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 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., 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 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 xTfx 1, d fLT Meantime, for any we also defined 1 2ππ || 0 ,:gfx g, 2π,, sup mm n nm ylnn l fx lnfygf y 1 ˆ (2) where ,m mi ki 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 are non-linear. We will see that for any , d T , m 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 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 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 D Xx with the following finite l norm 1 :,1;:. sup p p p 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 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 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 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 fA 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 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 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 , 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 , 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 jk k Rf fke 00 ˆ :0 fe, 1, ,,, d kkkkZ 12 d, 12 max,, ,kkkk. d We define the seminorm 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 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 Lemma 3.1. For the d variable trigonometric polyno- mial n of degreee n abov, we have 1) If d T then 0 nx ; 2) If π n then 1 nx ; 3) 1 0 n 1n || 2π l ln C , wheC1 is a constant in- dedent of n; re pen 4) 2 d d dn T 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 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 nn dp l l n a (6) Proof. For the integral properties of n nl p xlnaC 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 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 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 , m Tfx is a linear combina- tion of at most 2m exponentials , k ex d kZ. When p , 2q , , 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, , for ansq the definition of Besov classes, there exists a sby Rx of the trigonometric polyno coordinate degree 2j such that 0 :, j j mials of xRx and 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 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 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 Ulnn p pp p pp TT yUx n hx x hyhxxhxxh nh For a fixed 12 ,, , d d set 12 || 12 . d d D xx By the mathematical induction d, it is easy to see that on || ||1 . d ss1,1 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 2qp 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. 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 by , 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 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 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 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 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 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 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. REFERENCES lyakov, “Nonlinear ums,” Journal of Fourier Analysis Application, Vol. 2, No. 1, 1995, pp. 29-48. hen 2.p This finishes the proof of Theorem [1] proximation by Trigonometric S R. A. Devore and V. N. Tem Ap- doi:10.1007/s00041-001-4021-8 . N. Temlyakov, “Greedy Algorithm and m-Term Trigo- nometric Approximation,” Constructive Approximation Vol. 14, No. 4, 1998, pp. 569-587. oi:10.1007/s003659900090 [2] V , d [3] Estimations of of Function Classes DetermineAdvance in Mathematics (008, pp. 211-221. Series, Vol. 26, No. 5, 2010, pp. 975-984. doi:10.1007/s10114-009-6478-3 [5] T. Ganelius, “On One-Sided Approximation by Trigono- metrical 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 Approxi- mations of Besov Classes of Periodic Functions of Sev- eral 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 Math- ematics, Vol. 8, No. 3, 1998, pp. 249-265. doi:10.1023/A:1018900431309 [10] R. Li and Y. Liu, “The Asymptotic Estimations of Best .089 m-Term Approximation and Greedy Algorithm for Mul- tiplier Function Classes Defined by Fourier Series,” Chi- nese Journal of Engineering Mathematics, Vol. 25, No. 1, 2008, pp. 89-96. doi:10.3901/JME.2008.10 R. S. Li and Y. P. Liu, “The Asymptotic Best m-Term One-Sided Approximation d by Fourier Coefficients,” China), Vol. 37, No. 2, 2 [4] R. Li and Y. Liu, “Best m-Term One-Sided Trigonomet- ric Approximation of Some Function Classes Defined by a Kind of Multipliers,” Acta Mathematica Sinica, English mation of Periodic Func- One-Sided Approximation of Mul- , Heidelberg, [11] V. A. Popov, “Onesided Approxi tions of Serveral Variables,” Comptes Rendus de Academie Bulgare Sciences, Vol. 35, No. 12, 1982, pp. 1639-1642. [12] V. A. Popov, “On the tivariate 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 Uni- versity Press, New York, 1959. [14] R. A. Devore and G. G. Lorentz, “Constructive Approxi- mation,” Spring-Verlag, New York, Berlin 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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