Ul’yanov Type Inequalities For Moduli Of Smoothness
Sadulla Jafarov
yReceived 8 May 2012
Abstract
LetT denote the interval[ ; ]. In this work we investigate the inequality of Ul’yanov type for moduli of smoothness of an integer order in theLp(T); p 1 spaces. In particular, we study(p; q) inequalities for moduli of smoothness of a derivative of a function via the modulus of smoothness of the function itself.
1 Introduction
Letf be2 -periodic and letf 2Lp[0;2 ] =Lp forp 1. Throughout this work,k kp
will denote theLp-norm and will be de…ned by
kfkp= 8<
: 1 2
Z2 0
jf(x)jpdx 9=
;
1=p
; f2Lp; 1 p <1:
The modulus of smoothness!k(f; )pof a functionf 2Lp;1 p 1, of fractional orderk >0are de…ned by
!k(f; )p= supjhj khf(x) p (1) where
k hf(x) =
X1
=0
( 1) k
f(x+ (k )h); k >0:
Note that, the following(p; q)inequalities between moduli of smoothness, nowadays called Ul’yanov-type inequalities, are known:
!k f(r);
q C
0
@Z
0
u !k+r(f; t)p q1 du u
1 A
1=q1
; (2)
Mathematics Sub ject Classi…cations: 26D15, 41A25, 41A63, 42B15.
yDepartment of Mathematics, Faculty of Art and Sciences, Pamukkale University, 20017, Deni- zli, Turkey; Mathematics and Mechanics Institute, Azerbaijan National Academy of Sciences, 9, B.Vahabzade St., Az-1141, Baku, Azerbaijan
221
where
r2N[ f0g; 0< p < q 1; = 1 p
1 q; q1= q ifq <1;
1 ifq=1:
In the case r= 0; p 1 the inequality (2) was proved by Ul’yanov [15]. In other cases, (p; q) estimates (the modulus of smoothness!k(f; )p of an integer order, the r-the derivative, r 2 N and the fractional derivative of order r >0 of the function) were obtained in references [3], [4], [14].
Note that the inequality between moduli of smoothness of various orders in di¤erent metrics was investigated by [6].
We denote byEn(f)p the best approximation off 2Lp(T)by trigonometric poly- nomials of degree not exceedingn, i.e.,
En(f)p:= infTn2 nkf Tnkp; n= 0;1;2; ::::;
where n denotes the class of trigonometric polynomials of degree at mostn.
LetWpr[0;2 ] =Wpr;(r= 1;2; :::)be the linear space of functions for whichf(r 1) is absolutely continuous andf(r)2Lp(T); p >1. It becomes a Banach space with the norm
kfkWpr :=kfkp+ f(r)
p: Letf 2Lp. For >0, theK-functional is de…ned by
K ; f;Lp; Wpr := inf kf kp+ (r)
p: 2Wpr : Let1< p <1. We de…ne an operator on Lp(T)by
( hg)(x) := 1 2h
Zh h
g(x+t)dt; 0< h < ; x2T:
Thek-modulus of smoothness k(; g)p;(k= 1;2; :::), ofg2Lp(T)is de…ned by
k( ; g)p := sup0<hi<
1 i k
Yk i=1
(I hi)g
Lp(T)
; >0; (3)
where Iis the identity operator [1], [5], [7].
In the case ofk= 0we set k( ; g)p:=kgkLp(T)and ifk= 1we write ( ; g)p:=
1( ; g)p.
It can be shown easily that the modulus of smoothness k(; g)pis a nondecreasing, nonnegative, continuous function satisfying the conditions
lim !0 k( ; g)p = 0;
k( ; f+g)p k( ; f)p+ k( ; g)p forf; g2Lp(T),
f (f) := a0(f)
2 +
X1 k=1
(ak(f) coskx+bk(f) sinkx) (4)
is the Fourier series of the functionf 2L1(T):
Then-thpartial sums and de La Vallée-Poussin sum of the series (4) are de…ned, respectively, as
Sn(x; f) :=a0(f)
2 +
Xn k=1
(ak(f) coskx+bk(f) sinkx)
and
Vn(f) :=Vn(x; f) := 1 n
2nX1
=n
S (x; f):
The following Lemma holds.
LEMMA 1. Forf 2Lp;1 p 1;andk= 1;2; :::we have
c1(p; k) k
1 n; f
p
n 2k Vn(2k)(f; x)
p+kf(x) Vn(f; x)kp
c2(p; k) k 1 n; f
p
:
PROOF. Considering reference [7], the inequality
k
1 n; Tn
p
c3(p; k)n 2k Tn(2k)
p (5)
holds, where Tn is a trigonometric polynomial of order n. Using the properties of smoothness k(; f)p [5 ], [7] and (5), we have
k
1 n; f
p
c4(p; k) k 1 n; Tn
p
+kf Tnkp
!
c5(p; k) n 2k Tn(2k)
p+kf Tnkp : By reference [7] the Jackson inequality
En(f)p c6 k
1 n+ 1; f
p
; k= 1;2; :::; (6)
holds, with a constantc6>0 independent ofn.
Note that, to estimate k
1 n; f
p
from below we shall use the following inequality in [7]
n 2k Tn(2k)
p c7(p; k) k 1
n; Tn p: (7)
LetVn(f)be de La Vallée-Poussin sum of the series (4).
We denote byTn(x; f)the best approximating polynomial of degree at most nto f in Lp(T). In this case, from the boundedness ofVn inLp(T);we obtain
kf Vn(f)kp kf(x) Tn(x; f)kp+kTn(x; f) Vn(x; f)kp
c7(p)En(f)p+kVn(x; Tn(x; f) f(x))kp
c8(p; k)En(f)p: (8)
Using (7) and (8) we reach
n 2k Vn(2k)(x; f)
p+kf(x) Vn(x; f)kp
c9(p; k) k
1 n; Vn
p
+En(f)p
!
c10(p; k) k 1 n; f
p
+ k 1
n; f Vn
p
!
c11(p; k) k
1 n; f
p
: Thus the proof of Lemma 1 is completed.
In this work we study(p; q)-inequalities of Ul’yanov type for the modulus of smooth- ness k f(r);
p; k= 1;2; :::,r= 1;2; :::de…ned in the form (3). To prove we use the method of the proof given in the study [14].
Main result in the present work is the following theorem.
THEOREM 1. Letf 2Lp;1 < p < q <1; = 1 p
1
q. Then for anyk= 1;2; :::, r= 1;2; :::the following estimate holds:
k ; f(r)
q C
0
@Z
0
u ( +r) r+k(u; f)p q du u
1 A
1=q
: (9)
2 Proof of the Main Result
According to reference [7] for1< q <1the following equivalence holds:
k
1 2n; f(r)
q
K 1
2n; f(r); Lq(T); Wq2k(T)
= inf f(r)
q+ 2 2nk (2k)
q : 2Wq2k(T) : (10)
IfVn is the de La Vallée-Poussin sum of the functionf using Lemma 1 we get
K 1
2n; f(r); Lq(T); Wq2k f(r) V2n f(r)
q+ 2 2nk V2(2k)n
q :=I1+I2: (11) Taking account of (8) we have
kf Vn(f)kp c12En(f)p: (12)
Considering [16] and [4], the following(p; q)-inequality holds:
(V2l)(r) (V2n)(r)
q c13 l 1
X
m=n
2m q (V2m+1)(r) (V2m)(r)
q p
!q
: (13)
Using the Bernstein-type inequality [7], [9], [14] we obtain
V2(r)m+1 V2(r)m
p c142mrkV2m+1 V2mkp: (14) Taking into account the relations (13), (14) and Jackson inequality [6] we have
I1 = f(r) V2n f(r)
q
c15 X1 m=n
2m q2mqrE2qm(f)
c16 X1 m=n
2m q2mqr k+r 1 2m; f
q
p
!1=q
c17 0 B@
2 n
Z
0
u ( +r) k+r(u; f)p qdu u
1 CA
1=q
: (15)
On the order hand, for 1 2 the following equivalence holds:
k( 1; f)p k( 2; f)p: (16)
It is known that for trigonometric polynomials of degreenthe following Nikol’skii inequality holds [4], [8], [10] :
kTnkq c18n1p 1qkTnkp; 0< p q 1: (17)
Use of inequality (17) gives us I2 = 2 nk V2(k)n
(r)
q
c192 nk2n V2(k+r)n
p
c202n 2nr k+r 1 2n; f
p
c21
0 B@
2 n
Z
0
u u r k+r(u; f)p
q du u
1 CA
1=q
= c21 0 B@
2 n
Z
0
u ( +r) k+r(f; u)p q du u
1 CA
1=q
: (18)
Using (10), (11), (15) and (18), we have (9).
Acknowledgment. The author wishes to express deep gratitude to the referee for valuable suggestions.
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