Convergence of the weak dual greedy algorithm in Lp-spaces

We prove that the weak dual greedy algorithm converges in any subspace of a quotient of Lp when 1opoN: r 2003 Elsevier Inc. All rights reserved. A subset D of a (real) Banach space X is called a dictionary if (i) D is normalized i.e. if gAD implies jjgjj 1⁄4 1: (ii) D is symmetric i.e. D 1⁄4 D: (iii) D is fundamental i.e. 1⁄2D 1⁄4 X : Given xAX we are interested in algorithms which generate a sequence of approximations by n-term linear combinations of members of the dictionary. Many examples of such algorithms have been introduced and studied in approximation theory. We refer to the paper of Temlyakov [11] for a survey of possible algorithms. A desirable feature of a given algorithm is that the sequence of approximations always converge to x (i.e. the algorithm converges). Surprisingly, relatively few general convergence theorems are known for most of the basic algorithms available. In this paper we consider the so-called weak dual greedy algorithm (WDGA). ARTICLE IN PRESS The authors were partially supported by NSF Grants DMS-9870027 and DMS-0244515. Corresponding author. Fax: +573-882-1869. E-mail addresses: [email protected] (M. Ganichev), [email protected] (N.J. Kalton). 0021-9045/$ see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9045(03)00133-3 The weak dual greedy algorithm is a natural generalization to Banach spaces of the so-called pure greedy algorithm (PGA) and its modification the weak greedy algorithm (WGA) for Hilbert spaces. The (PGA) was introduced and first studied by Huber [3]; its convergence was shown by Jones [4]. For the fact that the (WGA) converges in a Hilbert space see [8]; more general results are given in [9] and [7]. Very little is known about the convergence of the (WDGA) for an arbitrary dictionary in a Banach space; see [2]. In [11] it is conjectured that the (WDGA) converges whenever X is a uniformly smooth Banach space with power-type modulus of smoothness. Our main theorem in this paper is that for any subspace of a quotient of Lp when 1opoN the (WDGA) converges for any dictionary, thus proving a special case of the conjecture in [11]. As noted by one of the referees the convergence of the (WDGA) in Lp for 1opoN was previously unknown even for the dictionary consisting of the Haar basis. For any xAX we define the descent rate associated to the dictionary D by rDðxÞ 1⁄4 sup t40 sup gAD jjxjj jjx tgjj t 1⁄4 sup gAD lim t-0þ jjxjj jjx tgjj t : ð1Þ By the Hahn–Banach theorem rDðxÞ 1⁄4 sup jjx jj 1⁄4 1 x ðxÞ 1⁄4 jjxjj sup gAD x ðgÞ: ð2Þ We will usually deal with Banach spaces with a Gateaux differentiable norm, i.e. such that for each xAX \f0g there is a unique x AX with x ðxÞ 1⁄4 jjxjj and jjx jj 1⁄4 1: We denote this functional by Fx: The map x-Fx is norm to weak -continuous on X \f0g; see [1, p. 7]. We set F0 1⁄4 0 for notational convenience. Thus in this case we have rDðxÞ 1⁄4 sup gAD FxðgÞ: ð3Þ Suppose X has a Gateaux differentiable norm. Let us describe the weak dual greedy algorithm (WDGA) with parameter 0oco1: Suppose xAX : We construct a sequence ðgnÞn1⁄41 with gnAD and a sequence ðtnÞ N n1⁄41 of reals with tnX0: Let x0 1⁄4 x and construct ðxnÞn1⁄40; ðgnÞ N n1⁄41; ðtnÞ N n1⁄41 inductively as follows. For each nX1 pick gnAD so that Fxn 1ðgnÞXcrDðxn 1Þ: ð4Þ Pick tnX0 so that jjxn 1 tngnjj 1⁄4 min tX0 jjxn 1 tgnjj: ð5Þ Finally set xn 1⁄4 xn 1 tngn: ð6Þ ARTICLE IN PRESS M. Ganichev, N.J. Kalton / Journal of Approximation Theory 124 (2003) 89–95 90 Thus the n-term approximation to x is given by Pn k1⁄41tkgk and the error is given by xn: The (WDGA) is said to converge at x if limn-Nxn 1⁄4 0 and hence x 1⁄4 PN n1⁄41tngn: The (WDGA) (with parameter c) is said to converge if it converges for every xAX : Let us remark that Temlyakov [11] considers this algorithm for a sequence of parameters ðcnÞn1⁄41 with cn40 replacing c: Thus in place of (4) one has Fxn 1ðgnÞXcnrDðxn 1Þ: ð7Þ A necessary and sufficient condition in Hilbert spaces for convergence of the (WDGA) with a sequence ðcnÞn1⁄41 of parameters is given in [10]. Lemma 1. Let X be a Banach space with a Gateaux differentiable norm and let D be a dictionary in X. Suppose x 1⁄4 x0AX and 0oco1: Suppose further that ðxnÞn1⁄40; ðgnÞn1⁄41 and ðtnÞn1⁄41 are sequences with gnAD; tn40 which satisfy (4) and (6) but not necessarily (5). Suppose that jjxn 1jj jjxnjj tn XcrDðxn 1Þ nX1: ð8Þ Then if PN n1⁄41tn 1⁄4 N we have limn-Nxn 1⁄4 0 and x 1⁄4 XN