Greedy Algorithm for General Biorthogonal Systems

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1 Journal of Approximation Theory 107, (2000) doi jath , available online at httpwww.idealibrary.com on Greedy Algorithm for General Biorthogonal Systems P. Wojtaszczy 1 nstytut Matematyi, Uniwersytet Warszawsi, Banacha 2, Warsaw, Poland wojtaszczymimuw.edu.pl Communicated by Allan Pinus Received June 23, 1999; accepted in revised form July 17, 2000; published online November 28, 2000 We consider biorthogonal systems in quasi-banach spaces such that the greedy algorithm converges for each x # X (quasi-greedy systems). We construct quasigreedy conditional bases in a wide range of Banach spaces. We also compare the greedy algorithm for the multidimensional Haar system with the optimal m-term approximation for this system. This substantiates a conjecture by Temlyaov Academic Press Key Words quasi-greedy basis; conditional basis; biorthogonal system; Haar system. 1. NTRODUCTON We consider a general quasi-banach space X with the norm &}& such that for all x, y # X we have &x+ y&(&x&+&y&). The letter will always in this paper denote this constant. t is well nown (cf. [3] Lemma 1.1.) that in such a situation there is a p, 0<p1, such that & n x n & 4 ( n &x n & p ). Recall that a biorthogonal system in a quasi-banach space X is a family (x n, x n *) n # F /X_X* such that x n *(x m ) equals zero whenever n{m and equals one if n=m. Here F is any countable index set. We fix a biorthogonal system (x n, x n *) n # F in X such that span(x n ) n # F =X and inf n # F &x n &>0 and sup n # F &x n *&<. This implies that for each x # X we have lim n x n *(x)=0. For each x # X and m=1, 2,... we define G m (x)= n # A x n *(x) x n, (1) where A/F is a set of cardinality m such that x n *(x) x *(x) whenever n # A and A. The above set may not be uniquely defined but if this happens we tae any such set. The operator G m (x) is a non-linear and 1 This research was partially supported by the Grant 2 PO3A from Komitet Badan Nauowych and was completed while the author visited Erwin Schro dinger nstitute in Wien Copyright 2000 by Academic Press All rights of reproduction in any form reserved.

2 294 P. WOJTASZCZYK discontinuous operator. We will use linear projection operators P A defined for any finite subset A/F by the formula P A (x)= n # A x n *(x) x n. This simple theoretical algorithm is a model for a procedure which is widely used in numerical applications. t also raises many interesting questions in functional analysis. The reader can find in [9][11] a more detailed description of the connections with purely numerical questions and results for some concrete systems. n this paper we will use standard Banach space notation as explained in detail in [12] or [6]. The basic reference for simple facts we are going to use about quasi-banach space is [3]. Definition 1. A biorthogonal system (x n, x n *) n # F is called a quasigreedy system if for each x # X the sequence G m (x) converges to x in norm. f this system is a basis we will use the phrase quasi-greedy basis. Clearly every unconditional basis is a quasi-greedy basis. Let us recall the definition of the best m-term approximation. For x # X and m=0,1,... we put _ m (x)=inf {"x& n # A a n x n" A m and a n's are scalars =. (2) Definition 2. A basis (x n, x n *) n # F is called greedy if there exists a constant C such that for every x # X we have &x&g m (x)&c_ m (x). After the research reported in this paper was practically completed received the preprint [4] where the above terminology was introduced, so decided to follow this terminology in this note. t is shown in [4] that each greedy basis is unconditional and an example of a conditional quasi-greedy basis is given. The author expresses his gratitude to Professor Alesander Pe*czyn si for many helpful conversations about the research reported in this paper. 2. QUAS-GREEDY SYSTEMS Let us start with some general results. The following theorem gives some natural equivalent conditions for quasi-greedy systems. Theorem 1. The following conditions are equivalent 1. The system (x n, x n *) n # F is quasi-greedy. 2. For each x # X the series n=1 x* _(n) (x) x _(n) converges to x where _ is an ordering of F such that ( x* _(n) (x) ) n=1 is a decreasing sequence.

3 GENERAL BORTHOGONAL SYSTEMS There exists a constant C such that for any x # X and m=1, 2,... we have &G m (x)&c &x&. This theorem is basically a uniform boundedness result. However, since the operator G m is non-linear and discontinuous we have to give a direct proof. Proof. Clearly O 1. Since the convergence is clear for x's with finite expansion in the biorthogonal system, let us assume that x has an infinite expansion. Tae x 0 = n # A a n x n such that &x&x 0 &<= where A is a finite set and a n {0 for n # A. f we tae m big enough we can ensure that G m (x&x 0 )= n # B x n *(x&x 0 ) x n with B#A and G m (x)= n # B x n *(x) x n. Then This gives 1. &x&g m (x)&(&x&x 0 &+&x 0 &G m (x)&) (=+&G m (x 0 &x)&)(c+1) =. 1 O 3. Let us start with the following lemma. Lemma 1. f 3 does not hold, then for each constant K and each finite set A/F there exists a finite set B/F disjoint from A and a vector x= n # B a n x n such that &x&=1 and &G m (x)&k for some m. Proof. Let us fix M to be the maximum of the norms of the (linear) projections P 0 (x)= n # 0 x n *(x) x n where 0/A. Let us start with a vector x 1 such that &x 1 &=1 and &G m (x 1 )&K 1 where K 1 is a big constant to be specified later. Without loss of generality we can assume that all numbers x*(x 1 n ) are different. For x 2 =x 1 & n # A x n *(x 1 ) x n we have &x 2 & (M+1) and G m (x 1 )=G (x 2 )+P 0 (x 1 ) for some m and 0/A. Thus &G (x 2 )& K 1 &M and for x3 =x 2 } &x 2 & &1 we have &G (x 3 )&(K 1 &M) (1+M). Let us put $=inf[ x n *(G (x 3 )) x n *(G (x 3 )){0] and tae a finite set B 1 such that for n B 1 we have x*(x 3 n ) $2. Let us tae ' very small with respect to B 1 and A and find x 4 with finite expansion such that &x 3 &x 4 &<'. f' is small enough we can modify all coefficients of x 4 from B 1 and A so that the resulting x 5 will have its biggest coefficients the same as x 3 and &x 4 &x 5 &<$. Moreover x 5 will have the form x 5 = n # B x n *(x 5 ) x n with B finite and disjoint from A. Since &x 5 &(&x 4 &+$)((1+')+$)C and G (x 5 )=G (x 3 ), for x=x 5 } &x 5 & &1 we get &G (x)&( K 1 &M)(+M) &1 (C) &1 which can be made K if we tae K 1 big enough. K

4 296 P. WOJTASZCZYK Using Lemma 1 we can apply the standard gliding hump argument to get a sequence of vectors y n = # Bn a x with sets B n disjoint and &y n &=1, a decreasing sequence of positive numbers = n 2 &n such that if x *( y n ){0 then x *( y n ) = n and a sequence of integers m n such that &G mn ( y n )&2 n > n&1 j=1 =&1 j. Now we put x= n=1 (>n&1 = j=1 j) y n. This series is clearly convergent in X. f we write xt n # F b n x n we infer that j inf { b n n #. s=1 B s and b n {0 = `j s=1 This implies that for = j&1 s=1 B s +m j we have G (x)= n j\ n&1 ` s=1 j = s+ y n+g mj\ ` j = s max { b n n. s=1 = s+ y j+1 s=1 B s=. so &G (x)& &1 (> n&1 = s=1 s) &G mj y j+1 &&C2 j+1 &C. Thus G m (x) does not converge to x. K Let us now introduce the following definition Definition 3. A system (x n, x n *) n # F is called unconditional for constant coefficients if there exist constants C and c>0 such that for each finite A/F and each sequence of signs (= n ) n # A =\1 we have c " n # A x n" " n # A = n x n" C " n # A Definition 3 is justified by the following observation. x n". (3) Proposition 2. coefficients. Every quasi-greedy system is unconditional for constant Proof. For a given sequence of signs (= n ) n # A let us define the set A 1 = [n # A = n =1]. For each $>0 and $<1 we apply Theorem 1 and we get " n # A 1 x n" C " n # A 1 x n + n # A"A 1 (1&$) x n". Since this is true for each $>0 we easily obtain the right hand side inequality in (3). The other inequality follows by analogous arguments. K Remar. Let us clarify a bit the problem of non-uniqueness of G m (x). n our definition of quasi-greedy system we require that for each x # X we can

5 GENERAL BORTHOGONAL SYSTEMS 297 choose (if there is a choice) a G m (x) such that G m (x) x. The statements 2 and 3 of Theorem 1 are also to be understood in this wayin 2 we thin about one convergent decreasing rearrangement and in 3 we thin about one good G m (x). However Proposition 2 immediately imply that those reservations are not essential. t shows that if (x n, x n *) n # F is quasi-greedy, then any series n=1 x* _(n)(x) x _(n) such that ( x* _(n) (x) ) is decreasing, converges to x. This implies that we have convergence for any choice of G m (x). Our definitions of a quasi-greedy system and of the operator G m depend on the normalisation of the system considered. This, however, is not essential. Namely we have Proposition 3. Suppose that (x n, x n *) n # F is a quasi-greedy system as discussed. Let (* n ) n # F be a sequence of numbers such that 0<a= inf n # F * n b= sup n # F * n <. Then the system (* n x n, x n ** n ) n # F is also quasi-greedy. Proof. By homogeneity we can and will assume that b=1. Let G 1 be m the greedy approximation operator corresponding to the system (* n x n, x n ** n ) n # F. Let us fix x # X and a natural number m. Explicitly we have G 1 (x)= m n # A x n *(x) x n where A/F is a set of cardinality m such that x*(x)* n n x s *(x)* s whenever n # A and s A. Let us write '=inf n # A x*(x) n and let V=[n # F x n *(x)'] and U=[n # F x*(x) 'a]. n We put V = and U =l. Clearly U/V so l. Using those notations we can write G 1 (x)=g m (x)& x s *(x) x s s # B =G l (x)+ \G (x)&g l (x)& x*(x) s x s+, (4) where B is a certain subset of V"U and so we now that for s # B we have s # B ' x s *(x) 'a. (5) Clearly G (x)&g l (x)= n # V"U x n *(x) x n. Note that for each finite set D/F and each set of numbers (a n ) n # D we have " a n x n # D n" C " x n # D n". (6)

6 298 P. WOJTASZCZYK To see (6) we write a dyadic expansion of each a n namely a n = \ s=1 a(n, s) 2&s where a(n, s)=0, 1. Then from Proposition 2 we have " a n x n # D n" = " 2 &s s) x s=1 n # D\a(n, n" 4 s=1 Thus we infer from (5) and (6) that 2 " &sp \a(n, s) x n # D n" C. " G (x)&g l (x)& x*(x) n x s # B n" C " ('a) x n # V"U n" C 2 a " x n *(x) x n # V"U n" C 2 a &G (x)&g l (x)&. (7) Comparing (4) and (7) we infer that &G 1 m (x)&c$ &x& so by Theorem 1 the system (* n x n, x** n n ) n # F is a quasi-greedy system. K Remar. Proposition 2 allows us to show that the trigonometric system in L p (T) with is quasi-greedy only if p=2, because only then it is unconditional for constant coefficients. To see this observe that for 1<p we have & N n=1 eint & p tn p&1 p and for p=1 we have & N n=1 eint &tlog(n+1). On the other hand for < the average over all signs \ of & N n=1 \eint & p can be written as p 1 0 &N r n=1 n(s) e int & p ds p where r n (s) are classical Rademacher functions. t is well nown in the theory of type and cotype of Banach spaces (see [12]) and easily follows from the Khintchine's inequality that 1 0 &N r n=1 n(s) e int & p dstn p2 p so the trigonometric system in L p (T) (<) an be quasi-greedy only when p&1=p2 i.e. when p=2. n the case p= we can invoe the RudinShapiro polynomials i.e. polynomials, N = N n=1 \eint such that &, N & C - N. This argument reproves results from [11] remar 2. Now we will discuss examples of conditional quasi-greedy bases. Let us recall Definition 4. A biorthogonal system (resp. basis) (x n, x n *) n # F is a p-besselian system, 0<p< if there exists a constant C such that for each x # X we have \ p+ x n *(x) C &x&. n # F

7 GENERAL BORTHOGONAL SYSTEMS 299 A 2-Besselian system (resp. basis) will be called Besselian. Theorem 2. Suppose X is a quasi-banach space with Besselian basis (x n, x n *) n=1. The space Xl 2 has a quasi-greedy basis. f the basis (x n, x n *) is conditional we get a conditional quasi-greedy basis in Xl n=1 2. Before we start the proof let us recall some classical notions from Banach space theory. f X and Y are Banach spaces then the symbol XY denotes the direct sum of those spaces i.e. the space of all pairs (x, y) with x # X and y # Y. This is a linear space with coordinatewise addition and scalar multiplication. As a norm on XY we can tae &(x, y)&=(&x& 2 + &y& 2 ) 12. We will identify an element x # X with a pair (x, 0)#XY, soin particular x+ y means (x, y) whenever x # X and y # Y. f we have a sequence of Banach spaces (X n ) n=1 and a number < then ( n=1 X n) p denotes the space of all sequences (x n ) n=1 such that x n # X n for n=1, 2,... and &(x n )&=( n=1 &x n& p ) <. Proof. Let us recall some facts about Olevsii matrices (cf. [7]). For =1, 2,... we define 2 _2 matrices A =(a () ij ) 2 i, j=1 by the following formulas a () i1 =2&2 for i=1,2,...,2 and for j=2 s +& with 1&2 s and s=0,1,2,...,&1 we put 2 (s&)2 for (&&1) 2 &s <i(2&&1) 2 &s&1 a () ij ={&2 (s&)2 for (2&&1) 2 &s&1 <i&2 &s 0 otherwise. One easily checs that the A are orthonormal matrices and there exists a constant C p such that for all i, we have j a () ij p C p for p>0. (8) Note that A is a matrix which maps an orthonormal Haar-lie system in R 2 onto the unit vector basis. We put N =2 10 and define S so that S 1 =N 1 &1 and S +1 &S =N &1. Let (e r ) r=1 denote the unit vector basis in l 2. Let us denote by (g s ) /Xl s=1 2 the following basis x 1, e 1,..., e S1, x 2, e S1 +1,..., e S2, x 3, e S2 +1,..., e S3, x 4,...

8 300 P. WOJTASZCZYK To each bloc [x, e S&1 +1,..., e S ] we apply the matrix A 10 to get a new system i = x N + - N j=2 a 10 ij e S&1 + j. (9) The system 1 1,..., 1 N 1, 2 1,..., 2 N 2,..., ordered in this fashion will be denoted by ( j ) j=1. t is clear that 0<inf j & j &sup j & j &< and that ( j ) j=1 is a complete biorthogonal system in Xl 2 with the biorthogonal functionals given by the formula V i = x N n* + - N j=2 a 10 ij e* S&1 + j. t is also a basis in Xl 2. Since the system (g j ) j=1 is a basis it suffices to chec that for each the system ( i )N i=1 have uniformly bounded basis constant. But on each subspace span[x, e S&1 +1,..., e S ] the l N 2 norm and the norm in Xl 2 are uniformly equivalent, so any orthonormal basis in this finite dimensional space has uniformly bounded basis constant in Xl 2.f(x n ) n=1 is a conditional basis then ( j ) is also conditional because (x j=1 n) n=1 is a bloc basis of ( j ). j=1 Thus we still have to show f f = a j=1 j j # Xl 2 with & f &=1, and _ is a permutation such that a _( j) is a decreasing sequence, then the series a j=1 _( j) _( j) converges in Xl 2. First observe that (g s ) is a Besselian basis in Xl s=1 2, so we can define an operator Xl 2 l 2 as ( a s=1 s g s )=(a s ). Since ( s=1 j) j=1 is obtained from (g s ) s=1 by the action of a unitary matrix we infer that ( j ) is also Besselian and ( j=1 j) is an orthonormal basis in l j=1 2.LetP denote the natural projection from Xl 2 onto l 2. Note that [0]l 2 is an isometric embedding. Let Q denote the orthogonal projection onto ([0]l 2 ). Let us write f as a double sum f = =1 N i=1 b i. Since ( i j) is j=1 Besselian we get =1 N i=1 b i 2 C. (10) This implies that the series N =1 i=1 b i () converges in l i 2 in any order so also =1 N i=1 b i Q( i ) converges in any order. This implies that also the series =1 N i=1 b i P i converges in Xl 2 in any order.

9 GENERAL BORTHOGONAL SYSTEMS 301 Thus we have to study the convergence of the series =1 N i=1 b (&P) i i = =1 N i=1 b(x i - N ) but ordered in such a way that coefficients b i form a decreasing sequence. Let us denote 4 =[i N &1 < b &110 i <N ] 4$ =[i b i 4" =[i b i N &1 ] N &110 ]. Note that for each we have i # 4$ b - N i N }1N }1-N N &12. Thus the series =1 i # 4$ (b - N i ) x is absolutely convergent. t follows from (10) that for each, N i=1 b 2 i C so C i # 4" b 2 i 4" N &15 which gives 4" CN 15. From this we get =1 b i i # 4" - N =1 1 4" - N =1 N &310 < so the series =1 i # 4" (b - N i ) x is absolutely convergent. Since N &110 &1 +1 N we see that the decreasing permutation of the series =1 i # 4 b i i has to tae place inside each 4. But in this case (since ( ) is a basis in Xl 2 ) the series over converges in Xl 2. From the Schwarz inequality and (10) we get b i \ i # 4 - N b i i # 4 ( 4 N &1 )12 =o(1) so we see that the series =1 i # 4 (b - N ) x converges when rearranged in decreasing order of the coefficients (b ). K i This proof is a modification of an argument used in the main result in [5]. Let us note some corollaries from the above construction. Corollary 4. A separable, infinite dimensional Hilbert space has a quasi-greedy conditional basis. Proof. t is nown (cf. e.g. [7]) that a Hilbert space has a conditional Besselian basis, so writing l 2 =l 2 l 2 we get the claim. K Corollary 5. The space l p for 1<p< has a conditional quasi-greedy basis. Proof. t is well nown (cf. [8]) that l p is isomorphic to ( n=1 l n 2) p. Let us fix ( j ), a conditional quasi-greedy basis in l j=1 2 which exists by

10 302 P. WOJTASZCZYK Corollary 4. Then l n is isometric to span( 2 j) n j=1 so the obvious basis in ( span( n=1 j) n ) j=1 p is a conditional quasi-greedy basis in l p. K Corollary 6. f X has an unconditional basis and contains a complemented subspace isomorphic to l p with 1<p< then X has a conditional quasi-greedy basis. Proof. We write XtYl p tyl p l p txl p so taing the unconditional basis in the first summand and a conditional quasi-greedy basis in l p (cf. Corollary 5) we get a conditional quasi-greedy basis in X. K Note that the above Corollary 6 gives the existence of conditional quasi-greedy bases in L p [0, 1] with 1<p< and also in H 1. The following theorem shows that quasi-greedy bases in a Hilbert space are rather close to unconditional bases. Theorem 3. Let (x n ) n=1 be a normalized quasi-greedy basis in a Hilbert space H. Then there exist constants 0<cC< such that for each x= a n=1 nx n we have c &(a n )& 2, &x& 2 C &(a n )& 2, 1, (11) where &}& 2, and &}& 2, 1 are natural Lorentz sequence space norms. n particular such a basis is p-besselian for each p>2. Let us recall the definition of the relevant Lorentz norms. For a sequence (a n ) we denote by (a n=1 n*) n=1 the non-increasing rearrangement of the sequence ( a n ). Then &(a n=1 n) & n=1 2, =sup n - na n * and &(a n ) & n=1 2, 1= n=1 n&12 a n *. Proof. First observe that applying Khintchine's inequality (or type and co-type 2 of the Hilbert space) we infer from Proposition 2 that for each finite set of indices A and each choice of signs we have & n # A \x n &t - A. Now let us denote n = [n a n 2 & ]. Reordering the series n a n x n so that a n z0 we have " n a n x n" 2 C n=1 n 2 " & x s=1 s" C 1 - n a n. 2 & - n To prove the other inequality observe that from the Abel transform we obtain that if n=1 y n converges in a Banach space X and sup N & N y n=1 n&=c and n z0 then the series n=1 n y n converges and

11 GENERAL BORTHOGONAL SYSTEMS 303 sup N & n=1 n y n &C 1. Now we consider the series a n=1 nx n and assume that a n z0. Since the basis is quasi-greedy this series converges and sup N & N a n=1 nx n &C &x&, so for each N we have sup & a s=1 N+1&sx N+1&s &2C &x&. Applying our observation to the sequence = a n a N+1& &1 we get " N s=1 a N+1&s a N a N+1&s x N+1&s" 2C &x&. From this, using the unconditionality for constant coefficients we get a N " N s=1 x s" C &x& which gives sup n a n - nc &x& which completes the proof. K 3. OPTMALTY Suppose now that X is a quasi-banach space with an unconditional basis (x n, x n *) and let us assume that inf n # F &x n &>0 so sup n # F &x n *&<. An unconditional basis is called a lattice basis if & n a n x n && n b n x n & whenever a n b n for all n. f we have an unconditional basis we can always introduce an equivalent lattice norm by _x_=sup {" n a n x n *(x) x n" a n 1 =. With this norm we have &x&_x_c &x& and _x n _=&x n &. Proposition 7. Let X be a quasi-banach space with lattice basis (x n, x n *). For each x and each m=1, 2,... there exists an element T m (x) of best m-term approximation i.e. T m (x)= n # A a n x n with A =m and &x&t m (x)&=_ m (x). Proof. Let x = n # A a n x n with A =m be such that &x&x & _ m (x). Using a standard diagonal procedure we can assume that for each n lim a n =a n. Clearly the a n are not zero for at most m indices n. Write x = n a n x n = n # A a n x n where A =m. f we tae B a finite set, B#A, then &x&x &&P B x&p B x & &P B x&x &.

12 304 P. WOJTASZCZYK Thus for each such set B we have _ m (x)&p B x&x &=&P B (x&x )&. Taing a sequence of B's exhausting the whole index set we obtain _ m (x) &x&x &, so we can put T m (x)=x. K Let us recall the following quantities essentially considered in [10] &x&g e m =sup m (x)& x # X _ m (x) \ with 0 0 =1 + sup[& n # A x n & A =] + m = sup m inf[& n # A x n & A =]. The importance of those quantities is clear. The sequence e m estimates the error between the greedy algorithm G m and the best possible m-term approximation. The quantity + m measures some sort of asymmetry of the basis. The important fact is that they are closely connected. Theorem 4. Let (x n, x n *) be a lattice basis in a quasi-banach space X. Then for each m=1, 2,... we have me m 2+ m. (12) The proof of this theorem follows the ideas from [10]. Proof. Let us fix m and x= n a n x n # X. LetT m (x)= n # A b n x n be the best m-term approximation. Let G m (x)= n # B a n x n. Note that &x&p A x&=&x&t m (x)+p A T m (x)&p A x&=&(d&p A )(x&t m (x))& &x&t m (x)&=_ m (x). (13) Thus we can tae T m (x)=p A (x). n order to estimate &x&g m (x)& write x&g m (x)=x&p A x+p A x&p B x=(x&p A x)+p A"B x&p B"A x =P F"B (x&p A x)+p A"B x so &x&g m (x)&(&x&t m (x)&+&p A"B x&)(_ m (x)+&p A"B x&). Note now that max[ x n *(x) n # A"B] =cmin[ x n *(x) n # B"A] and also A"B = B"A m. This implies that &P A"B x&c & n # A"B x n & and &P B"A x& c & n # B"A x n &. Thus estimating c from the second inequality and substituting it into the first we get &P B"Ax& &P A"B x& & n # B"A x n & } &P A"Bx&+ m &P B"A x&+ m _ m (x) (14)

13 GENERAL BORTHOGONAL SYSTEMS 305 so we get &x&g m (x)&_ m (x)(1++ m )2+ m _ m (x). n order to prove the other inequality we will need the following Lemma 8. For each m there exist disjoint sets A and B with A = B m such that & n # A x n && n # B x n & &1 (2) &1 + m. Proof. f + m 2 the claim is obvious. Otherwise tae sets A and B with A = B m such that & n # A x n && n # B x n & &1 >max(2, + m &=). For simplicity write a= " n # A a 1 = " n # A & B x n" b= " n # B x n" x n" a 2 = " n # A"B x n". With this notation we have 2<(1)(ab)(1)(aa 1 ) so a 1 <(12) a. This implies a b (a 1+a 2 ) = a 1 b b + a 2 b < a 2b + a 2 b so a 2 b>(12)(ab). Thus it suffices to replace A by any set of proper cardinality which contains A"B and is disjoint with B. K Now let us tae sets as in Lemma 8 and denote A = B =m. Let C#A be a set of cardinality m disjoint with B. Consider x =(1+=) x n +(1+=2) n # B n # C"A x n + n # A x n. (15) Then G m (x)=x& n # A x n so &x&g m (x)&=& n # A x n &. From (13) we see that This and Lemma 8 give _ m (x)=min[&p S x& S/B _ C and S =] e m & n # A x n & _ m (x) &P B x&(1+=) " n # B x n". & n # A x n & (1+=) & n # B x n & 1 (1+=) 2 + m. Since = was arbitrary we get the claim. K

14 306 P. WOJTASZCZYK Remar. Actually one can show that for x defined in (15) we have _ m (x)t& n # B x n &. Namely &P S x& 1 & 1+= n # S x n & and since + m & n # A x n && n # S x n & &1 from Lemma 8 we get " x n # S n" +&1 m " x n # A n" 1 2 so _ m (x) 1 2(1+=) & n # B x n &. & n # B x n & & n # A x n &" x n # A n" 2" 1 x n # B n" Remar. Observe that we need to have + m defined as sup m in order to have the above estimate. As an example tae l n l 1 with the natural basis. Let ( f j ) n be the basis in j=1 ln and (e ) the basis in l =1 1.Form>n and x =2 n f j=1 j+ m e =1 we have _ m (x)=2 and G m (x)=2 n f j=1 j+ m&n e =1 so &x&g m (x)&=n which gives e m n. Also + m =n. But! m = sup[& n # A x n & A =m] inf[& n # A x n & A =m] = m m&n so no estimate of the form e m C! m is valid for all m unless Cn. Butn can be arbitrary. For general biorthogonal systems we have the following result. Theorem 5. Suppose (x n, x n *) n # F is a complete biorthogonal system in a quasi-banach space X with &x n &=1 for n # F. Assume that for some 0<cC and 0<pq we have c \ n # F q+ 1q x n *(x) &x&c \ p+ x n *(x). (16) n # F Then e m Km &1q where K depends only on, C and c. The proof is similar to the proof of Theorem 4 and Theorem 2.1 from [11]. Proof. Let us fix an x # X and m=1, 2,... For any given =>0 we fix almost best m-term approximation i.e. T m = n # A b n x n such that &x&t m &_ m (x)+=. First note that for any finite subset V/F we have &P V x&c \ n # V p+ x n *( p) C V \ q+ 1q &1q x n *(x) n # V C c V &1q &x&. (17)

15 GENERAL BORTHOGONAL SYSTEMS 307 Let G m (x)= n # B a n x n. We write &x&g m (x)&=&x&p A x+p A x&g m (x)& (18) (&x&p A x&+&p A x&g m (x)&). (19) The first summand is estimated as &x&p A x&(&x&t m &+&P A (x&t m )&) (_ m (x)+=+&p A &(_ m (x)+=)) so &x&p A x& \_ m(x)+=+ C c A &1q (_ m (x)+=) +. (20) The second summand we write as and obtain &P A (x)&p B x&(&p A"B x&+&p B"A x&) &P B"A x&=&p B"A (x&t m )& C c B"A &1q (_ m (x)+=). (21) To estimate the other summand we note that B"A = A"B and x n *(x) x s *(x) whenever n # B"A and s # A"B. Thus &P A"B x&c \ n # A"B C B"A &1q =C B"A &1q p+ x n *(x) \ n # B"A \ n # B"A \ n # B"A C c B"A &1q &x&t m & q+ 1q x n *(x) p+ x n *(x) q+ 1q x n *(x&t m ) C c B"A &1q (_ m (x)+=). (22) Since = was arbitrary and B"A m= A from (20), (21) and (22) we obtain &x&g m (x)&k(c, c, ) _ m (x)}m &1q. K

16 308 P. WOJTASZCZYK Remar. Using Theorem 3 and arguing lie in the above proof we can get that for each quasi-greedy basis in a Hilbert space we have e m C ln(m+1). Now we will list some immediate consequence of Theorem 5. Corollaries. (a) f (x n ) n=1 is a complete, uniformly bounded orthonormal system (in particular the trigonometric system) then in L p [0, 1] with we have e m Km 12&. This follows immediately from F. Riesz inequality which says that for 2p we have \ 2+ (x n, f) 12& f & p M \ p$+ $ (x n, f) n=1 n=1 where p p$ =1, and also the dual inequality valid for 2. This proves Theorem 2.1 from [11]. This is an optimal inequality as was shown for the trigonometric system in [11] Remar 2. For p>1 it also follows from Remar 2. (b) Since for any semi-normalized biorthogonal system (x n, x n *) in a Banach space X we have c sup n x*(x) &x&c n x n *(x) we infer that for each such system we have e m Cm. This estimate is also optimal, even for unconditional bases. One easily checs that for the natural unconditional basis in l 1 c 0 one has e m m. (c) n each super-reflexive space, in particular in L p [0, 1] with 1< p<, each semi-normalized basis (x n, x*) n satisfies equation (16) for some 1<qp< (see [1]). So we obtain that for each semi-normalized basis in a super-reflexive space we have e m Km ; with ;<1. (d) f (x n, x n *) is a semi-normalized unconditional basis in L p with 1<p<, then for p2 it satisfies c \ n # F p+ x n *(x) &x& p C \ x n *(x) n # F and the dual inequality for 1<p2. Thus for an unconditional basis in L p we have e m Km 12&. Also this estimate is optimal. To see it consider L p as being isomorphic to l 2 L p and tae the natural basis in l 2 and the Haar basis in L p. n

17 GENERAL BORTHOGONAL SYSTEMS MULTPLE HAAR SYSTEM n this section we will discuss the efficiency of the greedy algorithm with respect to the multi-dimensional Haar wavelet. For a more detailed exposition of the general bacground setched below the reader may consult [13]. We will argue in the context of the square function for 0<p<. To start we define 1 if t #[0, 1), 2 H(t)={&1 if t #[ 1 2,1), (23) 0 otherwise. For a dyadic interval =[2 &n,(+1) 2 &n ) we put h (t)=2 np H(2 n t&). For a dyadic rectangle in J= 1 _}}}_ d /R d we put h (d) J (t)=h 1 (t 1 )}}}h d (t d ). (24) The set of all dyadic intervals in R will be denoted by D(1) and the set of all dyadic rectangles in R d will be denoted by D(d). The system (h (d) i ) # D(d) is a complete orthogonal system in L 2 (R d ) and is normalized in L p (R d ). Note that formally the definition of this system depends on p but since p will be fixed in our future arguments we will not indicate this dependence explicitely. A function f = # D(d) a h (d) is in H p (R d ) if the norm _ f _= \ R d\ 2+ p2 a h (d) (t) dt + # D(d) (25) is finite. t is nown by the LittlewoodPaley theory that for 1<p< this norm is equivalent to the usual L p norm and we have H p (R d )=L p (R d ). For 0<p1 we get the dyadic H p -space. The main result of this section is the following Theorem 6. H p (R d ) we have For 0<p< and d=1,2,...for the system (h (d) ) # D(d) in e m t(log m) (d&1) 12&. (26) This result substantiates the conjecture formulated (for p>1) in [10] and extends results from [9] and [10]. Our argument is a modification of the argument from [2]. Let us start with a lemma which summarises the argument from the first few lines of the proof of Proposition 3.3 from [2]. We repeat the short proof of this lemma for the convenience of the reader.

18 310 P. WOJTASZCZYK Lemma 9. For 0<p< and any finite subset B/D(1) we have 2 & B }}} h }}}. Proof. Let us denote 2 M(t) =max h (t) p. From the definition of the Haar system we infer that 2 M(t) 1 2 h (t) p so h }}} }}} 2 \ M(t) dt + \ 1 2 R h (t) p dt + =2 & B. K R Proposition 10. For any d=1, 2,..., any finite B/D(d), B =m and any numbers (a ) we have (a) (b) if 0<p2 then (log m) (12&) d if 2p< then \ p+ a }}} \ p+ a }}} a h (d) }}} \ p+ a (27) a h (d) d (log m)(12&) }}} \ p+ a. (28) Proof. The right hand side inequality in (27) is easy (it is actually the type of H p ). We simply apply the Ho lder's inequality with exponent 2 1 p to the inside sum and we get \ R d\ 2+ p2 a h (d) (t) dt + \ R d = \ a h (d) (t) p dt + p+ a. Now let d=1 and 0< p2. Let _ [1, 2,..., B ] B be such that a _(i) is a decreasing sequence. Fix s such that 2 s&1 <m2 s and put f = ( 2 a j=2 &1 +1 _( j)h _( j) 2 ) 12. Then s p2 a }}} h }}} = \ R \ f 2 (t) dt + + = \ =0 \\ s 2p p2 =0\ f p (t) dt R = \ s 2 =0}}} j=2 & a _( j) h _( j)}}} R \ + 12 (t))2p+ s p2 ( f p dt + =0

19 \ s 2 =0}}} j=2 &1 +1 and from Lemma 9 \ s GENERAL BORTHOGONAL SYSTEMS = a _(2 ) h _( j)}}} (&1)p a _(2 ). 311 Since we get B a p = j=1 s a _( j) p =0 2 a _(2 ) p s 1&p2 \ 2+ s p2 2 2p a _(2 ) =0 a }}} h }}} 2& &(1& p2) (log m) \ p+ a =2 & (log m) 12& \ p+ a. Now we will prove the left hand side inequality in (27) by induction on d. Suppose we have (27) valid for d&1. Given a finite set B/D(d) we write each as =J_K with J # D(1) and K # D(d&1) and then h (d) (t)= h J (t 1 )}h (d&1) K (!) where!=(t 2,..., t d ). Now we estimate }}} a h (d) }}} = \ R R d&1\ a h J (t 1 ) 2 h (d&1) = \ R \ R d&1\ K \ J 2+ p2 K (!) dt 1 d! + a h J (t 1 ) h(d&1) K (!) p2 d! + 1+ dt. (29) For each t 1 we apply the inductive hypothesis (note that the number of different K's is at most B ) and we continue the estimates C(d&1, p)(log B ) (d&1)(12&) _ \ R C(d&1, p)(log B ) (d&1)(12&) _ \ K R \ \ K J J 2+ p2 a h J (t 1 ) dt p2 a h J (t 1 ) dt 1+. (30)

20 312 P. WOJTASZCZYK Now we apply the estimate (27) for d=1 and we continue as C(d&1, p)(log B ) \ (d&1)(12&) K J =C(d, p)(log B ) \ d(12&) p+ a C(1, p)(log B ) (12&) p+ a. (31) The inequality (28) follows by duality from (27) for 1< p2. K Proposition 11. (a) if 0<p2 then For every finite set B/D(d) we have C(d, p)(log B ) (12&)(d&1) B }}} h (d) }}} B (32) (b) if 2p< then B }}} h (d) }}} C(d, p)(log B )(12&)(d&1) B. (33) Proof. As in the previous Proposition 10 inequality (33) follows by duality from (32). Note also that (32) for d=1 is Lemma 9. For d>1 we proceed lie in the proof of Proposition 10. We write each as J_K and estimate _ h (d) i _ exactly lie in (29) and (30). Since a =1 instead of (27) for d=1 we apply Lemma 9 and we obtain }}} h (d) }}} C(d&1, p)(log B )(12&)(d&1) 2 & B. K Proof of Theorem 6. The estimate e m C(log m) 12& (d&1) follows immediately from Theorem 4 and Proposition 11. The estimate from below was proved in [9]. K Theorem 6 covers and extends the main results about the Haar system proved in [9] and [10]. n particular it gives a new proof that the Haar wavelet is a greedy basis in L p (R). One can note that the Haar system is not the only such basis in L p for 2< p<. Let us recall the definition of the Rosenthal space (cf. [6] p. 169). Fix 0<;1 and define a norm on sequences (a n ) n=1 as &(a n )& ; = \ n=1 p+ a n + \ a n w ; n n=1

21 GENERAL BORTHOGONAL SYSTEMS 313 with w ; n =n&;( p&2)2p. t is nown that for all ;'s we get the same space X (called Rosenthal space) and that XL p is isomorphic to L p. However for different ;'s we get different unconditional bases in X. f(e n ) n=1 denotes the unit vector basis then for each finite set A/N we have ; " e n # A n" = A + \ n # A A + \ A n= w ; n w ; n A +C A (1&;)2&;p For ;=1 we get & n # A e n & 1 t A so this basis and the Haar basis give an unconditional greedy basis in L p which is not equivalent to the Haar basis. For 0<;<1 we get & N n=1 e n& ; tn (1&;)2+;p so for this basis in X and the Haar basis in L p we get an unconditional basis in L p with + m tm (1&;)2+;p m & tm (1&;)(12&) so we get all possible power type behaviours of e m (c.f. Corollary (d) after Theorem 5). REFERENCES 1. V.. Gurarii and N.. Gurarii, Bases in uniformly convex and uniformly smooth Banach spaces, zv. Acad. Nau SSSR Ser. Mat. 35 (1971), [in Russian] 2. N. J. Kalton, C. Leranoz, and P. Wojtaszczy, Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces, srael J. Math. 72 (1990), N. J. Kalton, N. T. Pec, and J. W. Roberts, ``An F-space Sampler,'' London Math. Soc. Lecture Notes, Vol. 89, Cambridge University Press, Cambridge, UK, S. V. Konyagin and V. N. Temlyaov, A remar on greedy approximation in Banach spaces, preprint. 5. S. Kostyuovsy and A. Olevsii, Note on decreasing rearrangement of Fourier series, J. Appl. Anal. 3 (1997), J. Lindenstrauss and L. Tzafriri, ``Classical Banach Spaces,'' Springer-Verlag, Berlin, A. M. Olevsii, ``Fourier Series with Respect to General Orthonormal Systems,'' Springer- Verlag, BerlinHeidelbergNew Yor, A. Pe*czyn si, Projections in certain Banach spaces, Studia Math. 19 (1960), V. N. Temlyaov, The best m-term approximation and greedy algorithms, Adv. Comput. Math. 8 (1998), V. N. Temlyaov, Non-linear m-term approximation with regard to the multivariate Haar system, East. J. Approx. 4 (1998),

22 314 P. WOJTASZCZYK 11. V. N. Temlyaov, Greedy algorithm and m-term trigonometric approximation, Constr. Approx. 14 (1998), P. Wojtaszczy, ``Banach Spaces for Analysts,'' Cambridge Studies in Advanced Math., Vol. 25, Cambridge University Press, Cambridge, UK, P. Wojtaszczy, ``A Mathematical ntroduction to Wavelets,'' London Math. Soc. Student Texts, Cambridge University Press, Cambridge, UK, 1997.