Advances in Pure Mathematics, 2012, 2, 183-189
http://dx.doi.org/10.4236/apm.2012.23025 Published Online May 2012 (http://www.SciRP.org/journal/apm)
The Best m-Term One-Sided Approximation of Besov
Classes by the Trigonometric Polynomials*
Rensuo Li1, Yongping Liu2#
1School of Information and Technology, Shandong Agricultural University, Tai’an, China
2School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems,
Ministry of Education, Beijing Normal University, Beijing, China
Email: rensuoli@sdau.edu.cn, #ypliu@bnu.edu.cn
Received December 20, 2011; revised February 5, 2012; accepted February 15, 2012
ABSTRACT
In this paper, we continue studying the so called best m-term one-sided approximation and Greedy-liked one-sided ap-
proximation by the trigonometric polynomials. The asymptotic estimations of the best m-terms one-sided approximation
by the trigonometric polynomials on some classes of Besov spaces in the metric
1
d
p
LT p
are given.
Keywords: Besov Classes; m-Term Approximation; One-Sided Approximation; Trigonometric Polynomial; Greedy
Algorithm
1. Introduction
In [1,2], R. A. Devore and V. N. Temlyakov gave the
asymptotic estimations of the best m-term approximation
and the m-term Greedy approximation in the Besov
spaces, respectively. In [3,4], by combining Ganelius’
ideas on the one-sided approximation [5] and Schmidt’s
ideas on m-term approximation [6], we introduced two
new concepts of the best m-term one-sided approxima-
tion (Definition 2.2) and the m-term Greedy-liked one-
sided approximation (Definition 2.3) and studied the
problems on classes of some periodic functions defined
by some multipliers. We know that the best m-term ap-
proximation has many applications in adaptive PDE solvers,
compression of images and signal, statistical classifica-
tion, and so on, and the one-sided approximation has
wide applications in conformal algorithm and operational
research, etc. Hence, we are interested in the problems of
the best m-term one-sided approximation and corre-
sponding m-term Greedy-liked one-sided approximation.
As a continuity of works in [3,4], we will study the
same kinds of problems on some Besov classes in the
paper.
There are a lot of papers on the best m term approxi-
mation problem and the best onee-sided approximation
problem, we may see the papers [7-10] on the best m
term approximation problem and see [11,12] on the best
one-sided approximation problem.
1
:0,2π0, 2π
d
d
TT

12
,,,
d
be the d dimensional Let
x
xx x, torus. For any two elements
12
,,, d
d
yyyy R

:ikx
ex e, set k,
,,, d
kkkkZ
11 2.
12 d, where xy denotes the inner prod-
uct of x and y, i.e.,
x
2dd
yxy xyxy

1
d
LT p
p
Denote by
 the space of all 2π-
periodic and measurable functions f on Rd for which the
following quantity

1
:d,1,
p
p
d
pT
ffxxp

,,
sup
d
xT
fessfxp

d
p
LT is a Banach space with the norm is finite.
.
For any
,
d
p
fLT
 
we denote by
 

1
ˆd, ,
2π
d
dk
dT
fkfxe x xkZ

the Fourier coefficients of f (see [13]).
For any positive integer m, set 1
:.
d
nnm m

For any
,
d
fLT

1 as Popov in [11,12], by using the
multivariate Fejér kernels,


2
2
1
12
sin 2
:π2,
sin 2
,,, ,
d
di
n
ii
d
d
nx
xnx
xxxx T

 


*This paper is supported by Shandong Province Higher Educational
Science and Technology Program; National Natural Science Founda-
tion of China (project No. 110771019); partly by the research fund for
the Doctoral Program of Higher Education and partly by Beijing Natu-
ral Science Foundation (Project 1102011).
#Corresponding author.
we defined
C
opyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU
184
 
  
1
2ππ
|| 0
,: ,
2πsup
mm
n
ylnn
l
Tfx Tfx
,,
nm
x
lnf y

 
Tf y

,Tfx1
|| 0
n
l
(1)
and called it to be the best m-term one-sided trigonomet-
ric approximation operators, where and in the sequel the
operator m is the best m-term trigonometric
approximation operators and
denotes
12
11
00 d
nn
ll


 
,.
m
1
0
.
n
l
 It is easy to see that

f
xTfx

1,
d
fLT
Meantime, for any we also defined
 
  
1
2ππ
|| 0
,:gfx g,
2π,,
sup
mm
n
nm
ylnn
l
fx
x
lnfygf y

  
1
ˆ
(2)
where

,m
mi



ki
g
fxf

1
ˆ
i
fkikie and
is
a sequence determined by the Fourier coefficients
of f in the decreasing rearrangement, i.e.,


ˆ
d
kZ
fk




12fk fk
m
m
.
It is easy to see that two operators and
T
g
are
non-linear. We will see that for any ,
d
x
T

,
m
g
fxfx

,,Tfx

,Tfx

,
(see Lemma 3.1 2)).
The main results of this paper are Theorems 2.5 and
2.6. In Theorem 2.5, by using the properties of the op-
erator m we give the asymptotic estimations of
the best m-term one-sided approximations of some Besov
classes under the trigonometric function system. From
this it can be seen easily that the approximation operator
m is the ideal one. In Theorem 2.6, by using the
properties of the approximation operator m
g
fx

x
, the
asymptotic estimations of the one-sided Greedy-liked
algorithm of the best m-term one-sided approximation of
Besov spaces under the trigonometric function system
are given.
2. Preliminaries
For each positive integer m, denote by m the non-
linear manifold consists of complex trigonometric poly-
nomials T, where each trigonometric polynomial T can
be written as a linear combination of at most m exponen-
tials , . Thus mk
ed
kZT
if and only if there
exits d
Z
 such that m and


,
kk
k
ce x

Tx
where .
 
1p

j
is the cardinality of the set
Let D be a finite or infinite denumerable set. Denote
by the space of all subset of some
p
lD
complex numbers
j
D
Xx with the following finite
l norm

1
:,1;:.
sup
p
p
p
X
j
j
lDl jD
jD
xpXx




For any
1,
d
fLT let


ˆ
d
kZ
fk be the set of
Foficients of fthe page urier coef. As in 19 of [14], de-
note by


ˆ
dd
pp
ll
ffk
the
l norm of the set of Fourier coefficients of f.
rf the
tri d
k
Th oughout this paper, let n
denote the set o
gonometric polynomials of variables and degree n
with the form

||
ˆ
kn
TTke
and

qn
 denote
the set of all triomials n
such
that
gonometric polyn T in



ˆ
:1.
dd
qn q
kZ lZ
TTk

Here we take as
ˆ0Tk
if ,kn
, .
d
kk k
Definition 2.1. (see cf. [For a given function f, we
ca
12
:max, ,k
1])
ll
:inf
m
mp
f
T
fT


the best m-term approximation error of f with trigono-
metric polynomials under the norm Lp. For the function
set
,
d
p
ALT we call
:s

up
mm
pp
fA
A
f
the best m-term approximation error of the function class
set
A with trigonometric polynomials under the norm Lp.
Definition 2.2. (see cf. [3,4]) For given function f,
:,.TTTf
 
The quantity
2mm
:inf
m
mpp
T
f
fT


is called to be the best m-term one-sided approximation
error of f with trigonometric polynomials under the norm
Lp. For given function set

,
d
p
ALT the quantity

:s
up
mm
p
p
fA
A
f
is called to be the best m-term one-sided approximation
,4]) For given function f, we
ca


error of the function class A with trigonometric polyno-
mials under the norm Lp.
Definition 2.3. (see cf. [3
ll
,
m
g
fx
(given by relation (2)) the Greedy-liked
algorithe best m-term one-sided approximation of
f under trigonometric function system. For given function
set
m of th
,
d
p
ALT we call

:sup
p,
mm
fA
A
fg fx

Copyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU
Copyright © 2012 SciRes. APM
185
the Greedy-liked one-sided approximationthe
best m-term one-sided approximation of function class A
0,1, 2,
 1,q
. In the case we can error of Here
given by trigonometric polynomials with norm Lp.
As in [1,15], denote by
,
s
q
BL
0
, 0,qs
,
the Besov space. The definition of the Besov space is
gi equivalent chacterizationven by using the followingar.
A function f is in the unit ball


sq
UB L
of the Besov
space
,
s
q
BL if and only iftrigonometric
polynomials ():Rx ch that
there exist

,e x suc

j
R x and
||2
j
jjkk
k

0
:j
fx

0
21.
s
jl
(3)
j
jq
R
take
1
2||2
ˆ
:,
jj
j
jk
k
Rf fke


00
ˆ
:0
f
fe,
1,
j
,,, d
kkkkZ
12 d,

12
max,, ,kkkk.
d
We define the seminorm

s
B
q
L
f
as the infimum
over all decompositions (3) and denote by
sq
UB L
the unit ball with respect to this seminorm.
Throughout this paper, for any two given sequences of
1
nn
,
non-negative numbers 1
nn
n
c
if there is a non-
negative constant c independent of all n, such that
n
, then we write n.
n
nn
If both
n
and n
. hold, then we write nn
1p 0,qs, For any , set

11,0 2and1,
max, ,otherwise,
2
dqppq
qp
dd
q







(4)
and
,:pq


,,02and1,
, ,otherwise.
dpq qppq
pq
 
(5)


sq
L of the Besov spaces
,:pq
,,pq

we have
For the unit ball UB

s
q
L
, Devore anv in [1] gave the follow
BTemlyako -
ing result:
d
Theorem 2.4. (c.f. [1]) For any 1p, 0q
,
s, let

,pq
be defined as in (4). Then, for

,pq
the estimate



111
max ,d
q
UB L
2qp
ms p
m








is valid.
In this paper, we give the following results about the
one-sided approximation and corresponding
G
best m-term
reedy-liked one-sided algorithm of some Besov classes
by taking the m-term trigonometric polynomials as the
approximation tools. Our results is the following theo-
rems.
Theorem 2.5. For any 1p, 0q, s
, let
,pq
be defined as in (5). Then, for
,,pq

w



e have
11
maxd
1
,2.
qp
ms
UBL
q
p
m




2.6. For
, 0s
and for

 
11 112
,
dqpdq
ms p
m

q
mUBL
 

when 12,p and
1p
, Theorem 1q



112 max1,12,
dq dq
sq
BLm
 
when 2.p
m
mU

e Proofsf the Main Results
2.6, we need
3. Th o
In order to prove Theorem 2.5 and Theorem
following lemmas for
.
n
x
Lemma 3.1. For the d variable trigonometric polyno-
mial
n
x
of degreee n abov, we have
1) If d
x
T then
0
nx
;
2) If π
x
n then
1
nx
;
3)

1
0
n
1n
||
2π
l
x
ln C

, wheC1 is a constant in-
dedent of n;
re
pen
4)
2
d
d
dn
T
x
xC
de dent of n.
n
, where C2 is a constant in-
of. We only prove 4).
pen
Pro
If 0π,t
then from πsin 22,ttt we have
  
2
2
sin
d2 d
22
πππ2
2
22
000
111
2in 2sin
1 d
sin 2
d
ddn
dd
iii
niii
Tiii
iii
nx nxysd
d
dd
x
x
xn ny
nxx y
xnn


 
 
 

 
4) follows from above equalities.
Similarly, we have
Lemma 3.2. For 1p


, 0
j
a, d
lZ there is
positive constant C independent of n, such that
R. S. LI, Y. P. LIU
186
 
1
11
|| 0
π.
|| 0
2π2
p
nn
dp
l
l
n a


(6)
Proof. For the integral properties of

n
nl
p
xlnaC
x
l
mainly
determined by the properties of free variables in the
neighborhood of zero, we have





1
2πd
p
p
l
1
1d
n



1
|| 0|| 0
1
2
12
2
|| 01
2
12π
0
|| 01
2π
2π
2π2
sin
1π2d
2πsin 2π2
2π2
sin
2
d
d
n
nl n
T
ll
p
p
p
dd
ndii
l
Tliii
d
nii
p
l
lii
xlna
nxl n
ax
nx ln
nxl n
a
nx

  

 





 

 





xlnax




1
2
11
2
11
ππ
1
2
π
|| 0|| 0
1
d
π2
sin d2π.
i
i
p
p
i
i
p
pp
d
nn
nl d
pp
i
lil
l
ll
ii
x
ln
y
anyn a
y












 









The proof of Lemma 3.2 is finished.
Proof of Theorem 2.5. First, we consider the upper
estimation. For a given function

d
p
f
LT, mm
T
set

  
1n
|2π|π
|| 0
:2π.
sup
mn m
ylnn
l
Tx
ln fyTy

 
,
m
Tfx
By Lemma 3.1 2) and Remark 1.1, we have
,
m
xTfx
and
f
,
m
Tfx
is a linear combina-
tion of at most 2m exponentials

,
k
ex d
kZ.
When p
, 2q
,
s
, by Definition
ve
2.2, we
ha











 
2
2
1
|2π|π
|| 0
1
212
2 2
|2π|π
|| 0
inf up
2π:,
sup sup
md
md md
n
Tylnn
l
n
n
ylnn
l
fUB L
f TlnT
Lxlnf TfSS
 




 







(7)
ere we hm
(7).
y the for any given natu-
have
2
2
2sup
md
fU
B L
UBL
UB
2πs
nx f

wh
B
ave written
conditions
2 in
eorem 2.5,
n
of Th
ral number m, we
,2.pqd

 Notice that

1max1,120qp in Theorem 2.4. Thus,

12
2
:2.
md
m
SUBL
For any
(8)
,
sq
fUBL
by Lemma 3.2, under the
condition of Theorem 2.5, we have
 



 
1
2
|2π|π
:2π
sup sup
q
n
ylnn
fUB L
ddm
Sx lnf

1
||=0 ||=
2
2
sup
2π,2,
md
n
n
l l
fU
B L
md
y T
lnnfyTfyfT
20
,2π
md
q
nl
f yxlna
(9)



where


2
|2π|π
:,.
sup md
l
ylnn
afyTfy


By the monotonicity of m
and (8), (9), we have



.
d
msq
UB Lm


(10)
y


,fUBL
then,
e-
quence

0
j
When pq,
s
 , for ansq
the definition of Besov classes, there exists a sby
j
Rx
of the trigonometric polyno
coordinate degree 2j such that
 
0
:,
j
j
mials of
xRx
and
f
1
21.
j
jpjl
R
In particular, take

01
,Rx Tfx,
1
22
,,
jj
j
RxTfx Tfx
 , 1, 2,3,.j
Copyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU 187
Here er

,
mfx ar trigo-
nometric approximation operators in (1). From the rela-



the opator Te the best m-term tioapproximation and non-linear ap-
proximation and Lemma 3.2, whave


n between linear
e





 






2
2md
md
2md p
pp
1
|2π|π
||
=0 1
1
=1 ||=0
1
=1 ||=0
inf
sup
2π
sup sup
22 2π
sup sup
22π
q
q
qq
ppg
fUB L
n
nj
ylnn
ljm
fUB Lp
n
jj
jnl
p
jm l
fUB LfUB L
p
n
d
j
jm l
UB Lfg
lnR y
Rlna
na
















EUB L
1
12
:.
p
p
lSS





(11)
Here

|2π|π1.
sup
lj
ylnn jm
aRy
 
Under the
condition of Theorem 2.5, it is easy to see that
22.
jm
1
1
jm
S

(12)
. Set

hy

Next,

1,
j
jm R y

and
we will estimate 2
S


 
1
1
,2π
,1d .
sup
d
p
p
pTyUx n
hnhyhxx





Since the measure of the neighborhood

1
2π2π
ππ
2π,π,
dii
i
jj
Ujnn nnnn

is

2πd
n,
so, by the definition of Besov classes and Minkowskii
inequality, we have

 




 



 
11
1
22π,ππ,π2π,π
|| 0|| 0
1
2π,π2π,π
|| 0
1
1
2π,π2π,π|| 0
:d d
sup
sup
d
sup
pp
n
U lnnlnn
yUln n
ll
p
n
Ulnn
yUln n
l
p
p
n n
UlnnyUlnnl
Sfahyx
hy x
hy hxx

 


 
 












1
2
:
n
p
lU
x
1
h


|| 0l
d
p
hx x
p



 


1
1
2π,π
1
1
1
,2π
d
dd,1.
sup
dd
p
p
Ulnn
p
pp
p
pp
TT
yUx n
hx x
hyhxxhxxh nh




 






For a fixed

12
,, ,
d
d
 
 set
12
||
12
.
d
d
D
x
xx

 By the mathematical induction
d, it is easy to see that on

||
||1
.
d
ss1,1 p
p
Rn
Notice that 2m
n
n DR

. From
modulus [12] and Bernstein i
the properties of smooth
nequality, we have



|||| ||
11
11||1 1||1
1||1
,1 2
.
dd
s
ss
p
|| ||
,1
222
s
1
2 2
p
sm smsm
p
jd
sp
sm
hnnn DRnR
R
 


 

  
 



 

By the conditions of Theorem 2.5, and α > d, we have
dsss
R
n
  


m
j


1,12 m
p
hn
From (11), (12) and (3), we have
2.
m
p
h
Copyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU
188
So


 
221
:,
sup
q
p
fU
B L
SSfh
12.
m
p
nh


 (13)
For sufficiently large m, by (12), (13) and the
mcity of ,
m
onotoni
we have



.
d
UB Lm


(14)
mqp
r cases can be ob-
tai In detail, we may
show them in the following.
,en
The upper estimations for the othe
ned by the embedding Theore.m
If 2qp th.
p
 So for any
B


q
L, by fU
0
21,
j
jqjl
f
we have
s
21,
j
jq
f
for all .j Thus,
2
221.
jj
jj
q
ff

 Hence we have

20
21,
j
jjl
f
i.e.,


2.
f
UB L
So, we have
following embedding relation

2.
q
L L

ave


UB
UB
By (10), we h



2
p
LUBL

then for any j and
(3), we hav
.
d
m
02 ,qp


,
q
L by
mq
m
UB



If
fUB
e21
j
jq
f
different valu, repl
(if q takes
es acing
j
f
by ,
j
T does not influ-
ii inequality (see [1], p.
102) for the inequality), we have
ence the proof). So by Nikol’sk

112
2
22
jd q 2 1.
jj
jj
q
ff

Hence



112dq
sq
L
 and we have folw-
ing embedding formula
fUB
lo





112dq
q sL

2.
s
UB LUB
By (10) we can get







11
2
.
dq
m
112
2
dq
msq ms
p
UB LUBL




0< 2,qp then for any j and

L
If
sq
UB
, we have f
=0
21
s
jqjl
.
jf
By
Nikol’skii inequality we have

11
00
222 .
s
s
jd qp
jj
jj
pq
nj
ll
ff




Thus we have following embedding formula


11 .
dq p
sq sp
UB LUBL

By (14), we have





11
/11 .
dqp
ms pmsq
p
dqp
UB L
m


If 1,pq
UBL

,
sq
fUBL
by

then, for any
=0 s
jqjl
f
21
j
, we have 2
j
n
f
1,
q for any
.jN 221
j
jj
pq
ff and
j

So there hold
0
21.
j
jpjl
f
Therefore, we have


.
sqsp
UB LUB L

By (14) we have



.
d
mspmsq
pp
UB LUB Lm


 

The upper estimation is finished.
By the definition of m
and m
, the lower estima-
tion can be gotten from Theorem 2.4, and the following
relation


2.
msqmsq
p
p
UBB L


Proof of Theorem.
Proof of Theorem 2.6. First, we consider the case
1p
L U
2.5 is finished
2 .qition 2.2 and 2.3, we ha By Definve


q msq






2
2
.
ms
p
p
ms
msq
UBL
UBL
UBL



(15)
By Theorem 2.5, we have
q
UB L

11 .
dqp
UB Lm

ms
q
p
When 12,p
for 2q1 ,
by Theorem 2.5, the
upper estimation is

112
2
.
dq
msq
UB Lm

From (15) we can get
Copyright © 2012 SciRes. APM
R. S. LI, Y. P. LIU
Copyright © 2012 SciRes. APM
189





112dq
2
.
ms
q msq
p
mUB LUBL



)
, 12q, by the relation between
-tximation and Greedy algorithm [7], we
have

(16
Whe
best m
n 2p
erm appro
 
1.
p
mm
12
p
p
f
gf mf
(17)
For
,
sq
fUBL
and any l, by (16), (17) and
Theorem 2.4, we have
 
|2π|π
121 12
sup
lm
ylnn
dq
gfy


(18)
:,
.
m
afy
fgmmm

From Lemma 3.2 and relation (17), (18), w


e have

1
2πsup
n
n
|2π|π
|| 0
1 112/1 121
sup
.
sq
ms
q mm
p
pylnn
l
fUB L
p
dqp dq dq
f gf g
mm mmmm


  






r the case 2.q
xy ln

UBL
When 2,p we conside

By the 2,
jj
q
ff we have



2.
ssq
UB LUB L

(20)
By (19) and (20), we can get





12
2.
d
msqms p
p
UB LUB Lm


 
(21)
In the following we will give the lower estimation. By
on 2.3, we have
.
Definiti



2msqmsq
p
p
UBL
UBL


And by Theorem 2.4, we have




11dqp

when 12,p and



ms
q
p
UB Lm

112dq
msq
p
UB Lm

w
2.6.
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