The Best m-Term One-Sided Approximation of Besov

1 downloads 0 Views 229KB Size Report
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 ...
Advances in Pure Mathematics, 2012, 2, 183-189 doi: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#

1

School of Information and Technology, Shandong Agricultural University, Tai’an, China School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, China Email: [email protected], #[email protected]

2

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 approximation 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 L p T d 1  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 approximation (Definition 2.2) and the m-term Greedy-liked onesided 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 approximation has many applications in adaptive PDE solvers, compression of images and signal, statistical classification, 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 corresponding 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 approximation 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. d Let T d :  0, 2π  T 1   0, 2π  be the d dimensional torus. For any two elements x   x1 , x2 , , xd  , y   y1 , y2 , , yd   R d , set ek  x  : eikx , k   k1 , k2 , , kd   Z d , where xy denotes the inner product of x and y, i.e., xy  x1 y1  x2 y2    xd yd . Denote by L p T d 1  p    the space of all 2πperiodic and measurable functions f on Rd for which the following quantity



 

f

Copyright © 2012 SciRes.

p

: f

 



f  x  dx p

Td



1 p

,1  p  ,

 ess sup f  x  , p  ,



xT d

is finite. L p T d is a Banach space with the norm  p . For any f  L p T d , we denote by fˆ  k  

 

1

f  x  ek  x  dx,  k  Z d  ,  Td  2π  d

the Fourier coefficients of f (see [13]). For any positive integer m, set n  n  m  :  m1 d  . For any f  L1 T d , as Popov in [11,12], by using the multivariate Fejér kernels,

 

2

 sin nxi 2   n  x  :  π 2     , i 1  n sin xi 2  x   x1 , x2 , , xd   T d , 2d

*

This paper is supported by Shandong Province Higher Educational Science and Technology Program; National Natural Science Foundation of China (project No.110771019); partly by The research Fund for the Doctoral Program of Higher Education and partly by Beijing Natural Science Foundation (Project 1102011). # Corresponding author.



d

we defined APM

184

R. S. LI, Y. P. LIU

Tm  f , x  : Tm  f , x 

l p norm

n 1

   n  x  2πl n  |l |  0

sup y  2πl n  π n

f  y   Tm  f , y  ,

(1)

and called it to be the best m-term one-sided trigonometric approximation operators, where and in the sequel the operator Tm  f , x  is the best m-term trigonometric n 1 approximation operators and  |l | 0 denotes

 l 0 l  0  l n 1

n 1

n 1

1

2

d

f  x  T

 f , x.

 m

. It is easy to see that

0

 

g m  f , x  : g m  f , x  n 1

|l |  0

sup y  2πl n  π n

f  y   gm  f , y  ,

m where g m  f , x    i 1 fˆ  k  i   ek  i  and

 fˆ  k  i 



i 1

(2)

is

a sequence determined by the Fourier coefficients fˆ  k  of f in the decreasing rearrangement, i.e., d





k Z

f  k 1   f  k  2     . It is easy to see that two operators Tm and g m are non-linear. We will see that for any x  T d , g m  f , x   f  x  (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 operator Tm  f , x  , 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 Tm  f , x  is the ideal one. In Theorem 2.6, by using the properties of the approximation operator g m  f , 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 nonlinear manifold consists of complex trigonometric polynomials T, where each trigonometric polynomial T can be written as a linear combination of at most m exponenttials ek  x  , k  Z d . Thus T  m if and only if there exits   Z d such that   m and

T  x 

 ck ek  x  ,

k 

where  is the cardinality of the set . Let D be a finite or infinite denumerable set. Denote by l p  D 1  p    the space of all subset of some with the following finite complex numbers X   x j  j D

Copyright © 2012 SciRes.

,1  p  ; X

 

For any f  L1 T d , let

 fˆ  k 

: sup x j .

l

k Z d

j D

be the set of

Fourier coefficients of f. As in the page 19 of [14], denote by f  fˆ  k  lp

Meantime, for any f  L1 T d , we also defined    n  x  2πl n 

1 p

p   X l  D  :   x j  p  j D 





 

 d l p d

the l p norm of the set of Fourier coefficients of f. Throughout this paper, let  n denote the set of the trigonometric polynomials of d variables and degree n with the form T   |k | n Tˆ  k  ek and q   n  denote the set of all trigonometric polynomials T in  n such that T : Tˆ  k   1.



q   n 



 

k Z d l Z d q

Here we take as Tˆ  k   0 if k  n, k : max  k1 , k2 , , kd  . Definition 2.1. (see cf. [1]) For a given function f, we call  m  f  p : inf f  T p T m

the best m-term approximation error of f with trigonometric polynomials under the norm Lp. For the function set A  L p T d , we call

 

 m  A  p : sup  m  f  p f A

the best m-term approximation error of the function class A with trigonometric polynomials under the norm Lp. Definition 2.2. (see cf. [3,4]) For given function f, set

 m : T T   2m , T  



f . The quantity

 m  f  p : inf f  T T m

p

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 A  L p T d , the quantity

 



 m

 A p : sup   f  p f A

 m

is called to be the best m-term one-sided approximation error of the function class A with trigonometric polynomials under the norm Lp. Definition 2.3. (see cf. [3,4]) For given function f, we call g m  f , x  (given by relation (2)) the Greedy-liked algorithm of the best m-term one-sided approximation of f under trigonometric function system. For given function set A  L p T d , we call

 

 m  A  p : sup f  g m  f , x  f A

p

APM

185

R. S. LI, Y. P. LIU

the Greedy-liked one-sided approximation error of the best m-term one-sided approximation of function class A given by trigonometric polynomials with norm Lp. As in [1,15], denote by Bs  Lq  ,   0, 0  q, s   , the Besov space. The definition of the Besov space is given by using the following equivalent characterization.



A function f is in the unit ball U Bs  Lq 



of the Besov

space Bs  Lq  , if and only if there exist trigonometric polynomials R j ( x) :  |k | 2 j c jk ek  x  , such that f  x  :  j  0 R j  x  and 

2

j

Here    0,1, 2, . In the case 1  q  , we can take R j  f j :  fˆ  k  ek , j  1, f 0 : fˆ  0  e0 , 2 j 1  |k | 2 j d d

k   k1 , k2 , , k   Z , k  max  k1 , k2 , , kd  . We define the seminorm f B  L  as the infimum s

q



over all decompositions (3) and denote by U Bs  Lq 

the unit ball with respect to this seminorm. Throughout this paper, for any two given sequences of   non-negative numbers  n n 1,  n n 1 if there is a nonnegative constant c independent of all n, such that

Rj

q



 j 0 l  s 

 1.

(3)

 n  c  n , then we write  n   n . If both  n   n and  n   n hold, then we write  n   n . For any 1  p   , 0  q, s   , set

 1 1  d    , 0  q  p  2 and 1  p  q  ,   q p    p, q  :  max  d , d  , otherwise,    q 2 

(4)

 d    p, q  , 0  q  p  2 and 1  p  q  , otherwise.    p, q  ,

(5)

and

  p, q  : 



For the unit ball U Bs  Lq 



of the Besov spaces

Bs  Lq  , Devore and Temlyakov in [1] gave the following result : Theorem 2.4. (c.f. [1]) For any 1  p   , 0  q , s   , let   p, q  be defined as in (4). Then, for     p, q  the estimate

 

 m U Bs  Lq 



p

m

1  1 1   d    max  ,    p 2   q

 



p

m

1  1 1   d    max  ,    p 2   q

T

2d

d

2



 0 i 1

 d  1 q 1 p  

when 1  p  2, and m

 d  1 q 1 2  

 



 



  m U Bs  Lq 

p

m

  m U Bs  Lq   m

 d  1 q 1 2  

 d  max1 q ,1 2

,

,

when 2  p  .

In order to prove Theorem 2.5 and Theorem 2.6, we need following lemmas for  n  x  . Lemma 3.1. For the d variable trigonometric polynomial  n  x  of degree n above, we have 1) If x  T d then  n  x   0 ; 2) If x  π n then  n  x   1 ; 3)

n1

  n  x  2πl n   C1 , where C1 is a constant in-

|l |  0

dependent of n; 4)  d  n  x  dx  C2 n d , where C2 is a constant independent of n. Proof. We only prove 4). If 0  t  π, then from t π  sin t 2  t 2, we have 2

2

d d  sin nxi 2  nπ 2  sin yi  π  sin nxi 2  2d 2 d d   dxi  1 n   0   dxi  n n 0   dyi  n n sin x 2 x y   i i 1 1      i  i i

4) follows from above equalities. Similarly, we have Copyright © 2012 SciRes.

m

T

.

Theorem 2.6. For 1  p   , 1  q   , 0  s   d  n  x  dx    2 

and for     p, q  , we have

3. The Proofs of the Main Results

is valid. In this paper, we give the following results about the best m-term one-sided approximation and corresponding Greedy-liked one-sided algorithm of some Besov classes by taking the m-term trigonometric polynomials as the approximation tools. Our results is the following theorems. Theorem 2.5. For any 1  p   , 0  q , s   , let   p, q  be defined as in (5). Then, for     p, q  , we have

 m U Bs  Lq 



Lemma 3.2. For 1  p   , a j  0 , l  Z d there is positive constant C independent of n, such that APM

186

R. S. LI, Y. P. LIU

Proof. For the integral properties of  n  x  mainly determined by the properties of free variables in the neighborhood of zero, we have

1 p

n 1   d  C  2 π n   alp  |l |  0  

n 1

  n  x  2πl n  al

|l |  0

p

n1

  n  x  2πl n  al

|l |  0

p

. (6)

1 p

 1  d    2π  

p    n 1 x l n a x 2π d         d n l T |l|0   

 d  1      2π  

p   n 1 d sin 2  n  x  2πl n  2   2d  i i   dx   π 2  al  T d  | 2 l |0 i 1  n sin  x  2πl n  2    i i   

1 p

1 p

p  n 1   2  d 2π sin  n  xi  2πli n  2   p  dxi     al  0  2 |l |0 i 1   n  xi  2πli n  2      1 p

p 2 d nπ  πli  sin y   n 1     alp  n 1  πl  2 i  dyi  i  yi  |l |0 i 1 

 

n1

|l |  0



 2md U  B  L2  





sup

| y  2 πl n| π n



tion of at most 2m exponentials ek  x  , k  Z d . When p   , q  2 , s   , by Definition 2.2, we have

f  y   Tm  y  .

  inf  f  T T    L f U  B  2    2md  sup

 

  2md U B  L2  











sup



n 1

  n  x  2πl n  | y  2sπlupn| π n

sup  L f U  B  q 

  2πl n  n d

 f U B  L2 

n 1



where al :

sup

| y  2πl n| π n

|l |=0

2



 f  T2md

 m U Bs  Lq 





 m  d .





2

 m

quence







: S1  S 2 , 

 2 m .

(8)



sup  f U  B  Lq  



n 1

  n  x  2πl n  al

|l |= 0



(9)

,

R  x 



j

j 0

of the trigonometric polynomials of 

(10)



When p  q , s   , for any f  U Bs  Lq  , then, 

f  T2md  f 

coordinate degree 2j such that f  x  :  j  0 R j  x  , and

By the monotonicity of  m and (8), (9), we have

by the definition of Besov classes, there exists a seCopyright © 2012 SciRes.



(7)

For any f  U Bs  Lq  , by Lemma 3.2, under the condition of Theorem 2.5, we have

md

f  y   T md  f , y 

 

      

S1 :  2md U B  L2 

n 1

f  y   T2md  f , y  .

 

  n  x  2πl n  | y  2πsup l n|  π n

|l |  0

f  y   T2  f , y    n  x  2πl n  | y  2πsup l n| π n

d

f T

|l |  0

where we have written n  2m in (7). By the conditions of Theorem 2.5, for any given natural number m, we have     p, q   d 2. Notice that 1 q  max 1 p ,1 2  0 in Theorem 2.4. Thus, S2 :

.

By Lemma 3.1 2) and Remark 1.1, we have f  x   Tm  f , x  and Tm  f , x  is a linear combina-

The proof of Lemma 3.2 is finished. Proof of Theorem 2.5. First, we consider the upper estimation. For a given function f  L p T d , Tm   m set Tm  f , x  : Tm    n  x  2πl n 

1 p

n 1   d   2π n   alp  |l |  0  

2

j

Rj

p



 j 1 l 

 1.

In particular, take R0  x   T1  f , x  , R j  x   T2 j  f , x   T2 j1  f , x  , j  1, 2, 3,. APM

187

R. S. LI, Y. P. LIU

Here the operator Tm  f , x  are the best m-term trigonometric approximation operators in (1). From the rela-

 

 2md U B  L p 



 sup  inf f  g   L f U  B  q    g2md



p

tion between linear approximation and non-linear approximation and Lemma 3.2, we have



sup  L f U  B  q 

p

   E2md U B  L p   

 

n 1



|l |=0

j  m 1

sup  2 j 2 j R j = j     m1

Here

al  sup | y  2πl





2 j  2 m .

and 1 p

 n 1  S2  f  :    a p dx  U  2πl n ,π n  l  |l |  0 

1 p





1 p

p       T d  sup h  y   h  x   dx     yU  x ,2π n  





1 p

 dx  

p

1 p

p  n 1      h  y   h  x   dx  sup  U  2πl n ,π n   yU  2πl n ,π n   |l |0 

1 p

 n 1  p   h  x  dx  U  2πl n ,π n  |l | 0 

T

 D    . By the mathematical induction 1 x1 x2 2  xdd on d, it is easy to see that



 p







s  m 1|  | 1

By the conditions of Theorem 2.5, and α > d, we have

 1  h,1 n  p  2 m Copyright © 2012 SciRes.

d

s  m 1|  | 1

d

 1  h,1 n  p  h p . d

 n |  |

|  | 1

D  Rs

p

.

Notice that n  2m . From the properties of smooth modulus [12] and Bernstein inequality, we have

  n |  |

  n| | 2s| | 2 s  2s

1 p

 1  Rs ,1 n  p 

| |

 Rs ,1 n 



h  x  dx p

d

For a fixed    1 , 2 ,  d    d , set

 s  m 1

.

 n 1  p :    h  y  dx  sup U  2 πl n ,π n  yU  2πl n ,π n   |l |  0 

 n 1    U 2πl n,π n h  y   h  x  h  x sup   yU  2πl n,π n   |l |  0



p        d  sup h  y   h  x   dx  T    yU  x ,2π n  

Since the measure of the neighborhood π 2 j π π d 2 j π d U  2πj n , π n    i 1  i  , i   is  2 π n  , n n n  n so, by the definition of Besov classes and Minkowskii inequality, we have

(12) 



p

: S1  S2 .

 1  h,1 n  p

j  m 1

 1  h,1 n  p   1 

|l |=0

1 p

Under the

Next, we will estimate S2 . Set h  y    j  m 1R j  y  ,



  n 2πl n  al

sup   

n 1   d 2 j   2π n   alp  |l |=0 j = m 1  

condition of Theorem 2.5, it is easy to see that S1 

(11)

1 p



 j  m 1 R j  y  .

p

n 1

 f U B Lq





n|  π n



p

 f U B Lq



p

  n  2πl n  | y  2πsup  Rj  y l n| π n







Rs

p

D  Rs

2

 m

p





d

  n |  | 2 s |  |

s  m 1|  | 1

Rs

p



 2  j   d . j 1

From (11), (12) and (3), we have h

p

 2 m .

APM

188

R. S. LI, Y. P. LIU

2

So sup S2  f   1  h,1 n  p  h  f U  B  Lq  

S2 :

 2 m . (13)

p



U  B  L  



q

m

p

 d

(14)

.

The upper estimations for the other cases can be obtained by the embedding Theorem. In detail, we may show them in the following. If 2  q  p  , then  p    . So for any



2



f  U B  Lq  , by 2 j f j 2



j



q

fj

2

2 j f j

j

fj

q





 1, we have

j 0 l s

 1, for all j  . Thus, 2 2



j

fj

q

 1. Hence we have



 1, i.e., f  U B  L2  . So, we have

j 0 l 









 

  m U B  L2 

p

 

 m U Bs  Lq 







 m  d .

q

different values, replacing f j by T j , does not influence the proof). So by Nikol’skii inequality (see [1], p. 102) for the inequality), we have  jd 1 q 1 2 



fj

2

 2 j f j

 L 

  d 1 q 1 2 

Hence f  U Bs



q

 1.

and we have follow-

q





  d 1 q 1 2 



j 0 l s

.

p

 L  p

p

.





2

j

fj

2

j



q

 j =0 l s

 1 , we have 2 j f n

So there hold 2 j f j fj



p



p

q

 1, for any

 2 j f j

q

 1 and

 1. Therefore, we have

j 0 l 









p

 



  m U Bs  L p 

p

 m  d .

The upper estimation is finished. By the definition of  m and  m , the lower estimation can be gotten from Theorem 2.4, and the following relation

 

 m U Bs  Lq 



 

  2 m U Bs  Lq 

p



p

.

Proof of Theorem 2.5 is finished. Proof of Theorem 2.6. First, we consider the case 1  p  2  q  . By Definition 2.2 and 2.3, we have

 

 m U Bs  Lq 



p

   U  B  L  U  B  L  .

  m U Bs  Lq 

  m

p



s

q

2

(15)



s

q

2

By Theorem 2.5, we have p

 

  d 1 q 1 2 

  m U Bs

m

 d  1 q 1 2 



f  U Bs  Lq  , we have Nikol’skii inequality we have Copyright © 2012 SciRes.

2

j

 L2   

.

fj

 

 m U Bs  Lq 



p

m

 d  1 q 1 p 

.

When 1  p  2, for 1  q  2, by Theorem 2.5, the upper estimation is

If 0 < q  p  2, then for any j and





If 1  p  q  , then, for any f  U Bs  Lq  , by

 L2   .

By (10) we can get

 



 L .

  d 1 q 1 p  

 / d  1 q 1 p 

  m

U Bs  Lq   U Bs

 m U Bs  Lq 

  d 1 q 1 p 

 

ing embedding formula



q

  m U Bs

p

m

 

f  U B  Lq  , by (3), we have 2 j f j  1 (if q takes

2 j 2

 2 j f j

By (14) we have

If 0  q  2  p  , then for any j and







 m U Bs  Lq 







p n 0 ls



By (10), we have

 



U Bs  Lq   U Bs  L p  .

U B  Lq   U B  L2  .

 m U B  Lq 

fj

By (14), we have

following embedding relation



 jd 1 q 1 p 

U Bs  Lq   U Bs

j  N.





2

Thus we have following embedding formula

For sufficiently large m, by (12), (13) and the monotonicity of  m , we have  m

j

q



 j =0 l s

 1. By

 

 m U Bs  Lq 



2

m

 d  1 q 1 2  

.

From (15) we can get APM

189

R. S. LI, Y. P. LIU

 

 m U Bs  Lq 



p

 

  m U Bs  Lq 



2

m

 d  1 q 1 2 



Theorem 2.4, we have

(16)

al :

When 2  p   , 1  q  2 , by the relation between best m-term approximation and Greedy algorithm [7], we have f  gm  f 

m

p

1 2 1 p

 

 m U Bs  Lq 



m  f p .



p

m

 d

1 q 1 p

m

When 2  p  , we consider the case 2  q  . By the f j  f j , we have 2



q







U Bs  L2   U Bs  Lq  .

(20)

By (19) and (20), we can get

 

 U Bs  Lq   m





p



 m

U  B



s

 L2    p  m

 d 1 2

p



n 1

|l |  0  / d

m1 q 1 2  m 

 



 

  2m U Bs  Lq 

p



p

.

 



p



p

when 1  p  2, and

 

 m U Bs  Lq 

m

 d  1 q 1 2  

when 2  p  . This finishes the proof of Theorem 2.6.

REFERENCES [1]

[2]

R. A. Devore and V. N. Temlyakov, “Nonlinear Approximation by Trigonometric Sums,” Journal of Fourier Analysis Application, Vol. 2, No. 1, 1995, pp. 29-48. doi:10.1007/s00041-001-4021-8 V. N. Temlyakov, “Greedy Algorithm and m-Term Trigonometric Approximation,” Constructive Approximation, Vol. 14, No. 4, 1998, pp. 569-587. doi:10.1007/s003659900090

[3]

R. S. Li and Y. P. Liu, “The Asymptotic Estimations of Best m-Term One-Sided Approximation of Function Classes Determined by Fourier Coefficients,” Advance in Mathematics (China), Vol. 37, No. 2, 2008, pp. 211-221.

[4]

R. Li and Y. Liu, “Best m-Term One-Sided Trigonometric Approximation of Some Function Classes Defined by a Kind of Multipliers,” Acta Mathematica Sinica, English

Copyright © 2012 SciRes.

   p

.

T. Ganelius, “On One-Sided Approximation by Trigonometrical Polynomials,” Mathematica Scandinavica, Vol. 4, 1956, pp. 247-258.

[6]

E. Schmidt, “Zur Theorie der Linearen und Nichtlinearen Integralgleichungen,” Annals of Mathematics, Vol. 63, 1907, pp. 433-476. doi:10.1007/BF01449770

[7]

A. S. Romanyuk, “Best m-Term Trigonometric Approximations of Besov Classes of Periodic Functions of Several Variables,” Izvestiya: Mathematics, Vol. 67, No. 2, 2003, pp. 265-302. doi:10.1070/IM2003v067n02ABEH000427

[8]

S. V. Konyagin and V. N. Temlyakov, “Convergence of Greedy Approximation II. The Trigonometric Systerm,” Studia Mathematica, Vol. 159, No. 2, 2003, pp. 161-184. doi:10.4064/sm159-2-1

[9]

V. N. Temlyakov, “The Best m-Term Approximation and Greedy Algorithms,” Advances in Computational Mathematics, Vol. 8, No. 3, 1998, pp. 249-265. doi:10.1023/A:1018900431309

 d  1 q 1 p 

m

d 1 q

f  gm

[5]

And by Theorem 2.4, we have

 m U Bs  Lq 

(18)

Series, Vol. 26, No. 5, 2010, pp. 975-984. doi:10.1007/s10114-009-6478-3

In the following we will give the lower estimation. By Definition 2.3, we have

 m U Bs  Lq 

 m1 2 m  d m1 q 1 2 .

  n  x y  2πl n  | y  2πsup l n| π n

.

(21)



f  y   gm  f , y 

From Lemma 3.2 and relation (17),(18), we have

m m 12

sup

| y  2πl n| π n

 f  gm

(17)

 sup  f  g m  f U  Bs  Lq   



For f  U Bs  Lq  , and any l, by (16), (17) and

.

[10] R. Li and Y. Liu, “The Asymptotic Estimations of Best m-Term Approximation and Greedy Algorithm for Multiplier Function Classes Defined by Fourier Series,” Chinese Journal of Engineering Mathematics, Vol. 25, No. 1, 2008, pp. 89-96. doi:10.3901/JME.2008.10.089 [11] V. A. Popov, “Onesided Approximation of Periodic Functions of Serveral Variables,” Comptes Rendus de Academie Bulgare Sciences, Vol. 35, No. 12, 1982, pp. 1639-1642. [12] V. A. Popov, “On the One-Sided Approximation of Multivariate Functions,” In: C. K. Chui, L. L. Schumaker and J. D. Ward, Eds., Approximation Theory IV, Academic Press, New York, 1983. [13] A. Zygmund, “Trigonometric Series II,” Cambridge University Press, New York, 1959. [14] R. A. Devore and G. G. Lorentz, “Constructive Approximation,” Spring-Verlag, New York, Berlin, Heidelberg, 1993. [15] R. A. Devore and V. Popov, “Interpolation of Besov Spaces,” American Mathematical Society, Vol. 305, No. 1, 1988, pp. 397-414.

APM