The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials *

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    1 d p L T p        are given.


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][8][9][10] on the best m term approximation problem and see [11,12] on the best one-sided approximation problem.
  1 : 0,2π 0, 2π , , , d be the d dimensional Let , where xy denotes the inner product of x and y, i.e., x is a Banach space with the norm is finite.

 p
For any the Fourier coefficients of f (see [13]).
For any positive integer m, set . [11,12], by using the multivariate Fejér kernels, , , , , and called it to be the best m-term one-sided trigonometric approximation operators, where and in the sequel the operator m is the best m-term trigonometric approximation operators and  denotes 1 2 Meantime, for any we also defined where It is easy to see that two operators and T g  are non-linear.We will see that for any , The main results of this paper are Theorems 2.5 and 2.6.In Theorem 2.5, by using the properties of the operator 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 , 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.

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 exponentials , .Thus Here we take as   [ For a given function f, we ca the best m-term approximation error of f with trigonometric polynomials under the norm L p .For the function set

  ,
is called to be the best m-term one-sided approximation error of f with trigonometric polynomials under the norm L p .For given function set is called to be the best m-term one-sided approximation

APM the Greedy-liked one-sided approximation the best m-term one-sided approximation of function class
In the case we can error of Here given by trigonometric polynomials with norm L p .
As in [1,15], denote by , the Besov space.The definition of the Besov space is gi equivalent ch acterization ven by using the following ar .
A function f is in the unit ball f  as the infimum over all decompositions (3) and denote by the unit ball with respect to this seminorm.Throughout this paper, for any two given sequences of , Devore an v in [1] gave the follow B Temlyako ing result:  be defined as in (4).
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 theorems.
Theorem 2.5.For any   be defined as in (5).Then, for   e Proofs f the Main Results 2.6, we need

Th o
In order to prove Theorem 2.5 and Theorem following lemmas for  .
of degre e e n abov , we have 1) If , where C 2 is a constant inof.We only prove 4). pen follows from above equalities.Similarly, we have Proof.For the integral properties of  mainly determined by the properties of free variables in the neighborhood of zero, we have The proof of Lemma 3.2 is finished.Proof of Theorem 2.5.First, we consider the upper estimation.For a given function ere we h m  (7).y the for any given natuhave ave written conditions 2 in eorem 2.5, by Lemma 3.2, under the condition of Theorem 2.5, we have where By the monotonicity of m   and ( 8), ( 9), we have the definition of Besov classes, there exists a s by 1 .
In particular, take , , Here er   , m f x ar trigonometric approximation operators in (1).From the rela- the op ator T e the best m-term tio approximation and non-linear approximation and Lemma 3.2, w have Here Under the condition of Theorem 2.5, it is easy to see that 2 2 .
. Set   so, by the definition of Besov classes and Minkowskii inequality, we have : , , , From modulus [12] and Bernstein i the properties of smooth nequality, we have .
For sufficiently large m, by ( 12), (13) and the m city of , r cases can be obtai In detail, we may show them in the following.
, en The upper estimations for the othe ned by the embedding Theore .m then for any j and (3), we hav different valu , repl (if q takes es acing j f by , j T does not influii inequality (see [1], p. 102) for the inequality), we have ence the proof).So by Nikol'sk and we have fol wing embedding formula   then for any j and Nikol'skii inequality we have Thus we have following embedding formula .
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 By Theorem 2.5, we have by Theorem 2.5, the upper estimation is the space of all 2πperiodic and measurable functions f on R d for which the following quantity by the Fourier coefficients of f in the decreasing rearrangement, i.e., -term approximation error of the function class set   A with trigonometric polynomials under the norm L p .Definition 2.2.(see cf. [3,4]) For given function f,  

, 4 ]
) For given function f, we ca     error of the function class A with trigonometric polynomials under the norm L p .Definition 2.3.(see cf. relation (2)) the Greedy-liked algorith e best m-term one-sided approximation of f under trigonometric function system.For given function set m of th Journal of Fourier Analysis Application, Vol. 2, No. 1, 1995, pp.29-48.
This finishes the proof of Theorem