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 ,
xT 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
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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
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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
2π
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 max1 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)
n1
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
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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
n1
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
n1
|l | 0
2md 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 j1 f , x , j 1, 2, 3,. APM
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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-
2md U B L p
sup inf f g L f U B q g2md
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 m1
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 yU x ,2π n
1 p
dx
p
1 p
p n 1 h y h x dx sup U 2πl n ,π n yU 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 xdd 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 yU 2πl n ,π n |l | 0
n 1 U 2πl n,π n h y h x h x sup yU 2πl n,π n |l | 0
p d sup h y h x dx T yU 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 .
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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
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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.
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[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
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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
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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.
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