Applied Mathematics E-Notes, 12(2012), 221-227 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Ul’yanov Type Inequalities For Moduli Of
Smoothness
Sadulla Jafarovy
Received 8 May 2012
Abstract
Let T denote the interval [ ; ]. In this work we investigate the inequality of
Ul’yanov type for moduli of smoothness of an integer order in the Lp (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
Let f be 2 -periodic and let f 2 Lp [0; 2 ] = Lp for p
will denote the Lp -norm and will be de…ned by
1. Throughout this work, k kp
8
91=p
< 1 Z2
=
p
kf kp =
jf (x)j dx
; f 2 Lp ; 1
:2
;
p < 1:
0
The modulus of smoothness ! k (f; )p of a function f 2 Lp ; 1
order k > 0 are de…ned by
! k (f; )p = sup jhj
where
k
hf
(x) =
1
X
k
( 1)
k
hf
(x)
f (x + (k
p
1, of fractional
(1)
p
)h) ;
k > 0:
=0
Note that, the following (p; q) inequalities between moduli of smoothness, nowadays
called Ul’yanov-type inequalities, are known:
! k f (r) ;
q
0
Z
C@
u
! k+r (f; t)p
0
q1
11=q1
du A
;
u
(2)
Mathematics Subject Classi…cations: 26D15, 41A25, 41A63, 42B15.
of Mathematics, Faculty of Art and Sciences, Pamukkale University, 20017, Denizli, Turkey; Mathematics and Mechanics Institute, Azerbaijan National Academy of Sciences, 9,
B.Vahabzade St., Az-1141, Baku, Azerbaijan
y Department
221
222
Ul’yanov Type Inequalities for Moduli of Smoothness
where
r 2 N [ f0g ; 0 < p < q
1;
=
1
p
1
;
q
q if q < 1;
1 if q = 1:
q1 =
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 by En (f )p the best approximation of f 2 Lp (T ) by trigonometric polynomials of degree not exceeding n, i.e.,
En (f )p := inf Tn 2
n
kf
Tn kp ;
n = 0; 1; 2; ::::;
where n denotes the class of trigonometric polynomials of degree at most n.
Let Wpr [0; 2 ] = Wpr ; (r = 1; 2; :::) be the linear space of functions for which f (r 1)
is absolutely continuous and f (r) 2 Lp (T ) ; p > 1. It becomes a Banach space with the
norm
kf kWpr := kf kp + f (r) :
p
Let f 2 Lp . For
> 0, the K-functional is de…ned by
r
K
; f ; Lp ; Wp
:= inf kf
kp +
(r)
p
2 Wpr :
:
Let 1 < p < 1. We de…ne an operator on Lp (T ) by
1
( h g)(x) :=
2h
Zh
g (x + t) dt; 0 < h < ; x 2 T:
h
The k-modulus of smoothness
k
k
( ; g)p ; (k = 1; 2; :::), of g 2 Lp (T ) is de…ned by
( ; g)p := sup 0<hi <
1 i k
k
Y
(I
hi ) g
i=1
;
> 0;
(3)
Lp (T )
where I is the identity operator [1], [5], [7].
In the case of k = 0 we set k ( ; g)p := kgkLp (T ) and if k = 1 we write ( ; g)p :=
1 ( ; g)p .
It can be shown easily that the modulus of smoothness k ( ; g)p is a nondecreasing,
nonnegative, continuous function satisfying the conditions
lim
!0
k
( ; g)p = 0 ;
S. Jafarov
223
k
( ; f + g)p
k
( ; f )p +
k
( ; g)p
for f; g 2 Lp (T ),
f
1
a0 (f ) X
+
(ak (f ) cos kx + bk (f ) sin kx)
2
(f ) :=
(4)
k=1
is the Fourier series of the function f 2 L1 (T ):
The n-th partial sums and de La Vallée-Poussin sum of the series (4) are de…ned,
respectively, as
n
Sn (x; f ) :=
a0 (f ) X
+
(ak (f ) cos kx + bk (f ) sin kx)
2
k=1
and
Vn (f ) := Vn (x; f ) :=
The following Lemma holds.
LEMMA 1. For f 2 Lp ; 1
c1 (p; k)
k
1
;f
n
p
2n 1
1 X
S (x; f ) :
n =n
1; and k = 1; 2; ::: we have
n
2k
Vn(2k) (f; x)
p
p
c2 (p; k)
1
;f
n
k
+ kf (x)
Vn (f; x)kp
:
p
PROOF. Considering reference [7], the inequality
k
1
; Tn
n
c3 (p; k) n
2k
Tn(2k)
p
(5)
p
holds, where Tn is a trigonometric polynomial of order n. Using the properties of
smoothness k ( ; f ) p [5 ], [7] and (5), we have
!
1
1
;f
c4 (p; k)
; Tn + kf Tn kp
k
k
n
n
p
p
c5 (p; k) n
2k
Tn(2k)
p
+ kf
Tn kp :
By reference [7] the Jackson inequality
En (f )p
c6
k
1
;f
n+1
holds, with a constant c6 > 0 independent of n.
;
p
k = 1; 2; :::;
(6)
224
Ul’yanov Type Inequalities for Moduli of Smoothness
Note that, to estimate
1
;f
n
k
from below we shall use the following inequality
p
in [7]
n
Tn(2k)
2k
c7 (p; k)
p
1
; Tn
n
k
p
:
(7)
Let Vn (f ) be de La Vallée-Poussin sum of the series (4).
We denote by Tn (x; f ) the best approximating polynomial of degree at most n to
f in Lp (T ). In this case, from the boundedness of Vn in Lp (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 )
c9 (p; k)
c11 (p; k)
1
; Vn
n
k
c10 (p; k)
1
;f
n
k
1
;f
n
k
+ kf (x)
p
Vn (x; f )kp
!
+ En (f )p
p
+
k
p
1
;f
n
Vn
p
!
:
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 smoothness 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.
1 1
THEOREM 1. Let f 2 Lp ; 1 < p < q < 1; =
. Then for any k = 1; 2; :::,
p q
r = 1; 2; ::: the following estimate holds:
11=q
0
Z
q du
A :
(9)
; f (r)
C@
u ( +r) r+k (u; f )p
k
u
q
0
2
Proof of the Main Result
According to reference [7] for 1 < q < 1 the following equivalence holds:
k
1 (r)
;f
2n
K
q
=
inf
1 (r)
; f ; Lq (T ) ; Wq2k (T )
2n
f (r)
q
+2
2nk
(2k)
q
:
2 Wq2k (T ) :
(10)
S. Jafarov
225
If Vn is the de La Vallée-Poussin sum of the function f using Lemma 1 we get
K
1 (r)
; f ; Lq (T ) ; Wq2k
2n
f (r)
V2n f (r)
q
(2k)
2nk
+2
V2n
q
:= I1 + I2 : (11)
Taking account of (8) we have
kf
Vn (f )kp
c12 En (f )p :
(12)
Considering [16] and [4], the following (p; q)-inequality holds:
(r)
(r)
(V2l )
(V2n )
l 1
X
c13
q
2
m q
(r)
(r)
(V2m+1 )
(V2m )
q
p
m=n
!q
:
(13)
Using the Bernstein-type inequality [7], [9], [14] we obtain
(r)
(r)
V2m+1
V2m
p
c14 2mr kV2m+1
V2m kp :
(14)
Taking into account the relations (13), (14) and Jackson inequality [6] we have
I1
f (r)
=
c15
V2n f (r)
1
X
q
2m q 2mqr E2qm (f )
m=n
c16
1
X
m q mqr
2
2
k+r
m=n
0
B
c17 @
On the order hand, for
1
2
2
Z
1
;f
2m
n
u
( +r)
q
k+r
(u; f )p
0
q
p
!1=q
11=q
du C
A
u
:
(15)
the following equivalence holds:
k
( 1 ; f )p
k
( 2 ; f )p :
(16)
It is known that for trigonometric polynomials of degree n the following Nikol’skii
inequality holds [4], [8], [10] :
kTn kq
1
c18 n p
1
q
kTn kp ;
0<p
q
1:
(17)
226
Ul’yanov Type Inequalities for Moduli of Smoothness
Use of inequality (17) gives us
I2
=
2
(k)
nk
(r)
V2n
q
c19 2
2
c20 2n 2nr
0
B
c21 @
0
B
= c21 @
(k+r)
V2n
nk n
2
Z
n
2
Z
n
k+r
u
u
p
1
;f
2n
r
p
q
k+r
(u; f )p
0
0
u
( +r)
q
k+r
(f; u)p
11=q
du C
A
u
11=q
du C
A
u
:
(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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