Weak Convergence of Greedy Algorithms in Banach Spaces

Transcription

1 J Fourier Anal Appl (2008) 14: DOI /s Weak Convergence of Greedy Algorithms in Banach Spaces S.J. Dilorth Denka Kutzarova Karen L. Shuman V.N. Temlyakov P. Wojtaszczyk Received: 30 March 2007 / Published online: 16 September 2008 Birkhäuser Boston 2008 Abstract We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the eak topology for general dictionaries in uniformly smooth Banach Communicated by Ronald A. DeVore. The research of the first author as supported by NSF Grant DMS The research of the third author as supported by the Grinnell College Committee for the Support of Faculty Scholarship. The research of the fourth author as supported by NSF Grant DMS and by ONR Grant N J1343. The research of the fifth author as supported by KBN grant 1 PO03A and the Foundation for the Polish Science. The fifth author thanks the Industrial Mathematics Institute of the University of South Carolina and Prof. R. DeVore and Prof. V.N. Temlyakov personally for their hospitality. The first and second authors ere supported by the Concentration Week on Frames, Banach Spaces, and Signals Processing and by the Workshop in Analysis and Probability at Texas A&M University in Current address of D. Kutzarova: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA S.J. Dilorth ( ) V.N. Temlyakov Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA dilorth@math.sc.edu V.N. Temlyakov temlyak@math.sc.edu D. Kutzarova Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria denka@math.uiuc.edu K.L. Shuman Department of Mathematics, Grinnell College, Ioa 50112, USA shumank@math.grinnell.edu P. Wojtaszczyk Institute of Applied Mathematics, Warsa University, Banacha 2, Warsa, Poland ojtaszczyk@mimu.edu.pl

2 610 J Fourier Anal Appl (2008) 14: spaces ith the WN property. We sho that reflexive spaces ith the uniform Opial property have the WN property. We sho that our results do not extend to algorithms hich employ a dictionary dual greedy step. Keyords Greedy algorithms Weak convergence Uniformly smooth Banach spaces Uniform Opial property Mathematics Subject Classification (2000) Primary 41A65 Secondary 42A10 46B20 1 Introduction We study convergence in the eak topology of different greedy algorithms acting in a uniformly smooth Banach space. The first result on convergence of a greedy algorithm is the result of Huber [10]. He proved convergence of the Pure Greedy Algorithm in the eak topology of a Hilbert space H and conjectured that the Pure Greedy Algorithm converges in the strong sense (in the norm of H ). L. Jones [12] proved this conjecture. The theory of greedy approximation in Hilbert spaces is no ell developed (see [20]). Our interest in this paper is in convergence results for greedy approximation in Banach spaces. There is a number of open problems on convergence (in the strong sense) of different greedy algorithms in Banach spaces (see [20]). Let X be a Banach space. We say that D X is a dictionary if the folloing conditions are satisfied: (i) the linear span of D is norm-dense in X; (ii) g =1for all g D; (iii) D is symmetric, i.e. g D g D. For some algorithms (the Weak Chebyshev Greedy Algorithm ([6, 19]), the Weak Greedy Algorithm ith Free Relaxation ([22]) it is knon that the uniform smoothness of X guarantees strong convergence of these algorithms for each element f X and any dictionary D in X. For other algorithms (the Weak Dual Greedy Algorithm, the X-Greedy Algorithm) it is not knon if uniform smoothness (even uniform smoothness ith a poer type modulus of smoothness) guarantees convergence in the strong sense for each f X and all D. There is a result of M. Ganichev and N. Kalton [8] that establishes strong convergence of the Weak Dual Greedy Algorithm in a uniformly smooth Banach space satisfying an extra condition Ɣ. In this paper e impose the folloing extra condition (the WN property) on a uniformly smooth Banach space and prove the eak convergence of some greedy algorithms. This condition as used implicitly in [6]. For f X, f 0, let F f X denote a norming functional of f,i.e, F f X = 1 and F f (f ) = f X. Definition 1.1 X has the WN property if every sequence {x n } X, ith x n X = 1, is eakly null in X henever the sequence {F xn } is eakly null in X. We prove eak convergence of the X-Greedy Algorithm for a uniformly smooth Banach space satisfying the WN property. We note that there are no results on

3 J Fourier Anal Appl (2008) 14: strong convergence of the X-Greedy Algorithm in nontrivial Banach spaces (infinitedimensional nonhilbertian spaces). We give an example that demonstrates that the WN property does not imply the Ɣ property. The use of the norming functional F fm of a residual f m of a greedy algorithm after m iterations for a search of an element from the dictionary to be added in approximation has proved to be very natural. Norming functionals for residuals are used in dual type greedy algorithms. At a greedy step of a dual type greedy algorithm e look for an element ϕ m D satisfying F fm 1 (ϕ m ) t sup F fm 1 (g), t (0, 1]. (1) g D In this paper e discuss the folloing modification of the greedy step (1): e look for an element ϕ m D satisfying F ϕm (f m 1 ) t sup F g (f m 1 ). (2) g D It is knon (see [21]) that the modification (2) is useful in the problem of exact recovery of sparsely represented elements in the case of dictionaries ith small coherence. We sho here that the modification (2) is not good for convergence of greedy approximations ith regard to general dictionaries. Let us no describe the organization of the paper. Section 2 recalls the relevant definitions from Banach space theory. In Sect. 3 e recall the definitions of the greedy algorithms hich are considered here and e present our main results on the eak convergence of these algorithms in uniformly smooth spaces ith the WN property. We also sho ho the algorithms can easily be modified to obtain strong convergence. In Sect. 4 e sho that many classes of Banach spaces have the WN property, e.g. the class of reflexive spaces ith the uniform Opial property, and e exhibit examples of spaces hich have the WN property but not the property Ɣ introduced by Ganichev and Kalton [8]. Finally, e sho that our results for general dictionaries do not apply to algorithms hich use (2) as the greedy step. In fact, e sho that if X is not isometric to a Hilbert space (and satisfies additional mild regularity conditions) then there exists a dictionary in X for hich such algorithms break don quite badly. 2 Definitions and Notation We use standard Banach space notation and terminology as in [15]. In this section e record some terminology hich may not be completely standard. 2.1 Bases of Banach Spaces and Related Notions Let {e i } be a (Schauder) basis for a real Banach space X and let {ei } be its sequence of biorthogonal functionals. For x,y X, e rite x<yif the support of x is to the left of the support of y,i.e.if max{n N: en (x) 0} < min{n N: e n (y) 0}.

4 612 J Fourier Anal Appl (2008) 14: Definition 2.1 A basis {e i } for a Banach space X is uniformly reverse monotone (URM) if there exists a function θ : (0, ) (0, ) such that for all x<y, ith y 1 and x ε, ehave y x + y θ(ε). In Definition 2.1, the function θ may and shall be taken to be nondecreasing. Let P m : X Span{e i } m i=1 denote the standard projection onto the span of the first m basis elements and let Q m = I P m. The basis is monotone if P n =1 for all n 1. Note that if y n 0, then for fixed m N, lim n P m(y n ) =0. (3) In much of hat follos, e shall consider 1-unconditional bases; these bases are unconditional bases ith unconditional basis constant equal to 1, i.e. such that a i e i = ±a i e i for all choices of scalars and signs. We note that hen {e i } is a 1-unconditional basis, then i A a ie i a i e i for all A N and scalars {a i }. 2.2 Smoothness Properties of Banach Spaces Here e recall some standard definitions pertaining to smoothness. Suppose X is a real Banach space. Define for f,g X and for u R f + ug + f ug 2 f ρ f,g (u) =. 2 The point f X is a point of Gâteaux smoothness if ρ f,g (u) lim = 0 for all g X. u 0 u By the Hahn-Banach theorem, each f X has at least one norming functional F f. It is knon that f is a point of Gâteaux smoothness if and only if its associated norming functional F f is unique. We say that a Banach space X is smooth (or X has a Gâteaux differentiable norm) hen every nonzero point in X is a point of Gâteaux smoothness. See [5] for more information on smoothness. The assumption that every nonzero point of X is a point of Gâteaux smoothness guarantees that for any nonzero f X and any dictionary D, inf f tg < f. t R g D Indeed, simply choose g D such that F f (g) > 0. Then f tg = f tf f (g) + o(t),

5 J Fourier Anal Appl (2008) 14: so f tg < f for all sufficiently small t>0. We ill also consider uniformly smooth Banach spaces. For each τ>0, the modulus of smoothness of a Banach space X is defined by { x + y + x y } ρ X (τ) = sup 1 : x,y X, x =1, y =τ 2 [16, p. 59]. We say that X is uniformly smooth if lim τ 0 ρ X (τ) τ = 0. There are Gâteaux smooth Banach spaces hich are not uniformly smooth. It is knon that a uniformly smooth Banach space is uniformly Fréchet differentiable,i.e. x + y =1 + F x (y) + ε(x,y) y (4) for all x,y X ith x =1, here ε(x,y) 0 uniformly for (x, y) {x : x = 1} X as y Uniform Opial Property This is a property hich relates eak convergence ith the geometry of the unit sphere S X := {x X : x =1}. We say that a Banach space has the uniform Opial property if there exists a function τ(ε) > 0 defined for all ε>0 such that for all sequences y n 0, ith yn =1, and for all x 0ehave lim inf n x + y n 1 + τ( x ). (5) Clearly e may and shall assume that the function τ is nondecreasing. Also e can obviously replace the condition y n =1 in the definition by the condition lim n y n =1. This property as introduced in [17] and turned out to be useful in geometric fixed point theory (see e.g. [18]). 3 Weak Convergence of Greedy Algorithms 3.1 The Weak Dual Greedy Algorithm (WDGA) We first revie the WDGA from [20, p. 66] (see also [6, p.491]).letd be a dictionary for X, and let 0 <t 1. To define the WDGA, first define f0 D D,t := f0 := f. Then, for each m 1, inductively define (1) φ D m := φd,t m D to be any element of D satisfying ( ) F f D φ D m 1 m t sup F f D (g). g D m 1 (2) Define a m via fm 1 D a mφm D = min fm 1 D aφd m. a R

6 614 J Fourier Anal Appl (2008) 14: (3) Denote f D m := f D,t m := f D m 1 a mφ D m. Using the same notation as in [6, p. 490], e define G D n to be and G D n = f f D n f 0 G D n = f GD n = f D n. The sequence { fm D } is nonnegative and nonincreasing since min fm D aφd m+1 fm D 0φD m+1. a R Therefore, { f m } converges to some α 0. We say that the algorithm converges strongly if α = 0. We note that the sequence {G D n } resulting from the WDGA ill be bounded above in norm: G D n f GD n f = f n f. In [6], Dilorth, Kutzarova, and Temlyakov prove that the WDGA converges strongly hen applied to dictionaries {±φ n }, here {φ n } is a strictly suppression 1-unconditional basis for a Banach space X hich has a Fréchet-differentiable norm. Also it follos from [6, Theorem 6] that if X is uniformly smooth and has the WN property then the WDGA converges eakly for any dictionary D. The strong convergence of the WDGA as proved under a different hypothesis in [8]. 3.2 The X-Greedy Algorithm (XGA) and the Weak X-Greedy Algorithm (WXGA) Let 0 <t 1 be a eakness parameter. The folloing algorithm is called the WXGA hen 0 <t<1 and is called the XGA hen t = 1. In the case of the XGA e have to add the assumption that the infimum in the first step is attained. Let D be a dictionary for X. (1) Choose φ m D and λ m R such that ( f m 1 f m 1 λ m φ m t f m 1 inf λ R φ D ) f m 1 λφ. (2) Define f m = f m 1 λ m φ m. We call f m the mth residual of f. The sequence { f n } is decreasing. (3) Set G m := f f m. Theorem 3.1 Suppose that X is uniformly smooth and has the WN property. When the XGA or the WXGA is applied to f X, then f n 0.

7 J Fourier Anal Appl (2008) 14: Proof If f n 0 then f n converges strongly to zero and e are done. So suppose that f n α, here α>0. If {F fn } is not eakly null, then some subsequence {F fnk } converges eakly to F 0 (since uniformly smooth spaces are reflexive). We shall no derive a contradiction by assuming that F fnk F 0. Choose g D such that F(g) = β>0; hence F fnk (g) > β/2 for all sufficiently large k. Noeuse the aforementioned fact (see (4)) that uniformly smooth spaces are uniformly Fréchet differentiable. For s>0, (4) yields ( ) fnk f nk sg = f nk sf fnk (g) + sε f nk, s g f nk f nk sβ ( ) 2 + sε fnk f nk, s g f nk for large k. Since f n α>0, it follos that there exists s 0 > 0 such that for large k. As a result, f nk s 0 g f nk s 0β 4 f nk f nk +1 t s 0β 4 for all sufficiently large k, hich contradicts the assumption that f n α. Thus {F fn } is eakly null, and hence {f n } is eakly null by the WN property. We no look at a convergence result hich does not require uniform smoothness. Let (X n, n ) be a Banach space for each n 1. The direct sum ( n=1 X n ) lp consists of all sequences {x n } n 1 ith x n X n such that ( ) 1 p {x n } = x n p n <. n=1 Theorem 3.2 Suppose that each X n is Gâteaux smooth and finite-dimensional. Then the WXGA converges eakly in ( n=1 X n ) lp,1<p<. Proof Suppose that {f n } is not eakly null. There is a subsequence {f nk } such that f nk g 0. Write fnk = g +x k here x k 0. By passing to a further subsequence, e may assume that x k β 0. Using the fact that ( n=1 X n ) lp is Gâteaux smooth, choose φ 0 D, s 0 > 0, and ε 0 > 0 such that g p g s 0 φ 0 p = ε p 0.

8 616 J Fourier Anal Appl (2008) 14: Then lim f n k s 0 φ 0 p = lim g s 0φ 0 + x k p k k = g s 0 φ 0 p + β p g p ε p 0 + βp = lim k f n k p ε p 0. Therefore, e can choose t 0 > 0 hich depends on the eakness parameter t such that f nk +1 p f nk p t 0 ε p 0 for all sufficiently large k. Hoever, this contradicts the fact that f n α>0. Let us remark that characterizations of Besov spaces on [0, 1] or T using coefficients of avelet or similar decompositions sho that Theorem 3.2 covers the case of Besov spaces equipped ith the appropriate norms. 3.3 Relaxed Greedy Algorithms The folloing to greedy algorithms have been introduced and studied in [22] (see [22] for historical remarks). We begin ith the Greedy Algorithm ith Weakness parameter t and Relaxation r (GAWR(t,r)). In addition to the acronym GAWR(t,r) e ill use the abbreviated acronym GAWR for the name of this algorithm. We give a general definition of the algorithm in the case of a eakness sequence τ GAWR(τ,r) Let τ := {t m } m=1, t m [0, 1], be a eakness sequence. We define f 0 := f and G 0 := 0. Then for each m 1 e inductively define 1). ϕ m D is any element of D satisfying 2). Find λ m 0 such that F fm 1 (ϕ m ) t m sup F fm 1 (g). g D f ((1 r m )G m 1 + λ m ϕ m ) = inf λ 0 f ((1 r m)g m 1 + λϕ m ) and define 3). Denote G m := (1 r m )G m 1 + λ m ϕ m. f m := f G m.

9 J Fourier Anal Appl (2008) 14: In the case τ ={t}, t (0, 1], e rite t instead of τ in the notation. We note that in the case r k = 0, k = 1,..., hen there is no relaxation, the GAWR(τ,0) coincides ith the Weak Dual Greedy Algorithm [20, p. 66] (see also [6, p. 491]). We ill also consider here a relaxation of the X-Greedy Algorithm (see [20, p. 39]) discussed above that corresponds to r = 0 in the definition that follos X-Greedy Algorithm ith Relaxation r (XGAR(r)) We define f 0 := f and G 0 := 0. Then for each m 1 e inductively define 1). ϕ m D and λ m 0 are such that and 2). Denote f ((1 r m )G m 1 + λ m ϕ m ) = inf f ((1 r m)g m 1 + λg) g D,λ 0 G m := (1 r m )G m 1 + λ m ϕ m. f m := f G m. We note that practically nothing is knon about convergence and rate of convergence of the X-Greedy Algorithm. The folloing convergence result as proved in [22]. Theorem 3.3 Let a sequence r := {r k } k=1, r k [0, 1), satisfy the conditions r k =, k=1 r k 0 as k. Then the GAWR(t,r) and the XGAR(r) converge in any uniformly smooth Banach space for each f X and for all dictionaries D. In this paper e discuss convergence of the GAWR(t,r) and the XGAR(r) under assumption k=1 r k < that is not covered by Theorem 3.3. Proposition 3.4 Suppose X is uniformly smooth and has the WN property. Let the relaxation r ={r k } be such that k=1 r k <. Then for the residual sequence {f m } of both the GAWR(t,r) and the XGAR(r) e have f m 0. Proof For both algorithms e have By the definition of λ m one has G m = (1 r m )G m 1 + λ m ϕ m. f G m (1 r m ) f m 1 +r m f (6)

10 618 J Fourier Anal Appl (2008) 14: and f m 1 f m r m f. The inequality (6) implies f m f. (7) We no need the folloing simple lemma. Lemma 3.5 Let a sequence {r k }, r k [0, 1], be such that k=1 r k <. Consider a sequence {a k }, a k (0,A], that satisfies the inequalities a m 1 a m Ar m, m= 2, 3,... Then either a m 0 as m or there exists an α>0 such that a m α for all m. Proof Suppose the sequence {a m } does not converge to 0. Then there exists a subsequence {a nk } such that lim a n k = γ>0. k Therefore, a nk γ/2fork k 0.Forn [n k 1,n k ) one has a n a nk A n k j=n+1 r j. (8) It is clear that there exists k 1 such that for n>n k1 (8) implies a n γ/4. This completes the proof. We return to the proof of Proposition 3.4. By Lemma 3.5 e obtain from (6) and (7) that either f m 0or f m α>0. It remains to prove that the inequality f m α>0 implies that {F fm } is eakly null and apply the WN property. The proof of this statement repeats the argument from the proof of Theorem Modified Algorithms First e recall the definition of the modulus of convexity δ X (ε) := δ(ε) of a Banach space X (see [16, pp ]): { x + y } δ(ε) := inf 1 : x,y X, x 1, y 1, x y ε, (9) 2 for 0 ε 2. X is said to be uniformly convex if δ(ε) > 0 for all ε>0. Suppose that X is a Banach space hich has a positive modulus of convexity δ(s) for at least one s (0, 1). We consider any greedy algorithm hich satisfies the folloing to conditions: the algorithm converges eakly (for every f 0inX, the residuals f n 0, or in other ords, G n f ); the norms of the residuals form a decreasing sequence { f n }.

11 J Fourier Anal Appl (2008) 14: No e shall modify a greedy algorithm hich satisfies these to conditions. The modified algorithm generates a sequence of residuals { f n } and a corresponding sequence of approximations { G n } such that f n 0 that is, the modified algorithm converges strongly. We proceed to describe ho to apply the modified algorithm to f = f 0 X. Set r 0 := f 0 and f 0 := f 0. We rite out the first to steps in detail and then proceed to the inductive step. Step 1: Apply the unmodified algorithm to f 0 and generate the sequence of residuals {(f 0 ) n }. We kno that (f 0 (f 0 ) n ) f 0. The norm of X is loer semicontinuous ith respect to eak convergence, so r 0 = f 0 lim inf n f 0 (f 0 ) n. Since s (0, 1), e can choose n 0 such that and e use n 0 to define the first residual sr 0 < f 0 (f 0 ) n0, (10) f 1 := f 0 + (f 0 ) n0. (11) 2 The first approximation of f is G 1 := f 0 f 1. We ill no find an upper bound for f 1 by considering the modulus of convexity δ(s). By definition, f 0 r 0 =1, and since the sequence of norms of residuals is decreasing, (f 0) n0 r 0 1. By (10), f 0 r 0 (f 0) n0 r 0 >s. Therefore, by (9), δ(s) 1 f 0 + (f 0) n0, 2r 0 2r 0 hich can be reritten as r 0 (1 δ(s)) f 0 + (f 0 ) n0. (12) 2 Set r 1 := f 1. We see that r 1 r 0 (1 δ(s)), and 0 < 1 δ(s) < 1 by hypothesis. Step 2: Apply the unmodified algorithm to f 1, generating a sequence of residuals {( f 1 ) n }. Because ( f 1 ( f 1 ) n ) f 1, e can choose n 1 such that sr 1 < f 1 ( f 1 ) n1. (13) Set f f 2 := 1 + ( f 1 ) n1 2 We note that G 2 = (f 0 f 1 ) + ( f 1 f 2 ). and G 2 = f 0 f 2.

12 620 J Fourier Anal Appl (2008) 14: As before, e bound f 2 using the modulus of convexity: r 1 (1 δ(s)) f 2. Referring back to (12), e see that r 0 (1 δ(s)) 2 f 2. Set r 2 := f 2. No e describe the inductive step. Step k + 1: Assuming that e have found { f 0, f 1,..., f k } and the accompanying norms {r 0,r 1,...,r k }, e apply the unmodified algorithm to f k in order to generate the sequence of residuals {( f k ) n }. Choose n k such that sr k < f k ( f k ) nk. Set f f k+1 := k +( f k ) nk 2 G k+1 := f 0 f k+1 r k+1 := f k+1. Finally, arguing as before, r 0 (1 δ(s)) k+1 f k+1. As a result f n 0asn, and the modified algorithm converges strongly. Remark 3.6 Note that the modification step can and should be omitted if ( f k ) nk f k + ( f k ) nk 2, because in this case e ould do better to set f k+1 := ( f k ) nk instead, and then continue ith the unmodified algorithm until the next modification step. So the modification step need only be applied hen it actually yields a better approximation than the unmodified algorithm. Note also that to apply the modified algorithm it is only necessary to store the value of f k in step k + 1 (in addition to the updated residuals). Remark 3.7 In all the results of this section e have applied the WN property directly, so in all results here this condition is assumed it can be dropped and the conclusion f n 0 replaced by either fn 0orF fn 0. 4 Comments and Remarks 4.1 Banach Spaces ith the WN Property In this section e discuss hich Banach spaces have the WN property. First of all it is knon that L p [0, 1] fails the WN property hen p 2(seee.g.[1]forp = 4). For completeness e give a short proof of this fact.

13 J Fourier Anal Appl (2008) 14: Example 4.1 Fix 1 <p< ith p 2 and let q be the Hölder conjugate index. Let {X n } be a sequence of independent identically distributed random variables defined on some separable probability space (,,P)such that P(X n = a) = 2/3 and P(X n = 2a) = 1/3, here a = (3/(2 + 2 q )) 1/q. Then X n q = 1, and, since E[X n ]=0, {X n } is a monotone basic sequence in L q ( ) (by Jensen s inequality). Since L q ( ) is a reflexive Banach space, it follos that {X n } is eakly null. No X n is the norming functional for Y n L p ( ), here Y n = X n q 1 sgn(x n ).ButE[Y n ]=(2 2 q 1 )a q 1 /3 0, so {Y n } is not eakly null in L p ( ). This shos that L p ( ) does not have the WN property. Theorem 4.2 Suppose X is a reflexive Banach space ith the uniform Opial property. Then X has the WN property. Proof Assume it is not true that x n 0 that is, suppose {xn } is not eakly null, here {x n } is as in Definition 1.1. By passing to a subsequence and relabelling, e may assume (by reflexivity of X) that x n x, here x 0. No, rite xn = x + y n, here y n 0. Once more passing to a subsequence and relabelling e may assume that y n c. If c>0 then from the uniform Opial property e get lim x n = lim x n c n c + y n 1 + τ( x /c), c hich gives 1 c + cτ( x /c), so c<1. (This is obviously also true if c = 0.) Hoever, since {F xn } is eakly null, e have 1 = lim F x n n (x n ) = lim F x n n (x) + lim F x n n (y n ) = 0 + lim F x n n (y n ) lim y n c<1, n hich is a contradiction. Therefore, {x n } S(X) is eakly null henever {F xn } is eakly null, as e ished to sho. Our next result connects bases ith the uniform Opial property. Proposition 4.3 A Banach space ith a URM basis has the uniform Opial property. Proof Let us take y n 0 ith yn =1forn = 1, 2,... and x 0. Let us fix a small δ>0 and N such that Q N (x) δ.wehave lim inf n x + y n =lim inf n P N(x) + Q N (y n ) + Q N (x) + P N (y n )

14 622 J Fourier Anal Appl (2008) 14: lim inf ( P N (x) + Q N (y n ) P N (y n ) Q N (x) ) n lim inf ( Q N (y n ) +θ( P N (x) )) lim sup P N (y n ) δ. n n Since y n 0 e have limn P N (y n ) =0 and lim n Q N (y n ) =1. We also have P N (x) x δ. Putting all this together e get lim inf x + y n 1 + θ( x δ). n Since δ as arbitrary e get the uniform Opial condition ith τ(ε)= θ(ε). Proposition 4.4 Suppose X is a uniformly convex Banach space ith a 1-unconditional basis {e i }. Then the basis is URM. Proof Fix 0 <ε 1. To verify the URM condition, suppose that x<y, x ε, and y 1. If x > 2 then x + y > 2 by 1-unconditionality, and hence y 1 x + y 1. So e may assume that x 2. By 1-unconditionality, ε x x + y = x + y, and since y = 1 2 ((x + y) + ( x + y)) it follos that ( ( )) 2 x y x + y 1 δ x + y ( ) 2ε x + y εδ. 3 Hence θ(ε) εδ(2ε/3). There are many examples of Banach spaces hich satisfy the hypotheses of Proposition 4.4. One obvious example is l p. Other examples are described in Sect belo Examples of Banach Spaces ith the Uniform Opial Property and URM Bases Some examples of spaces ith the uniform Opial property are presented in [14]. No e ill discuss the case of Orlicz sequence spaces. We thank Anna Kamińska for describing to us ho to construct the Orlicz sequence spaces in Example 4.5.For the relevant definitions and for more general theorems hich imply the facts stated belo, see Chen [2] and Lindenstrauss and Tzafriri [15, Chap. 4]. Example 4.5 (Orlicz sequence spaces) Let M be an Orlicz function, M its complement function, and l M its associated Orlicz sequence space. We equip l M ith the Luxemburg norm. If M satisfies the 2 condition at zero and M(1) = 1 then the unit vector basis {e i } of l M is a normalized 1-unconditional Schauder basis for l M.Wenostatesome important facts hich ensure that l M is both uniformly smooth and uniformly convex [2, 13]:

15 J Fourier Anal Appl (2008) 14: l M is uniformly convex if and only if M and M satisfy 2 and the function M is uniformly convex. l M is uniformly smooth if and only if M and M satisfy 2 and the function M is uniformly convex. If M and M satisfy 2, then there exists M 1 such that M 1 and M1 are uniformly convex, and M 1 is equivalent to M [2, Theorem 1.18, p. 12]. Putting these facts together, e can state the folloing. Suppose M(1) = 1. The Orlicz sequence space l M is uniformly smooth and uniformly convex if and only if M and M satisfy 2 and are uniformly convex. In addition, {e i } is a normalized 1-unconditional basis. Orlicz sequence spaces ith the uniform Opial property are described in [4] and more general sequence spaces in [3]. Example 4.6 L p [0, 1], 1<p<, equipped ith the square-function norm. Let f = n=0 a n h n be the expansion of f ith respect to the Haar basis. The norm defined by [ 1 ( ] 1 f = an 2 h n(t) 2) p p 2 dt 0 n=0 is equivalent to the usual L p norm by classical square function inequalities. Clearly, (L p [0, 1], ) is isometric to a subspace of the Lebesgue-Bochner space L p ([0, 1],l 2 ), hich in turn is isometric to a subspace of L p ([0, 1],L p [0, 1])) since l 2 is isometric to a subspace of L p [0, 1] (e.g. as the span of a sequence of independent identically distributed mean zero normal random variables). But L p ([0, 1],L p [0, 1]) is naturally isometrically isomorphic to L p ([0, 1] 2 ) (by Fubini s theorem), hich in turn is isometrically isomorphic to L p [0, 1] because [0, 1] and [0, 1] 2 have isomorphic measure algebras. In conclusion, (L p [0, 1], ) is isometric to a subspace of L p [0, 1]. Moreover, (L p [0, 1], ) contains a subspace isometric to l p spanned by disjointly supported Haar functions. Since l p and L p [0, 1] have identical moduli of convexity and smoothness, it follos that L p [0, 1] and (L p [0, 1], ) also have identical moduli of convexity and smoothness. In particular, (L p [0, 1], ) is uniformly convex and uniformly smooth. Finally, {h n } n=0 is obviously a 1-unconditional basis for (L p [0, 1], ). 4.2 Property Ɣ Let us recall that this property as used by Ganichev and Kalton [8] to prove the strong convergence of the WDGA. Our aim in this section is to give an example of a Banach space ith a URM basis (hence ith the WN property by Theorem 4.2 and Proposition 4.3) failing property Ɣ. On the other hand it is proven in [8] that L p [0, 1] has property Ɣ but (by Example 4.1) does not have the WN property for p 2. Definition 4.7 A smooth Banach space X has property Ɣ if there is a constant β>0 such that for any x,y X such that F x (y) = 0, e have x + y x +βf x+y (y). (14)

16 624 J Fourier Anal Appl (2008) 14: If e define ϕ(t) = x + ty x then e kno that x + (α + h)y x + αy F x+αy (y) = lim = ϕ (α), h 0 h so e can rerite (14)as ϕ(α) βαϕ (α). (15) The folloing lemma is proved for the sake of completeness. The result follos from the ork of Ganichev and Kalton [9] and is included here ith their permission. Lemma 4.8 If X is smooth and has property Ɣ then it is strictly convex. Proof Assume that X is not strictly convex. So there are points x 0 x 1 such that λx 0 + (1 λ)x 1 =1 henever 0 λ 1. Clearly F x0 = F x1,sofory = x 1 x 0 e have F x1 (y) = F x0 (y) = 0. We consider the function ϕ(t) = x 0 + ty x 0 for t 0. Clearly ϕ is increasing, differentiable, and satisfies ϕ(t) = 0for0 t 1. From (15) e get ϕ(t) βtϕ (t) βϕ (t) (16) for all t 1. Suppose ϕ(t 0 ) = 0forsomet 0 1. Integration of (16) yields β 2 ϕ(t 0 + β/2) β t0 +β/2 t 0 ϕ (t) dt = βϕ(t 0 + β/2), and hence ϕ(t 0 + β/2) = 0. By a repeated application of the latter, starting ith t 0 = 1, e get ϕ(t) = 0 for all t, hich is impossible since ϕ(t) as t. No let is fix a concave C 1 function H on [0, 1) hich has the folloing properties: { 1 if 0 t 0.1, H(t)= 1 t 2 1 if t 1, 2 and H(t)<1 if t>0.1. We define a Banach space E as R 2 equipped ith the norm such that (x, y) 1 y H( x ). The Banach space E is uniformly smooth (because H is C 1 ) and the basis e 1 = (1, 0), e 2 = (0, 1) is a 1-unconditional URM basis. From Lemma 4.8 e see that E does not have property Ɣ. If e ant to have an infinite-dimensional example it is enough to take X p = ( n=1 E) p ith 1 <p<. One easily checks that the natural basis in X p is a 1-unconditional URM basis. Clearly X p does not have property Ɣ and it is knon that it is uniformly smooth (because E is, see [7]).

17 J Fourier Anal Appl (2008) 14: Dictionary Dual Greedy Step In this section e ant to discuss the folloing greedy step (for an otherise unspecified greedy algorithm) hich e call the dictionary dual step. For a dictionary D X, eakness parameter 0 <t 1, and for x X e choose g 0 D to satisfy F g0 (x) t sup F g (x). (17) g D This greedy step as used in [21] for a special dictionary. One could also consider the algorithm obtained by replacing (1) in the definition of the WDGA (see Sect. 3.1) by (17). Since in the case of a Hilbert space e have F g (x) = g,x = x g, x x = x F x (g), e see that (hen t = 1) the dictionary dual greedy step in a Hilbert space coincides ith the dual greedy step. So the dictionary dual greedy step seems to be a good generalization of the pure greedy algorithm from the Hilbert space to the Banach space setting. Indeed, one could argue that it is easier to have the functionals F g computed once for each g belonging to the given dictionary than to have to compute F xn each time for vectors x n hich may be arbitrary (as e must do in dual greedy algorithms). The point hich e ant to make in this section is that for general dictionaries the dual greedy step (17) may create serious problems. Definition 4.9 We call a dictionary D a double dictionary if {F g : g D} X is a dictionary (i.e. linearly dense) for X. The Haar and Walsh systems are double dictionaries in L p [0, 1] for 1 <p<. Every dictionary in a Hilbert space is a double dictionary since Hilbert spaces are self-dual. Proposition 4.10 Let X be separable, reflexive, smooth, strictly convex, and not isometric to a Hilbert space ith dim X 3. Then X contains a countable non-double dictionary. Proof First note that the assumptions on X also apply to X by the ell-knon duality beteen smoothness and strict convexity. Let us take V X a subspace of codimension 1 and let x0 X be a functional of norm 1 ith ker x0 = V.Iftheset {F v } v V is not linearly dense in X then there exists a subspace Z X of codimension 1 such that {F v } v V Z. This implies that Z norms V,i.e.forv V e have v = sup z(v). (18) z S Z Let us consider a map q : X Z given by x x Z. From(18) e get that q V is an isometry from V into Z. Actually it must be onto Z because q is onto and

18 626 J Fourier Anal Appl (2008) 14: has one-dimensional kernel. This means that (q V) 1 q is a ell defined norm-one operator form X onto V hich is the identity on V, so it is a norm-one projection. No if X satisfies the assumptions of the theorem then by a theorem of James [11] X contains a one-codimensional subspace V hich is not 1-complemented in X. Thus e infer that the set {F v } v V is linearly dense. From this set e can choose a countable dictionary D hich is also linearly dense by separability of X. Since F Fx = x (by smoothness of X ) e see that {F g : g D} V. This means that X contains a non-double dictionary. Thus, X also contains a non-double dictionary by the self-duality of our assumptions. No suppose that e have a non-double dictionary D in a Banach space X. We have just proved in Proposition 4.10 that such a situation is quite common. Let us fix x 0 X, x 0 0 such that F g (x 0 ) = 0 for all g D. When e apply the dictionary dual greedy step e may choose an arbitrary element g 1 D. We do this and put x 1 = x 0 λg 1 ith the coefficient λ computed accordingly to our particular algorithm. Applying (17) tox 1 e get F g1 (x 1 ) = λ =sup F g (x 1 ), g D so g 1 is again an alloed (actually the best) choice. This means that using (17) e can not approximate x 0. Note that this problem appears already in nice finite-dimensional spaces (see Proposition 4.10). To complement the previous result let us construct a to-dimensional example. Let us considers l 2 p,1<p<, the space R2 ith the norm (x, y) p = ( x p + y p) 1/p. Let us recall hen e have equality in Hölder s inequality. We have x 1 x 2 + y 1 y 2 = ( x 1 p + y 1 p) 1/p ( x2 p + y 2 p) 1/p if and only if x 2 = C(sgn x 1 ) x 1 p 1 and y 2 = C(sgn y 1 ) y 1 p 1. This means that if a = (x, y) then F a = C(sgn x x p 1, sgn y y p 1 ). Lemma 4.11 Let p 2. For any vector a = (x, y) such that x y 0 and x y there are vectors g 1,g 2 such that: F g1 (a) = 0; (19) F a (g 2 ) = 0; (20) F g2 (a) 0. (21) Proof Without loss of generality a = (1,y) ith y>0 and y 1. Writing (19) for g 1 = (a 1,b 1 ) 0 (and assuming a 1 0) e get a p y(sgn b 1 ) b 1 p 1 = 0,

19 J Fourier Anal Appl (2008) 14: so a solution is given by g 1 = (1, y 1/(p 1) ). Analogously e solve (20) forg 2 = (a 2,b 2 ) and e get a solution g 2 = (1, y 1 p ). Using those values e rite the left-hand side of (21)as 1 y 1 (p 1)2. Since y 1 and the exponent of y is nonzero (because p 2) e get (21). No if e apply the dictionary dual greedy step (17) inl 2 p to vector a and dictionary D ={g 1,g 2 } as given in Lemma 4.11 e must choose g 2. But from (20) e get a + λg 2 > a henever λ 0. So any algorithm producing a decreasing sequence {f n } (as e assumed in Sect. 3.4) and using a dictionary dual greedy step cannot converge. Remark 4.12 We can make an infinite-dimensional example in l p by taking a dictionary g 1,g 2,e 3,e 4,... This is a basis and a double dictionary. Clearly, the vector (x, y, 0, 0,...)creates the same problem that as described above. References 1. Broder, F.E.: Fixed point theorems for nonlinear semi-contractive mappings in Banach spaces. Arch. Ration. Mech. Anal. 21, (1966) 2. Chen, S.: Geometry of Orlicz spaces. With a preface by Julian Musielak. Dissertationes Math. (Rozpray Mat.) 356 (1996) 3. Cui, Y., Hudzik, H.: Maluta s coefficient and Opial s properties in Musielak Orlicz sequence spaces equipped ith Luxemburg norm. Nonlinear Anal. 35, (1999) 4. Cui, Y., Hudzik, H., Yu, F.: On Opial property and Opial modulus for Orlicz sequence spaces. Nonlinear Anal. 55, (2003) 5. Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach Spaces. Pitman Monographs in Pure and Applied Mathematics, vol. 64. Longman Scientific & Technical, Harlo. Copublished in the United States by Wiley, Ne York (1993) 6. Dilorth, S.J., Kutzarova, D., Temlyakov, V.N.: Convergence of some greedy algorithms in Banach spaces. J. Fourier Anal. Appl. 8, (2002) 7. Figiel, T.: On the moduli of convexity and smoothness. Studia Math. 56, (1976) 8. Ganichev, M., Kalton, N.J.: Convergence of the eak dual greedy algorithm in L p -spaces. J. Approx. Theory 124, (2003) 9. Ganichev, M., Kalton, N.J.: Convergence of the dual greedy algorithm in Banach spaces. Preprint (2007) 10. Huber, P.J.: Projection pursuit. Ann. Stat. 13, (1985) 11. James, R.C.: Inner products in normed linear spaces. Bull. Am. Math. Soc. 53, (1947) 12. Jones, L.: On a conjecture of Huber concerning the convergence of projection pursuit regression. Ann. Stat. 15, (1987) 13. Kamińska, A.: Personal communication, April 6, Khamsi, M.A.: On uniform Opial condition and uniform Kadec-Klee property in Banach and metric spaces. Nonlinear Anal. 26, (1996) 15. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, Sequence Spaces. Ergebnisse der Mathematik, vol. 92. Springer, Berlin (1977) 16. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II, Function Spaces. Ergebnisse der Mathematik, vol. 97. Springer, Berlin (1979) 17. Prus, S.: Banach spaces ith the uniform Opial property. Nonlinear Anal. 18, (1992)

20 628 J Fourier Anal Appl (2008) 14: Prus, S.: Geometrical background of metric fixed point theory. In: Kirk, W.A., Sims, B. (eds.) Handbook of Metric Fixed Point Theory, pp Kluer Academic, Dordrecht (2001) 19. Temlyakov, V.N.: Greedy algorithms in Banach spaces. Adv. Comput. Math. 14, (2001) 20. Temlyakov, V.N.: Nonlinear methods of approximation. Found. Comput. Math. 3, (2003) 21. Temlyakov, V.N.: Greedy approximations. In: Found. Comput. Math., Santander London Math. Soc. Lecture Note Ser., vol. 331, pp Cambridge University Press, Cambridge (2006) 22. Temlyakov, V.N.: Relaxation in greedy approximation. Constr. Approx. 28, 1 25 (2008)