Approximation theorems for functions of Hôlder classes

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1991)

M. Gô r z e n s k a, M. Le s n i e w i c z and L . Re m p u l s k a (Poznan)

Approximation theorems for functions of Hôlder classes

Abstract. In the present paper we give some theorems on approximation of 2n-periodic continuous functions by general linear means of their Fourier series in metrics of special Banach spaces. For example the Cesàro (C, q) and Abel (A, p) means are considered in Sections 4 and 5.

The similar problem for the (C, l)-means was examined in [3].

Theorems on orders of approximation of functions f s H a by some special means of their Fourier series are given in [2 ] and [5].

1. Notations. Let C2„ be the set of all 2тс-periodic, real-valued functions continuous on [-7Г, л:], and let

Denote by Я®, 0 < a ^ 1, the class of those functions f e C 2n which satisfy the condition \f(x) — f(y)\ < M\x — y\a for all x, y e( — oo, +00), where M is a positive constant depending on f It is known that Я “ with the norm

is a Banach space, and Ha cz Hp if 0 < /? < a.

Denote by h*, 0 < a < 1, the class of all functions / е Я “ for which

We set H/IU* = ||/ ||h-, for f e h x (0 < a < 1).

Let h1 be the class of all functions / e C 2n having the derivative f ' e C 2n and let

(¹) Wf\\c = max\f(x)\.

(2)

II/II

«=

И“/ (x|y)|,

where

lim d “/( x I y) = 0 for every x .

(3) ll/ll* = ll/llc+ll/'llc-

It is known that Hp c= h* if 0 < a < ^ 1.

6 — Comment. Math. 30.2

Given any / e C 2n let

00 00

(4) a0/2+ £ (ak cos kx + bk sin kx) = £ Ak{x; f )

k = l fc=0

be its Fourier series and let £„(■; / ) be the nth partial sum of (4).

Let £ be a set of real numbers, let г0фЕ be a point of accumulation of E and let X — (2k(r)} be a sequence of real-valued functions, defined on E and such that

(5) A0(r) = 1 , lim Xk(r) = 1 (re£),

r~>r0

for each к = 1 , 2 , . . . and

(5') f |At (r)| < oo, re E .

= 0

We consider the means U(f, X),

(6) U(r, x; / , X) = X Ы г)Ак(х ’ /)»

k = 0

reE, x e ( — oo, +00), of the Fourier series (4) of / е С 2я.

In this paper we shall give theorems on approximation of functions / е Я “ and f Eh* by the means (6). For example in Sections 4 and 5 we shall consider the Cesàro (C, g)-means, q > 0, and the Abel {A, p)-means.

The problem of approximation of H* and h* functions by (C, l)-means of their Fourier series was examined in [3]. Theorems on orders of approximation of f eH* by some special means of its Fourier series are given in [2] and [5].

For / e C2n, by (4)-(6), we have

(7) U(r, x; / , X)-f{x) = n~l ] f x(t)K{r, t; X)dt

0

for г еЕ and every x, where

00

(8) K(r, t; X) = i + X Xk(r)coskt,

k = 1

fx(t) = f ( x + t ) + f ( x - t ) - 2 f ( x ) . We write

(9) L(r; Я) = 7i~1 j IK(r,t; X)\dt,

— П

(10) R,(x; / , 1) = U{r, x; f , X)-f{x),

(11) Rr(.x\y) = R r(x; / Л )

for r s E and all x, y.

From (7), (10) and (11) we get

(12) В Д у ) = n - ‘ J ( f J t ) - f y(tj)K{r, t; X)dt, 0

(13) \Rr(x \y )\^ 2 \\R r(- ; /,A)lie

for r e E and all x, y.

In the next sections by M k(p, ...), к = 1, 2, we shall denote suitable positive constants depending only on p, ... and possibly on M = M(f).

2. Approximation of Ha functions. It is clear that, for every fixed r e E , U(r, •; / , Л)еЯа if / е Я а, 0 < а < 1.

Th e o r e m 1. I f f e H a, 0 < а ^ 1 and 0 < fl < a, t/ien f/iere exists МДа, ff)

such that

ЦЯГ(-; / , 2)||н» « M i(a, P)||Rr(- ; / , ЩЬ~"'“(Hr; A))"'”, r e £ . P ro o f. By (1) and (2),

ЦЯД-; /,

X)\\Hfi =

1ВДу)|

\ x - y f ’ r e £ . Applying (10), (7) and (9), we get

ll*r(-; f, m h - pict(2\\f\\cL(r; reE .

For / e H a, 0 < a ^ 1, we have \fx(t)—f y(t)\ ^ 4M |x — y|a for all x, y, t.

Hence, by (12) and (9), we get

(14) \Rr(x\y )\^2 M \x -y \« L (r; X)

for r e E and all x, y.

Using (13), (14) and the identity |jRr(x|y)| = |Ær(x|y)|^/e|JRr(x|y)|1 ~plci we obtain

|Яг(х|у)| ^ M 2(a, P)\x -y |'(L (r; Л))"в||Яг(-; /, 2 ) | | ^ /а for r e £ and all x, y.

Summing up, we obtain our assertion.

3. Approximation of ha functions. It is easily verified that if f eha, 0 < а < 1 and r (reE) is a fixed number, then U(r,- ; /, Xjeh01. Moreover, if f eha, 0 < а < 1, then

(15) Ve > 0 3d > 0 Vx, у (|x -y | ^ д => |/( x ) - /( y ) | < e |x - y |a).

Applying (15) and the notations given above, we shall prove the following.

Th e o r e m 2. Suppose that X = {2k(r)}, re E, is a sequence having the properties (5) and (5') and such that

(16) L ( r ; l K M 3(l) for r eE

and, for every fixed <5e(0, тс),

(17) lim K{r, t; A) = 0 (reE)

Г-+ГО

uniformly in fe[<5 ,71]. I f f eha, 0 < a < 1, then

(18) ||Kr( s /, 2)||„«->0 as r ^ r 0 (reE).

P ro o f. By (2), (10) and (11),

(19) ||* , ( • ; / , 1)11*. = !*,(•; /, l)llc+sup reE.

х Ф у I * У I

As is well known ([6]), the conditions (5), (5') and (16) imply (20) ЦЯД-; / , A)||c -»0 as r-> r0 (reE),

for every / е С 2я.

• From (12) we get

\ R r ( x \ y ) \ ( / , « - / , ( ( ) ! |K (r, ^{t ;} 1^{) | d t} 0

for r e E and all x, y.

We choose e > 0 and apply (15). Then, for 0 < |x — yj < <5, we have

\fx(t)-fy(t)\ < 4 г |х - у Г , and hence by (16)

\Rr(x\y)\/\x-y\a < 2 M 3(X)s, reE.

In the case |x — y\ ^ <5, by (18), we have (21)

where

\К(х\у)\/\х~У\* ^ l ! + I 2> r e £ ,

I i = ^ J J f x ( t ) - f y ( t ) \ \ K ( r , t; Ц dt,

h = ^ JI/* (0 -/,(0 l|K (r, t; Я)|dt.

We remark that, for the integral J l5 we can apply (15). Hence, \fx(t)—f y(t)[

^ 4e|t|a and

Since \fx(t)—f y{t)\ ^ 8 ||/ ||c for all real x, y, t, we have l 2 * i- S - ° \\f\\c }\K(r,r, l)\dt, reE .

П Ô

But from (17) it follows that, for given 3 = <5(e), there exists p = p(3) such that if |r — r0| < pt (r e E ) and te[3, я], then |K(r, t; A)| < eôa. Consequently,

(23) I 2 ^ M f \ \ c

for r e E and |r — r0| < pi.

Collecting (19)—(23), we obtain (18).

Th e o r e m 3. I f f e h 1 and the sequence A = (Afc(r)}, reE, has the properties (5), (5') and (16), then ||JRr(*; / , А)||Л1 —>0 as r->r0.

P ro o f. The conditions (5), (5') and (16) imply (24) ||ЩГ5. ; / ', л ) _ Л |с -*0 as r->r0, (25) ||l7 (r,; ; / , A ) - /||c -+0 as r-> r0,

(reE) for every / e h 1. Moreover, U(r, x; / ' , A) = (d/dx)U(r, x; / , A) for r eE and every x. Hence, using (3), (24) and (25), we obtain the desired result.

4. The Cesàro means. Consider the Cesàro means (C, q), q > 0, of the Fourier series (4) of / e C2„, defined by the formula

(26) aq(x; f ) = - ^ Z An-kSk(x; f ) for n = 0, 1, ..., where

(q+l)(q + 2)...(q + n)

(q ф - 1 , - 2 , ...).

For every q > 0 and n = 0, 1, ..., we have (27) o*{x; / ) - / ( * ) = т Г 1 ] f x(t)Kq(t)dt,

о where

Kq(t) = -j- £ AqnZlDk(t), Dk(t) =

w Aqnk% feW kW 2sin(f/2)

([8], p. 157). Moreover ([8]), we have (28) \Kq(t)\ < n + 1, \Kq(t)\ ^ for n = 1 , 2 , . . . , te (0, я] and q > 0.

It is known ([8]) that if / e C2n and q > 0, then

(29) as n ->oo.

In [7] it is proved that if / е Я “,0 < о с < 1 and q > 0, then i(n+ l)~a if 0 < a < 1, (30) К С ; f ) - f \ \ c ^ M 5(q,a)

(n+1) 4og(n + 2) if a = 1, for n = 0, 1, ...

Applying (26)-(30) and arguing as in the proofs of Theorems 1-3, we obtain

Theorem 4. If f e H a, 0 < a ^ l , 0 < / J < o c and q>®, then

№(■ ;

I

Theorem 5. I f f eha, 0 < a ^ 1 and q > 0, then lk«(-; /)-/IU « -> 0 as n-+ 00.

5. The Abel means. Let p be a fixed non-negative integer. The papers [1]

and [5 ] investigate the approximation by the Abel means (A, p) of the Fourier series (4) of f e C 2n, defined by

OO

Щ г , x; / , A) = X Ы П P)Ak{x; / ) , ft = о

for r e E = [0, 1) and every x, where Я0(г; p) =1,

(31)

4 (r; p) = l - ( l - r )'’+1‘£ ( ^ +q Py ( k = U 2 , . . . ) .

The sequence X defined by (31) satisfies the conditions given for the means (6) in Section 1.

Clearly,

Up(r, x; f,X ) = n ~ l J / (t)Kp(r, t - x ; X)dt

— л

for r e E, x e ( — oo, + oo) and p = 0, 1, ..., where 00

K p{r, t; A) = i + ]T Xk{r; p)coskt.

ft = l Using the Abel transformation, we have

(32) Up(r, x ; f , a) - f i x ) = ( l - r y + l £ r l (Sk(x; f ) - f ( x ) )

- d -тУ*2 £ ( P+ 1 +‘k) Г 1 (*; / ) - / ( * ) ) for re E , x e ( —oo, +oo) and p = 0, 1 ,..., where a%(-; f ) is the Cesàro (C, p)-mean of the Fourier series (4).

In [1] it is proved that if / e C 2n and p — 0, 1, ..., then

\\Up{r, ■; / , A ) - /||c -*0 as r -*• 1 —.

This proves that

(33) Lp(r; Л) = тс-1 j |Кр(г, t\ X)\dt ^ M 7{p)

— Я

for r £[0., 1) and p = 0, 1,

Moreover, in [1] it is proved that if / £ Ha, 0 < a 1 and p = 0 , 1 , . . . , then

(34) \\Up(r, • ; / , Я ) - / | | С

f ( l — r)a i f 0 < a < l , p ^ l ,

^ M 8(p, r0, a) < or 0 < a < 1, p = 0, l ( l - r ) |l o g ( l - r ) | if a = 1, p = 0, for r £ [ r 0, 1), v0 > 0.

Applying (33), (34) and Theorem 1, we obtain

Theorem 6. / / / £ Ha, 0 < a < 1, 0 < /? < a and p = 0, 1 ,..., r/ien

\\Up(r, • ; f , X)—f\ \ Hp

C { l - r f ~ p if 0 < a < 1, p ^ 1,

^ M g -s o r 0 < a < l , p = 0, L ((l-r )|lo g (l- r)|)1-p if a = 1, p = 0, for re [ r 0, 1), w/iere M 9 = M 9(p, <?, r0, a, f) and r0 > 0.

Applying (32) and Theorem 5, we can prove

Theorem 7. I f f eha, 0 < a ^ 1 and p = 0, 1 ,. . . , then

\\Up(r, •; / , Л ) - / ||л«-+0 as r - > l - .

References

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

[2] M. G ô r z e n s k a , M. L e s n ie w ic z and L. R e m p u ls k a , On approximation of In-periodic functions belonging to the Holder classes, Fasc. Math, (in print).

[3] S. P r ô s s d o r f , Zur Konvergenz der Fourierreihen hôlderstetiger Funktionen, Math. Nachr. 69 (1975), 7-14.

[4] L. R e m p u ls k a , On the (A, p)-summability o f orthonormal series, Demonstratio Math. 13 (1980), 919-925.

[5] Z. S t y p in s k i, On a generalization o f the theorem o f Prôssdorf, Funct. Approx. Comment.

Math. 7 (1979), 101-104.

[6] A. F. T im a n , Theory o f Approximation o f Real Functions, Moscow 1960 (in Russian).

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

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