, ... hold the relations:

1° A[_u{r, (p)-v(r, (pY\ = [_Au(r, (p)] v(r, (p) + [Av(r, (p)'] u{r, <?)] + du dv 2 du dv

+ 2— -- ---1- ~ 2 —

dr dr г счр dcp du

J

d

1

du

2

d2u

~ [A u (r , + +

3° An[u(r, cp) + v (r, ç>)] = Anu(r, (p) + Anv (r, (p), 4° An(Àu{r, (p)) = XAnu(r, (p) (Л = conft),

r ) = r —— [Anu(r, (p)~\ + 2nAnu{r, (p), or ) dr

6

° An(r2u(r, <p)) = 12nAn~1u(r, cp) + r2Anu{r, q>)+ *

+ 4n r— \_A” 1u(r, (рУ\ —

8

d" *и(г, (p), dr

( ôu\ Ô

r

3

-^ ) = 2nr2Anu(r, cp) + r3 — [Anu(r, </>)] +

dr J dr

+ 2(n— 1) (12n —

8

) A" 1u(r, (p) + 4nr2 — [An 1u(r, <p)] +d2 dr

+ $(n2 + n— l)r — [An~ lu{r, (p)~] (A°u = u).

dr Now we shall give two lemmas.

Lemma 5. Suppose that u eC

2”+3

(\rel<p\ < 1), n = 1, 2, ... I f a function и is a solution o f equation AnX(r,(p) = 0 in the unit disc \reltp\ < 1, then the function

(1

+ r 2)r du (3.1) ' v(r,<p) = u(r,(p)+ — --- — is a solution o f An+1X (r, (p) = 0 in the unit disc \rel<p\ < 1.

) Summability o f series by harmonic methods 211

Proof. Applying Г -7

0

we have An v(r, ip) = An u{r, q>)+— {A

1

2 n

n +

1

^ÔU dr

ôu Hr

,, i f dAn+lu(r, ip) , ,

= A" lu(r, ф) + — < r --- г--- - + 2(n+ 1) A"+1u(r, (p) -

2

n or

— 2 (n + l)r An u{r, ip) — 2n[12(n + 1) —

8

] d"u(r, ip) jd A " +lu(r, <p) 1(n |

1)r2

e 2A"u(r,.q>)

— Г or or

-

8

г[(п +

1)2 2

+ л] , M nli(r, ip) or

Since Anu(r, (p) = 0, we obtain An+1v(r, (p) — 0 in the unit disc |rc‘>| < 1, Thus the lemma is proved.

у Similarly, we get easily the following

Lemma 6. Suppose that u e C n+l (\rel<p\ ^ 1), n = 2, 3, ... I f du\

dr)r= ! _or

1 0

,

r= 1

then the function v defined by ^(3.1) satisfies the conditions

dv \ / dnv

H r ) r = l V > " / r = 1

0 (0 ^ q> < 2k).

L et/ eC

1

(|re,,/,| = 1),/(1, (p) = f((p), and let ak, bk be the coefficients of the trigonometric Fourier series of /. It is known that

(3.2) /(<?) = i a 0 +

X

cos к(р + Ьк sin kip)

к = 1

(<pe<0, 2n)). Since (Dn(rk))r=l = 1 for к, n — 0, 1, ..., hence we can define

00

(3.3) H(r, ip; n ,f) = ^ a0+ £ Dn{rk) (ak cos k(p-\-bk sin kip)

k = 1

(n = 0, 1, ...) for re<0, 1) and 0 < ip < 2k. Moreover, by (3.2), we have (3.4) H(\, ip; n ,f) = f(ip) if ip e <0, 2k), n — 0, 1, ...

Clearly

(3.5) H(r, ip ; p, f ) = H(r, tp; p - \ J ) + -r - r

3

d H ( r , i p ; p - l , f )

2 p dr

in the unit disc \re,<p\ ^ 1 if H (p— l,/ )e C

1

(\rei<p\ ^ 1).

.212 Z. D o p i e r a l a and L. R e m p u lsk a

Applying Lemmas 5,

6

we shall prove

Theorem 5. I f f e C 2n{\reltp\ = 1), n —2 , 3 , . . . , then the function И (n—\, f ) defined by (3.3) has the following properties:

Г Я(и — l,/ )e C "“

1

(\rei<p\ ^ 1),

2° AnH{r, cp ; n— 1 , / ) = 0 in the unit disc \re,<p\ < 1, 3° H ( l , c p ; n - l , f ) = f ( ( p ) ,

( c?H{r, (p; n - l , f )

V c r p

I = o

r = 1

if p =

1

,

2

, . . n~

1

; <pe<

0

,

2

rc).

Proof. I f/ e C

2

"(ke'1 = 1), then ak, bk = 0 {k ~ 2n) ([10], p. 72). Hence H (0, f ) e C2n~2 (\reitp\ ^ 1) and H ( 0 , f ) e C x (\rei(p\ < 1), By (1.12) wé can write

(3.6) H(p, f ) e C2n~ 2~p(\rei<p\ ^ 1) and H ( p ,f ) e C x {\rei<p\ < 1) if p =

1

,

2

, ..., n —

1

.

1

As known AH{r, <p;0,/),= 0 in the unit disc Ire'll < 1 ([

8

], p. 279).

Applying (3.5) and Lemma 5 we obtain

ApH{i\ cp: p — 1,/) = 0 in the unit disc \re,<p\ < 1 (p — 2, 3, ..., n).

By (3.5), we have

ôH{r, q>; 1,/)

dr ⁼ 0

r = 1

if ( p e (

0

, 2n).

If n -> 3, then by (3.5) and (3.6) we can apply Lemma

6

for the function и = H (p ,f), p = 1, 2, ..., n — 2. Hence

c p H ( r, 2 , f )

c r p r = 1

if p =

1

,

2

and <pe<

0

, 2n),

f dpH(r, (jo; n - l , f )

V ‘ drp

l . r °

^{if p =}

¹

^,

²

, ..., n —

1

and <p e <

0

, 2k) . These results and (3.4) prove that conditions l°-3° hold.

References

[1] S. K a n iev, On the deviation o f biharm onic function in a disc fro m its boundary values (in Russian), Dokl. Akad. Nauk SSSR 153 (1963), p. 995-998.

[2 ] K. K n o p p , Szeregi nieskonczone, Warszawa 1965.

[3 ] L. L ei nd 1er, On summability o f Fourier series, Acta Sci. Math. Szeged 29 (1968), p. 147—

162.

[4 ] I. N a ta n s o n , On the order o f approxim ation o f continuous, In-periodic function by its Poisson integral (in Russian), Dokl. Akad. Nauk SSSR 72 (1950), p. 11-14.

Summability o f series by harmonic methods 213

[5 ] L. R e m p u lsk a , On the (A, p)-summability o f orthonorm al series, Demonstratio Math. 8, (4) (1980), p. 919-925.

[6 ] —, On som e means o f Fourier series, Functiones et Approximatif) 8 (1980), p. 3-7.

[7 ] R. T a b e r s k i , A theorem o f the SteCkin and Leindler type connected with A bel summability o f Fourier series, Demonstratio Math. 2 (1975), p. 215-225.

[8 ] A. T ic h o n o w , A. S a m a rs k i, Rôwnania fizyki matematycznej, Warszawa 1963.

[9 ] A. F. T im a n , T he theory o f approxim ation o f real functions (in Russian), Moscow 1960.

[10] A. Z y g m u n d , Trigonometrib series (Russian ed.), Moscow 1965.