У У I 1 1 Some properties of summability methods of the Abel type. IV

ANNALES SOCIETATIS MATHEMAT1CAE POLONAE Series I: COMMENTATIONES МАТНЕМАПСАЕ XXVI (1986) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria 1: PRACE MATEMATYCZNE XXVI (1986)

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

Some properties o f summability methods o f the Abel type. IV

Abstract. In the paper will be considered the summability o f double numerical series and double Fourier series o f continuous functions by the methods o f the Abel type introduced in paper [2].

1. Summability of double numerical series. Let R be the set of all real numbers, C°°<0, 1) be the class of all real functions defined in <0, 1) and having the derivatives of all orders in <0, 1 ), and let N be the set of all non­

negative integers. Let peR and

(

1

1

( - lo g Г

if РФ o, if p = 0 for re(0, 1). As in paper [2] we write

D°’ph(r) = h(r),

(1.2) rv(r; p) d .

Dmph{r) = D m- 1’ph(r) + —^ - ~ f ~ - D m~Uph{r)

m dr

(re(0, 1)) for m = 1, 2, ..., p e R and ZieC00 <0, 1).

We shall say that a real numerical series (1.3)

CO

I

1 = 0

is summable to U by the method (A; m, p; n, q) of the Abel type (m, neN,

OO CO

p, q e R ) or shortly {A; m, p; n, q)-summable to U if the series У У uktrksl is fc = о 1=0

convergent in Q = {(r, s): r, se(0, 1)] and if the function

(1.4) H ( r , s; m, p, n, q) = £ £ Dm'p(rk)D n'q{sl)ukl

k= 0 1=0

((r, s)eQ) satisfies the condition

(1.5) lim H(r, s; m, p, n, q) = U.

r,s-+1 -

108 L. R e m p u l s k a

The Abel method ([3], [5], p. 160) of the summability of series (1.3) we shall denote by (A \ 0, 0; 0, 0). The function H defined by (1.4) we shall call the (A; m, p; n, q)-mesm of series (1.3). From (1.4) it follows that the (A; m, p; n, q)-mean of (1.3) is defined in Q if the Abel mean of (1.3) is defined in Q. Moreover, definitions(1.4), (1.5) and (1.1), (1.2) formeiV and peR imply the linearity of all methods ( A ; m , p ; n , q ) . In view of (1.2) and (1.4),

(1.6)

rv(r; p) 8

H{r, s; m + 1, p, n, q) = H{r, s; m, p, n, q) + --- — — Я(г, s; m, p, n, q) m + 1 dr

and (1.7)

H ( r , s; m, p, n + 1, q) = H(r, s; m, p, n, q) + sv^s’ ^ -|-Я(г, s; m, p, n, q) n + 1 cs

for (r, s)eQ (m, neN, p, qeR).

Applying relation (1.6) we shall prove

Lem m a 1.1. I f the means (A; m+1, p; n, q) and (A; m, p; n, q) (m, neN, p, q e R) of series (1.3) are bounded in Q0 = {(r, s): 0 < r0 ^ r < 1, 0 < s0 ^ s

< 1} and if series (1.3) is (A; m+1, p; n, q)-summable to U, then it is (A; m, p; n, q)-summable to U.

Proof. Consider the case p = 0. By (1.1) and (1.6) we have ( 1.8)

r log r 8

H(r, s; m + 1, 0, n, q) = H(r, s; m, 0, n, q)---— — H ( r , s; m, 0, n, q) m+1 8r

for (r, s)e<2o and H (r 0, s; m, 0, n, q) = / (s; m, n, q) for se <s0, 1), where / is the function bounded in <s0, 1). The function H satisfying equation (1.8) in Q0 and the condition H (r 0, s; m, 0, n, q) = f ( s ; m, n, q) we can write

H(r, s; m, 0, n, q) = (log r)m+1 H { t , s ; m + l , 0 , n , q ) -(m + 1 ) --- :— -+2---“ t +

1 t ogm+2t

r0

+ log m V 0/(s; m, n, q)}.

Since the (A; m + 1, 0; n, q)-mean of (1.3) is the function continuous and bounded in Q0 and series (1.3) is (A; m + 1, 0; n, g)-summable to U, we get H(r, s; m + 1,0; n, q) = U + F(r, s; m, n, q), where F is the function

Summability methods of the Abel type. JV ¹⁰⁹

continuous and bounded in Q0 and lim F(r, s; m, n, q) = 0. Hence

r fs - * l —

lim

r,s 1 '

(log r)m+ 1 F(t, s; m, n, q) t log^{m + 2} dt = 0 ro

and lim # (r, s; m, 0, n, q) = U. The proof is completed.

r,s -*1 —

The proof in the case p # 0 is similar.

Analogously, applying (1.7) we obtain

Lem m a 1.2. // the means (A ; m, p; n + 1, q) and (A; m, p; n, q) (m, n e N, p, q e R ) о/series (1.3) are bounded in Q0 = {(r, s): 0 < r0 ^ r < 1, 0 < s0 ^ s

< 1 } and if series (1.3) is (A; m, p; n + 1, q)-summable to U, then it is (A; m, p; n, q)-summable to U.

In paper [2] was given

Lem m a 1.3. I f h e C ® <0, 1), peR and m — 1 ,2 ,..., then

m dk h m~ к

Dm pli(r) = h(r)+ £ (rv(n p)f X Bj(k, m, p)i^(r; p)

*=1 J=0

/or re(0, 1), where Bj(k, m, p) are some numbers depending on k, m, p only and such that B0(k, m, p) > 0.

Moreover, in [1], [2] was given the following formula

к - 1 , ..

Dm l (1) = 1, Dm’1(rk) = l - ( l - r ) m+1 £ Г J V ' j = о ' m / for к = 1, 2, ... and m — 0, 1, ...

By Lemma 1.3 and (1.4) we obtain

Co r o llar y 1.1. I f m, ne N and p, q e R , f/ien

(1.9) H (r, s; m, p, n, q)

m л

= Z Z p))*(si;(s; <?))' fc= 0 /=0

' л* + /

drk ds1^ ^ ' S ’ ^ Fkl(r, s; m, n, p, q) for (r, s )e g = {(r, s): r, se(0, 1)}, where F 00(r, s; m, n, p, q) = 1 and Fk/ are the functions continuous in Q and lim F kl(r, s; m, n, p, g) = /kJ ^ 0.

r,s “ ► 1 —

Applying Lemmas 1.1-1.2 and (1.1), (1.9), we obtain

Theorem 1.1. I f series (1.3) is (Л; m, p; n, q)-summable to U {m, neN\ p, qeR) and its (A; k, p; /, q)-means, /с = 0, l , . . . , m and l

= 0, 1, ..., n, are bounded in Q0 = {(r, s): 0 < r0 ^ r < 1, 0 < s0 < s < 1],

п о L. R e m p u l s ka

then the Abel mean satisfies the conditions t

(1.10) lim ( l - r ) * ( l - s ) '^ ^ i H ( r , s ; 0 , 0 , 0 , 0 ) U if k = l = 0,

= J 0 if к = 0, 1, . . m; / = 1, 2, . . n, I or к = 1, 2, ..., m; / = О, 1, ..., n.

Conversely, if the Abel mean of series (1.3) satisfies conditions (1.10) for some m, neN, then this series is summable to U by every method (А; к, p; l, q) with p, qeR, к = О, 1, ..., m and 1 = 0, i, ..., n.

Remark. There exist series (1.3) {A \ m, p; n, g)-summable (m, n e N and p, qeR) and

Г having the mean (A: m+\, p: n, q) bounded in Q0

= {(r, s): re <r0, 1), se <s0, 1)) but not {A; m+ 1, p; n, g)-summable,

2° having the mean (A; m, p\ n + 1, q) bounded in Q0 but not (A; m, p; n+ 1, </)-summable,

3° having the mean (A ; m + 1, p; n + 1, q) bounded in Q0 but not (Л; m + 1, p; n + 1, g)-summable.

We shall prove 1°. Consider series (1.3)

00 00 00 00 00 00 /£k + i p \ J

Z I “K = I 1"ч(т' п)=1 Z „ГГ/7

lt=0 i=0 k=0 1=0 k=0^/=0\dr os /r=ok.l.

s= 0

where

1 1

F(r, s) = (1 — r)m+1 (1 — s)"sin--- sin--- for r, se <0, 1) and m , n e N . 1—r 1 — s

Since the Abel mean of this series satisfies the condition H(r, s; 0, 0, 0, 0)

= F(r, s) for r, se(0, 1), we obtain

(f + l dk I 1 ) dl \ 1 1

s;

,

for к, le N and lim H(r, s; 0, 0, 0, 0) = 0.

r . s - l -

я* + /

Hence the function X kl(r, s) = (1 — r)*(l — s)1 -r-,^-,H(r, s; 0, 0, 0, 0) is crK os

bounded in Q0 if к = 0, 1,..., m + 1 and 1 = 0, 1, ..., n. Moreover, lim X kl (r, s) = 0 for к = 0, 1, ..., m and l = 0, 1 and

r,s —►1 —

lim X m + l n(r, s) does not exist. This proves, by Theorem 1.1 and (1.9), that

r,s -> 1 -

if p ,qeR, then the function H(r, s; m + 1, p, n, q) is bounded in Q0 and lim H(r, s; m, p, n, q) = 0 and lim H(r, s; m+ 1, p, n, q) does not exist.

r.s 1 - r.s -* 1 —

Summability methods of the Abel type. IV 111 Now we shall give the properties of the function Dm,ph, which will be applied in Section 2. Using the induction, we can prove

Lemma 1.4. //'/ieC®(0, 1), pe R and m = 1, 2, . . . , then

m dk h

/)"•'(( 1 — r)A(r)) = £ rkWk( r ; m , P) - j (re(0, 1 »,

к — 0 d t

where Wk are some functions of the class C°°(0, 1) depending on the

( dk \

parameters m, p and such that —r W.{r\ m, p) = 0 for j, к — 0, ..., m.

\dr Jr= i

Lemma 1.4 and the Taylor formula lead to

Co r o l l a r y 1.2. I f meN, pe R and r0e(0, 1)., then (1.11) |Dm-p( ( l - r ) r k)| 5$ M(m, p, г0) ( 1 - г ) т^1(к + \)тгк for к = 0, 1, ... and re ( r 0, 1), where

M(m, p, r0) = max

O^j^m max

Of)-1)

Jn+1

p) Moreover, Lemma 1.3 implies

Dm p(rk) -► 0 if к -* oo and re(0, 1),

П 12) ®

’ £ Dm,p(rk — rk + 1) = 1 if r e (0, 1)

k = о

for m eN and peR.

2. The ( A ; m , p ; n , q ) -means of double Fourier series of continuous function. Let C2n be the class of all real functions of two variables, 2k- periodic with respect to each variable and continuous everywhere. Let

(2^.1⁾

cok(u, 0;/) = sup <max

|J|| ^ « ( x.y Oh

Z (~1 r j ( . ) f ( x + j h , y )

j —0 4 /

(0, v ; f ) = sup I max £ ( - 1 )k~j ( k) f ( x , y+jh)

|й|$г ( x.y /=о V/'

(k = 1, 2, ...) be the partial moduli of smoothness of the function / e C 2n and let Fm,„(/) (m, neN) be the best approximation of / e C 2n by trigonometric polynomials of two variables of the order m, n at most ([4], p. 116, 126, 44).

Let akh hkl, ckl, dki be the coefficients of double Fourier series of / e C 2n and let

Tu(A% y'J) — 4 i(aki cos kx cos ly + bkl sin kx cos ly + ckl cos kx sin ly + + dki sin kx sin ly),

112 L. R e m p u l s k a

where

i if к = / = 0,

Ak< = J j if к = 0, / > 0 or к > 0, / = 0, ( 1 if к, l > 0.

The summability of double Fourier series of f e C 2n in the point (x0, y0) 00 00

we define by the summability of numerical series £ £ Tkl{x0, y0',f)- The

k=01 ⁼ 0

(A; m, p; n, q)-mean of the double Fourier series of f e C 2„, i.e., the function

H(r, s; x, y; m, p, n, q j ) = £ f D ^ ( r k) D ^ ( s l) Tkl(x, y ; f )

k=0 1=0

(r, .se(0, 1), x , y e R ) we can written in the form

П It

H(r, s; x, y; m, p, n, q , f ) = \ f (x, у ) К тгР(г, u - x ) K „ >q(s, v-y)dudv

with

(2.2)

K„,,p(r, t) = i + £ Z)m p(rk)cos kt,

k= 1

ao

K„>4(s, n) = ? + 2 /)л'4(5Л)С08 ku.

k= 1

Moreover, applying the Abel transformation, we have

00 00

H{r, s; x, y; m, p, n, q , f ) = X Z £>M’p(r * - r ‘ + l) D n'q(sl - s l + l)Skl(x, y ; f ), fc= 0 /=0

where Sk/(x, y ; f ) = £ £ Ти(х, y ; f ) . '

i = 0 j = 0

We shall estimate the quantity

max|/^;”(r, s; x, y;/)| = тах|Я(г, s; x, y; m, p, n, q , f ) - f ( x , y)|.

x, у x.y

it

Since J K m p(r, t)dt = n for re(0, 1), we get

— It

(2.3) P7,q{r, s; x, y ; f ) =

Я/»

У (и, v) Kmp(r, u)Kn>q(s, v)dudv, о о

Summability methods of the Abel type. IV 113

where

Fxy(u, v) = f ( x + u, y + i>) +

+ f { x + u, y - v ) + f { x - u , y + v ) + f { x - u , y - v ) - 4 f { x , y).

Moreover, by (1.12), (2.4) P ^ ( r , s ; x , y ; f )

QO 00

= £ £ Dm-4rt - r k+l) D n-4sl - s ‘+' ) [ S k,(x, y)\.

k = 0 1=0

Below M k(a, b) (k = 1, 2, ...) will denote positive constants depending on a, b, only.

Le m m a 2.1. I f pe R, 0 < r0 ^ r < 1 and j = 0, 1,2, then

(2.5) (1 -гУ 1^1*4 >p(r, 01 ^ Mi ij , p, r0).

о Proof. By (1.2) and (2.2),

K Up(r, I) = K 0'0(r, t) + rv{r; p)~j^ ê Ko,o(r> 0

for p e R , re(0, 1) and fe<0, тг). Since 1 — r2 K-o.o(r> 0 = 2(1 —2r cos r + r )>,--- T T ^ ’ we get

K i./Лг, 0 =

(1 — r)2 {1 — r2 + 2rt’ (r; p)] + 4r \ \—r2 — ( \ + r 2)v(r; p)) sin2(r/2) 2(1 — 2r cos t + r2)2

Observing that

1 — 2r cos t + r2 ^ j (1 ~ r ) 2 I г2/я2

for re <0, 1 — r), for f e (l — г, я ) and

| l- r 2 + 2ri;(r; p)| M 2(r0, p )(l- r ), U —r2 — (\ + r2)v(r; p)I =5$ M 3(r0, p) ( 1 r)2

8 — Pruce matematyczne 26.1

114 L. R e m p u l s ka

for r e ( r 0, 1) (p e R ), we obtain

Olrff = ( l j r+ J )t1lKUp(r,t)ldt

0 0 1 -r

1 - r n

^ M4(r0, p) { ( l - r ) -1 f tJdt + ( l - r ) 2 J (1 - r + t2)tj ~4dr]

0 l - r

< M 5(j, p, r0) ( l - / f (j = 0, 1, 2).

Hence the proof of (2.5) is completed.

Applying (2.1), (2.3) and (2.5), we obtain

Th e o r e m 2.1. I f f e C 2n and p, qeR , then

(2.6) max|Pj;J (r, s; x, y ; f )j < M% (co2( l - r , 0;/) + co2(0, 1 - s ; f ) \

x,y

for re ( r 0, 1), se <s0, 1) (r0, s0e(0, 1), M% = M 6(p, q, r0, s0)).

Proof. By (2.1),

max\FXty(u, v)\ ^ 2 \co2(u, 0 ; f ) + co2(0, v;f)}

x,y

for u, v ^ 0. Hence, by (2.3),

max|Р 1Р:Цг, s; x, y j)\ ^ (]\KUq(s, v)\dv)(]ca2(u, 0;f)\KUp(r, u)\du) +

x,y 0 0

+ ( j l x i,p(r’ u)\du)(]œ2(0, v;f )\Kl q (s, v)\ dv).

о 0

Applying the inequalities œ2(tu, 0;/) ^ (r+ l)2co2(u, 0;/) and to2(0, t v j ) ^ (t + 1)2 co2(0, v ; f ) for / e C 2n and r, u, v ^ 0 ([4], p. 116, 126), we get

J<w2(m, 0;f )\Kl p(r, u)\du ^ co2( l - r , 0;/)

j(l

0 0

я

and the similar estimation for the integral Jm2(0, v;f) \Kl q(s, t?)|dv. Now

о

inequality (2.6) follows by (2.5).

The estimation of max|Pp;4"(r, s; x, y;f)\ for m , n e N will be given in

x,y

the next theorem. In paper [3] was proved the inequality

Ik 21

(2.7) max { ( k + 1)~1 (/+ 1)” 1 £ £ (Sqix, y ; f ) - f ( x , y))2}112 ^ M 7 EKl{ f )

X,y i = k j = l

for k, 1 = 0, 1, ... and f e C 2n ( M 7 = const > 0). Applying this result, we shall prove

Summability methods of the Abel type. IV 115 Th e o r e m 2.2. I f m , n e N , p , q e R and f e C 2n, then

(2.8) max \P™(r, s; x, y;/)|

x,y

< М я ( т, n, p ,< f , r 0, s 0) ( l —r)" + 1 ( l —s)"+1 £ £ ( к + 1 Г ( 1 + 1 ) " Е и ( Л k = 0 1=0

for r e ( r 0, l ) , s e ( s 0, 1) with К = [1/(1 — r)] — 1 and L = [1/(1 — s)] — 1 (r0, soe(0, 1), [z] denotes the integral part of z).

2k' 21

Proof. Let k' = 2* — 1, U?f(r, s) = \2k + l £ £ (i+ l ) 2w(/ + 1)2W } 1/2 for k , l , m , n e N and K x = [log2 [1/(1 - r ) ] ] , L x = [log2 [1/(1 - s ) ] ] for re ( r 0, 1) and s e ( s 0, 1). By (2.4), (1.11) and (2.7),

\P^n(r, s; x, y, f) \ (1 ~ r ) m+1( l ~ s ) n+l

^ Mg(m, n, p, q, r0, s0) £ Z (k + l ) m{ l + l ) n rk sl \Skl{x, y ; f ) - f { x , y)\

k= 0 1=0

a M»(m, n, p, q, r0, s„) I I U?f(r, s) £„,,.(/) fc= 0 /=0

^1 ^*1 ^1 oo oo ^1 oo oo

= ( I I + I I + I I + I I

k = 0 1=0 k = 0 l = L x + l k = K x + l l = 0 k = K x + 1 l = L x + 1 4

■ l ï - I = 1 But

У , = S 4 £ ^£ 2 < " + l l ‘ + ( ” + l *' Jt= 0 /=0

n <£«*1 + .,-.,ц + .г(Л! I L

))T

! I

k= 0 1=0 k = K x + 1 l = L x + 1

^ M 10(m, n, r0, s0)(l —r)~m~ 1 (1 —s)~"~1 EK<L( f) ,

^1 oo 21' oo

Y2 ^ 2 £ 2(" + ш Е*..и.1 + „-(/) i L 2' £ ( j + 1)2"+ V } ‘'2 j £ 2- 1}1'2

k = 0 1=0 j = l ' l = L x + 1

* 1

< M 11(n ,s0) ( l - s ) - " - 1 I 2l” +ll,‘ £ l, (4 + „.(/),

k = 0

and, similarly

y3^M12(m, r0)(l—r)-"-‘ S 2<" + 111 £CKl

1 = 0

116 L . R e m p u l s k a

Applying the inequalities

2m— 1 2"- 1 2m + tt~2 Ezm_ 1'2„_ 1( f ) ^ X X £u ( A

k=2"»-l l=2n~1 2m— 1

I £*,0 (/)

fe= 2m~ 1 (m, n = 1, 2, ...), we obtain

л ,г 0,«о) Z t (* + 1Г(/+1)" £,.,(/) (/ = 1,2,3, 4).

fc=0 /=0

These results imply (2.8) (cf. [1], [3]).

Remark. It is known that

Еи ( Л ^ М ы (к, 0 (® 1(т ^ т ,О ;/ | + ю ,(о, т^т : / ')) for i , j e N ; к, l = 1 ,2 ,... and / e C 2„ ([4], p. 288). Hence, by (2.8),

max\Pp^(r, s; x, y;/)|

< M i5(m, n, p, q, r0, s0)(<um( l - r , 0;/) + co„(0, 1 —s;/)) for re<r0, 1), se<s0, 1), w, « = 1,2, ... and p,qeR. This proves that estimation (2.6) is not worse than (2.8) with m — n = 1.

References

[1 ] Z. D o p ie r a la , L. R e m p u ls k a , On the summahility o f series by harmonic methods, Comment. Math. 24 (1984), 15-18.

[2 ] L. R e m p u ls k a , Some properties and application o f summabilitу methods o f the Abel type, I, Functiones et Approx. 14 (1984), 17-22.

[3 ] R. T a b e r s k i, Strong summability o f double Fourier series, Bull. Acad. Polon. Sci. Sér. Sci.

Math. 18 (1969), 719-726.

[4 ] A. F. T im an, Theory o f approximation o f functions o f a real variable (in Russian), Moscow 1960.

[5 ] A. Z y g m u n d , Trigonometric series, Vol. I (in Russian), Moscow 1965.