Z I ниЛтЛ*) = { Z Wk^/](x)-/(*)lr}1/r> J Z IS„[ n-fn *«СЛ = гтт I Ш] S[/](*) = Z + sin

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

W l o d z im ie r z L e n s k i (Poznan)

On the rate of almost strong summability of Fourier series

Abstract. We estimate the rates of almost strong summability of Fourier series of functions belonging to the spaces Lp (1 < p < oo) and C, by matrix means. As corollaries, the norm and pointwise approximation of functions from the Lipschitz classes is examined. We also present connections of our results with some earlier ones.

1. Introduction. Let X = Lp (1 ^ p ^ oo) [respectively X = C] be the space of all measurable real-valued functions/= /( • ) , 27t-periodic, Lebesgue- integrable with pth power [continuous] and let Lip (a, X) denote the class of all functions f e X whose moduli of continuity

co(S)x = co(S,f)x = sup ||<jp.(t)||x, where (px(t) = f ( x + t)+f(x — t) — 2f(x), satisfy the condition

co(< 5 )x = 0{Sa) as <5->0+.

Consider the Fourier series

S[/](*) = ^{a0(f)/2 +} Z (ak(f)cos kx + ^bk(f) sin ^kx)

k = 1

and denote by Я Д /] the partial sums of S [ / ] , and by

i n + m

*«СЛ = гтт I Ш] rrl "Г 1 n = n

the generalized de la Vallée Poussin means of the sequence {ЯД/]}.

The aim of this paper is to estimate the quantities Я М [/]Х,7>== { Z tm,k\Wm, n i n - f \ \ x Y lr,

m = 0

H k , n \ f \ x , T , q , r — A Z *m,k m — 0

1 n + m \ 1 lq

J Z IS„[ ^{n - f n}

r ] 1/r

x\

ниЛтЛ*) = { Z Wk^/](x)-/(*)lr}1/r>

m = 0

о / 1

\r /<0 1

U u M \ t , , , ( x ) = •! Z «„.»(— T I |S „ [/]W -/(x )|«

^m = 0 уШ + m + 1 д = n 1 /

94 W. Le ns ki

where T = (rm k) is an arbitrary non-negative matrix and q, r > 0, by means of the quantity

, г 1 _ f v , Ш У \ lqV '

where g — g(-) will be either the modulus of continuity co(n/{- + 1 ),/)* , or the best approximation E.+n{f)x of / by trigonometric polynomials of degree at most n in the space X , or the function wJC(7i/(- + l))x, where

r sup

0 <u«<5

1 « ") 1 /P -$ \< P x(t)\pdt>

U 0 )

wx(<5)x = wx{ô,f)x = < ess sup {|<рх( 01 }

0 <t=$<

sup {\(pM}

Lo<t$<5

if X = Lp (1 ^ p < oo), if X = L°°,

if X = C.

We also show how some earlier results follow from ours.

By convention, the letter M will mean either an absolute constant or a constant depending on the parameters q, r, p, a, not necessarily the same at each occurrence.

2. Norm approximations. First we present a general estimate basing on the well-known results of Dahmen [1] and Steckin [ 8 ].

T heorem 1. I f f e X , then

H k , n L f ^ X , T , r ^ - M ^ f c , n [ 2 i - + n ( / ) x ] r , l , r , l '

P ro o f. The above mentioned results of Dahmen and Steckin may be written in the form

n + 2m

I K „ [ / ] - / l l * « ; M I (n + m + l ) - ' E r+n(f)x .

f i =

0

Therefore, using the monotonicity of the best approximation En(f), we immediately obtain our assertion.

The next theorem concerns a stronger estimate.

T heorem 2. I f f e X (here X = C when p = oo) and q(q—l) 1 ^ p ^ q (q ^ 2 ), then

H k , n \ f \ x , T , q , r ^ M h k ,n CO

+ 1 /X + Mhkf „

_ T , l , r , l

CO + 1 X j T , p , r , p / q

P ro o f. It is clear that

j n + m 'J l/q

m + Y I Is „[/](x )-/(*)l* ^< _{m +}

+

1 n + m A ц = п i n-fm

- I l

j п/(и + т+1)

Я ^о

в') l/q

m + ц = п i n + m

J я/(т+1) я/(n + m+ 1)

q") l/q

ц = п

= + Л2 + у 43, where D (0 denotes the Dirichlet kernel.

Since p ^ n + m, we have |/)Д01 ^ n + m + l, whence , n + m+ 1 n/(" + "I + 1>

< --- J |фх(01^

, 71 0

and so

ii . || ( n WAJx^ co

1 я

~ \ <Px(t)D,i(t)dt Пп/(т + 1)

q') i/q

n + m+ 1

To estimate A2 we use the inequality \D (t)\ ^ (7i/2) |t| 1 and obtain n/(m + 1 )

A , ^ f

dt,

whence n/(n + m+ 1)

я/(т+1) fn(t\ n + m \ CO I 1

I /Jr

cu i i ^/1+1 я/(и + т+1) 1 » = m A*+l ^ p = 0 P + ™+l

Next, we apply the following theorem of G. Hardy and J. E. Littlewood (cf. [10], p. 126 (5.20II)): I f /Д 0 = t~Qf( t ) , f ee X with 6 = 1/p + l / g - l and q i q - i y 1 ^ p ^ q (q> 2), then

Hence

In ( f\\9 °° ] !/«

Li4 r L+ I ( M / r + I M / r ) « M ||/B(-)ll*-

^ k= 1 J

1 m +

^ M and so

n + m i i _{A ^ = n} 1

1 <P,(2t) :

- f n ti/(2(m + l)) 2Sint sin ((2/i+l )t)dt

q") i/q

| 4 3 | | *

(m+ 1)р/9я/(Ш +1) 1

i Ш l +p/9 ь ]и’

(m + 1 )p/4 w = о

I

ay /1+1

^ ( / i + l ) 1 р/Ч/1 + т + 1 ) ^p/q /1+1

i/p

l/q

96 W. Le ns ki

Thus, in the notation of Section 1, we obtain the desired inequality.

3. Pointwise approximations. The first theorem of this section is similar to Theorem 1:

T heorem 3. I f f e X , then

Нк,„[ЛтЛх) < МЛ*.„ w.

+ 1

P ro o f. By the standard computation,

t t

n sin(m +l)-sin(2n + m + 1)- 0m.,,[ /] ( * ) - / ( * ) = T" 2 k 0 I --- :--- dt

(m + l)sin2-

, ./(.+m+ 1 ) „ sin(m+l)Um(2n + m + l ) -

~ 2n( I + J + J )<P*W---: dt

0 я/(п + т + 1 ) 1 \ «//».4-14 я / ( т + 1 ) . T 1

(m + l)sk+

— B1 + B2 + B3.

The terms B{(i = 1, 2, 3) will be estimated successively. Using partial integra­

tion we obtain

ID . 2n + m + l" /("+fm + 1\ /4IJ ( n

\Bt \ ^ ---- —---- J \(px(t)\ dt < w 2 k

*/(m+l) I I s 2l « i f

и + т + 1 /у ’

я/(и + m + 1 ) t m +

w .

ï h + i . Z

/i + l fi = m /1+1

m +

n + m w vx

î L + . l .

/i+ l

X f i ~ m /i + m+1

,B3| < —J î — î Af- Ц . £ wv

2(m+l )n/(w+D f m + l M=0 V/^+l/x

7Ü

a i

W

m *'/1 + 1

„ = o Z^ + m+1

Hence, by the monotonicity of we obtain our assertion.

In the case of stronger almost summability we have T heorem 4. I f f e X and q ( q - 1)-1 ^ p ^ q (q ^ 2), then Hk.n\f\T.q,r(X) ^ M Kn wv _{+ 1} + Mh k,m

T , l , r , l

W, + 1 T, p, r . p/ q

P roof. Under the notations of the proof of Theorem 2,

{ ^{J n + m} I | S , m ( * ) - / M I , j ^{■) 1 tq} « A t + A 2 + A 3 .

It is clear that

A x ^ w x .

\n + m+ l /x and, by the estimate of B2 from the previous proof,

/4, ^ ) л !

к

/* + l / x

p + m + 1 Next, integrating by parts, we get

A3 ^ M 1

( m + l ) ^ o Z (n + 1Y19' 1 wt

jU + 1

i I

f * W* \ u + l j ) 1/p

< 2 llqM { У _______ У1! 1'* ____ I { Д ( / i + l ^ - ^ + m + l ^ j

Summing up our partial estimates we get the desired result.

4. Corollaries. In this section we confine our attention to non-negative triangular matrices T which lead to regular methods of summability.

First, we give corollaries to the theorems of Section 2.

The following result is an improved version of [3] and [7] for 0 < a < 1.

C o r o l l a r y 1. Suppose that {£„,,*}„ = 0 is a non-decreasing sequence with respect to m and that /e L ip (a , X). Then, for ae(0, 1),

Г if 0 < ar < 1,

Hk,nLf]x,T,r ^ "S M(tM log(2 + Tfc))“ if otr = 1,

^ M (tk,k)llr if ar > 1, for all natural n, where xk II g rn ?r 1 1 _ 1

Proof. This is a consequence of Theorem 1. Indeed, the assumption /eL ip (a, X) implies, by Jackson’s theorem, that E^(f)x ^ M( g+ 1)_“,

whence

n + m

I

Z fo + n + ip ^ + m+l)-1

p + m + 1 u = o

^ M (m + 1) “max 2

1 1

1 — a 1 — a a

Commentationes Math. 30.1

9 8 W. Lenski

Consequently, by Theorem 1 and the monotonicity of {tm fc} with respect to m, we have the estimate

Я , , [ / ] И ,=? M [ £ W (« i+ l)" " } 1,r

m = 0

« M { X tm.k(m+\)~"+ I

m — 0 m = tjc

I (m + l)-"- + ((wr i W}1,r

m = 0 m = Tk

X (m + l)-“- + (tMr } 1,r, m = 0

ffk,n[/]x,7> ^ 4

from which our assertion follows at once.

If we consider the class Lip(l, X) then we additionally obtain C orollary 2. Under the assumptions of Corollary l, for a = 1,

Л/t*,к log (2 + тк) / 0 < г < 1 , Mtk>fc(log(2 + Tk))2 */r = l, ,M(fk>fc)1/rlog(2 + T fc) i f r > 1, /or a// natural n, where тк = min(fc, [(fk k)-1]).

P ro o f. As before, if /e L ip ( l, X), then n + m p / f\

[ < M (m + 1 )- i log(m + 2)>

„ = o ju + m + l

Hence, since the function (logu)/w is decreasing for u > e ,

Ht , [ / ] w „ < M{f,.,(log(2 + rt)X X (m + 1 )-■• + (rt.4 log(2 + 1 ^ m = 0

and the desired inequalities are obvious.

Another point of view on the considered question enables us to formulate

the next corollary which is an improvement on the results of [5, 6].

C orollary 3. 7 / / e Lip (a, X) then M ( k + l)1"— / £

Im-o "1 + 1

к t

1

lm,k

к f 1/r

M l M

= o m + 1 log(/c + 2)

(/c+ 1)

I i = 0 "*+l ^{1 m,fc} 1/r

M log (k + 2)< £ ^lm,k , = о Ю+1

1/r

г/ 0 < a < 1 and 0 < ar ^ 1, if 0 < a < 1 and <xr ^ 1, if a = 1 and 0 < r < 1, г/ a = 1 and r ^ 1, for all natural n.

Proof. These inequalities are direct cosequences of the assumption /eL ip (a, X) and Theorem 1, because then

[/]*,?> ^ ^

Г fe 1 /r

М / £ tm.k(m + I ) ' " '

Lm = 0 if 0 < a < 1,

Z w ( ( m+ l) 1log(m + 2))r V if a = 1.

m = 0

1 /r

An additional restriction on the considered matrix T gives the following improvement on the result of [4].

C orollary 4. 7 //e L ip ( a , X) and f/ien

Н к Л П х , т л <

for all natural n.

M(k+\)~*

M(k+ 1)_1 log(/c + 2)

if 0 < a < 1, if a = 1,

Further, we note that similar corollaries can be derived from Theorem 2.

We formulate only one of them.

C orollary 5. Suppose that f e Lip (a, X), 0 < a < 1, q(q — l)-1 ^ p ^ q (q ^ 2) and let /cZm=o Ln,k/(m+ 1) ^ Then

^м1/1х,Г,д,г

f° r all natural n.

M ( k + i y *

M(/c+ l)~alog 1/p(/c + 2) M ( k + \ ) - 1/(i

if 0 < a < l/q and ot ^ 1/r, i/ a = l/g ^ 1/r,

i/ 1/g < a < 1 and \/q ^ 1/r,

Finally, we give a corollary concerning pointwise almost strong summa­

bility.

100 W. Le ns ki

C o r o l l a r y 6. Let / e L 1. I f

and

lim - J \(px{u)\ du = 0 t-*o+ t g

uniformly in n, then uniformly in n, too.

P ro o f. This result, for r ^ 1, is a stronger form of Varma’s theorem [9]

and may be deduced from the proof of Theorem 3. More precisely, the first condition of our corollary implies B x = o(l) and B3 = o( 1), and the second one yields В2 = o(l). These relations, by regularity of the considered methods of summability, immediately give the desired statement.

5. Remarks, (a) It is obvious that Corollaries 1, 3, 4, 5 with ae(0, 1), hold automatically for the conjugate functions f.

(b) Applying Leindler’s and Totik’s ideas, it is possible to prove the sharpness of the obtained orders of approximation (cf. [2], pp. 49-52).

(c) We note that similar conditions to those of Corollary 6 for f e X (p > 1) together with some assumptions on p and q lead us to the relation

[1] W. D a h m e n , On best approximation and de la Vallée Poussin sums, Mat. Zametki 23 (1978), 671-683 (in Russian).

[2] L. L e in d le r , Strong Approximation by Fourier Series, Budapest 1985.

[3] K. P r a s a d , On the degree of approximation o f functions belonging to the class Lip a, Comment. Math. Prace Mat. 26 (1986), 97-105.

[4] K. Q u r e s h i, On the degree of approximation of a periodic function by almost Nôrlund means, Tamkang J. Math. 12 (1) (1981), 35-38.

[5] —, On the degree o f approximation o f functions belonging to the class Lip (a, p), Indian J. Pure Appl. Math. 13 (1982), 466-470.

[6] —, On the degree o f approximation o f functions belonging to the class Lip (a, q), Tamkang J.

Math. 15 (1) (1984), 5-11.

[7] P. L. S h a r m a and K. Q u r e s h i, On the degree of approximation o f a periodic function f by almost Riesz means, Ranchi Univ. Math. J. 11 (1980), 29-33.

[8] S. B. S te c к in, On the approximation o f periodic functions by de la Vallée Poussin sums, Anal.

Math. 4 (1978), 61-74.

[9] S. K. V arm a, On almost matrix summability of Fourier series, Tamkang J. Math. 15(1) (1984), 1-4.

[10] A. Z y g m u n d , Trigonometric Series II, Cambridge 1959.

INSTYTUT MATEMATYKI, UNIWERSYTET IM. A. MICKIEWICZA INSTITUTE OF MATHEMATICS, A. MICKIEWICZ UNIVERSITY MATF.JKI 48/49, 60-769 POZNAN, POLAND

k~* oo

which holds also uniformly in n.

lim HkJ f \ T<q,r(x) = 0,

References