The Threshold Algorithm

Transcription

1 Chapter The Threshold Algorithm. Greedy and Quasi Greedy Bases We start with the Threshold Algorithm: Definition... Let be a separable Banach space with a normalized M- basis (e i,e i ):i2n ; we mean by that ke ik =, for i 2 N) For n 2 N and x 2 let n N so that min e i (x) i2 n max e i (x), i2n\ n i.e. we are reordering (e i (x)) into (e (i)(x)), so that and put for n 2 N Then define for n 2 N e (x) e (x) 2 e (x)..., 3 n = {, 2,... n}. G T n (x) = (G T n ) is called the Threshold Algorithm. i2 n e i (x)e i. Definition..2. A normalized M-basis (e i ) is called Quasi-Greedy, if for all x (QG) x = lim n! GT n (x). A basis is called greedy if there is a constant C so that (G) x G T (x) apple C n (x), where we define n(x) = n x, (e j ) = inf inf kz N,# =n z2span(e j :j2 ) xk. 5

2 6 CHAPTER. THE THRESHOLD ALGORITHM In that case we say that (e i )isc-greedy. We call the smallest constant C for which (G) holds the greedy constant of (e n ) and denote it by C g. Remarks. Let (e i,e i ):i2n be a normalized M basis.. From the property that (e n ) is fundamental we obtain that for every x 2 n(x)! n! 0, it follows therefore that every greedy basis is quasi greedy. 2. If (e j ) is an unconditional basis of, and x = P i= a i 2, then x = lim n! n a (j) e (j), for any permutation : N! N and thus, in particular, also for a greedy permutation, i.e. a permutation, so that a () a (2) a (3)... Thus, an unconditional basis is always quasi-greedy. 3. Schauder bases have a special order and might be reordered so that the cease to be basis. But unconditional bases, M bases, quasi greedy M-bases, greedy bases keep their properties under any permutation, and can therefore be indexed by any countable set. 4. In order to obtain a quasi greedy M-Basis which is not a Schauder basis, one could take quasi greedy Schauder basis, which is not unconditional (its existence will be shown later), but admits a suitable reordering under which is not a Schauder basis anymore. Nevertheless, by the observations in (3), it will still be a quasi greedy M-basis. But it seems unknown whether or not there is a quasi greedy M-basis which cannot be reordered into a Schauder basis. Examples..3. -greedy.. If apple p<, then the unit vector basis (e i ) of `p is

3 .. GREEDY AND QUASI GREEDY BASES 7 2. The unit vector basis (e i )inc 0 is -greedy. 3. The summing basis s n of c 0 (s n = P n e j) is not quasi greedy. 4. The unit bias of (`p `q) is not greedy (but -unconditional and thus quasi greedy). Proof. To prove () let x = P x je j 2 `p, and let n N be of cardinality n so that min{ x j : j 2 n } max{ x j : j 2 N \ n } and N be any subset of cardinality n and z = P z i e i 2 `p with supp(z) ={i 2 N : z i 6=0}. Then kx zk p p = x j z j p + x j p j2 j2n\ x j z j p + x j p j2 j2n\ n x j p = kg T (x) xk p. j2n\ n Thus n(x) =inf{kz xk p :#supp(z) apple n} = kg T (x) xk p. (2) can be shown in the same way as (). In order to show (3) we choose sequences (" j ) (0, ), (n j ) N as follows: " 2j =2 j and " 2j =2 j +, j 3 for j 2 N and Note that the series n j = j2 j and = n n i for i 2 N 0. i= x = = i= + i= + (" 2j s 2i " 2j s 2i ) (" 2j " 2j )s 2i " 2j e 2i

4 8 CHAPTER. THE THRESHOLD ALGORITHM converges, because and i= + i= + (" 2j " 2j )s 2i = " 2j e 2i 2 c 0 n j (" 2j " 2j )= j 2 <. Now we compute for l 2 N 0 the vector x G T 2N l +n l+ (x): x G T 2N l +n l+ (x) = N l+ i=n l + " 2l+2 s 2i + j=l+2 i= + (" 2j s 2i " 2j s 2i ). From the monotonicity of (s i ) we deduce that j=l+2 i= + " 2j s 2i " 2j s 2i applekx. However, N l+ i=n l + " 2l+2 s 2i = N l+ i=n l + " 2l+2 = l +! l!, which implies that G T n (x) is not convergent. To show (4) assume w.l.o.g. p<q, and denote the unit vector basis of `p by (e i ) and the unit vector basis of `q by (f j ) for n 2 N and we put Thus Nevertheless G T n (x(n)) = x(n) = n 2 e j + n f j. n f j, and thus G T n (x(n)) x(n) = 2 n/p. x n 2 e j = n /q, and since 2 n/p /n /q %, for n %, the basis {e j : j 2 N} [{f j : j 2 N} cannot be greedy.

5 .. GREEDY AND QUASI GREEDY BASES 9 Remarks. With the arguments used in (4) Examples..3 one can show that the usual bases of n= `nq and `p(`q) = `p n= `q `p are also not greedy but of course unconditional. Now in [BCLT] it was shown that `p `q has up to permutation and up to isomorphic equivalence a unique unconditional basis, namely the one indicated above. Since, as it will be shown later, every greedy basis must be unconditional, the space does not have any greedy basis. Due to a result in [FDOS] however n= `nq has a greedy bases if < `p p, q <. More precisely, the following was shown: Let apple p, q apple. a) If <q< then the Banach space ( n=`np )`q has a greedy basis. b) If q = or q =, and p 6= q, then( n=`np )`q has not a greedy basis. Here we take c 0 -sum if q =. The question whether or not `p(`q) has a greedy basis is open and quite an interesting question. The following result by Wojtaszczyk can be seen the analogue of the characterization of Schauder bases by the uniform boundedness of the canonical projections for quasi-greedy bases. Theorem..4. [Wo2] AboundedM-basis (e i,e i ),withke ik =, i 2 N, ofa Banach space is quasi greedy if and only if there is a constant C so that for any x 2 and any m 2 N it follows that (.) kg T m(x)k appleckxk We call the smallest constant so that (.) is satisfied the Greedy Projection Constant. Remark. Theorem..4 is basically a uniform boundedness result. Nevertheless, since the G T m are nonlinear projections we need a direct proof. We need first the following Lemma: Lemma..5. Assume there is no positive number C so that kg T m(x)k appleckxk for all x 2 and all m 2 N. Then the following holds: For all finite A N all K>0 there is a finite B N, which is disjoint from A and a vector x, withx = P j2b x je j, such that kxk =and kg T m(x)k K, for some m 2 N. Proof. For a finite set F N, definep F to be the coordinate projection onto span(e i : i2f ), generated by the (e i ), i.e. P F :! span(e i : i 2 F ), x7! P F (x) = j2f e j(x)e j.

6 0 CHAPTER. THE THRESHOLD ALGORITHM Since there are only finitely many subsets of A we can put M = max kp F k = max F A sup F A x2b e j(x)e j apple ke jk ke j k <. j2a j2f Let K > so that (K M)/(M +) >K, and choose x 2 S \span(e j : j 2 N) and k 2 N so that so that kg T k (x )k K. We assume without loss of generality (after suitable small perturbation) that all the non zero numbers e n(x ) are di erent from each other. Then let x 2 = x P A (x ), and note that kx 2 kapplem + and G T k (x )= G T m(x 2 )+P F (x ) for some m apple k and F A. ThuskG T m(x 2 )k K M, and if we define x 3 = x 2 /kx 2 k,wehavekg T k (x 3)k (K M)/(M + ) >K. It follows that the support B of x = x 3 is disjoint from A and that kg T m(x)k > K. Proof of Theorem..4. Let b =sup i ke i k. ) Assume there is no positive number C so that kg T m(x)k appleckxk for all x 2 and all m 2 N. Applying Lemma..5 we can choose recursively vectors y,y 2,... in S \ span(e j : j 2 N) and numbers m n 2 N, so that the supports of the y n,which we denote by B n, are pairwise disjoint, (Recall that for z = P i= z ie i, we call supp(z) ={i 2 N : e i (z) 6= 0}, thesupport of z) and so that ny (.2) kg T m n (y n )k 2 n b n " j, where Then we let " j =min 2 j, min{ e i (y j ) : i 2 B j }. x = n n= Y (" j /b) y n, (which clearly converges) and write x as x = Since e i (y j) appleb, for i, j 2 N x j e j. n min x i : i 2 n[ o ny B j " j b " n = ny " j b b max n x i : i 2 N \ n[ B j o.

7 .. GREEDY AND QUASI GREEDY BASES We may assume w.l.o.g. that m n apple #B n, for n 2N. Letting k j = m j + P j i= #B i, it follows that G T k j (x) = j i i= Y (" s /b) y j Y i + G T mj (" i /b) y j!. s= i= and thus by (.2) G T k j (x) G T m j j Y! (" i /b) y j i= j i i= s= Y (" s /b) ky i k 2 j b, which implies that G T k j does not converge. ( Let C>0such that kg T m(x)k appleckxk for all m 2 N and all x 2. Let x 2 and assume w.l.o.g. that supp(x) is infinite. For ">0 choose x 0 with finite support A so that kx x 0 k <". Using small perturbations we can assume that A supp(x) and that A supp(x x 0 ). We can therefore choose m 2 N large enough so that G T m(x) and G T m(x x 0 ) are of the form G T m(x) = j2b e i (x)e i and G T m(x x 0 )= j2b e i (x x 0 )e i with B N such that A B. It follows therefore that kx G T m(x)k applekx x 0 k + kx 0 G T m(x)k = kx x 0 k + kg T m(x 0 x)k apple( + C)", which implies our claim by choosing ">0 to be arbitrarily small. Definition..6. An M basis (e j,e j ) is called unconditional for constant coefficients if there is a positive constant C so that for all finite sets A N and all ( n : n 2 A) {±} we have C n2a e n apple n2a ne n apple C n2a e n. Proposition..7. A quasi-greedy M basis (e n,e n) is unconditional for constant coe cients. Actually the constant in Definition..6 can be chosen to be equal to twice the projection constant in Theorem..4. Remark. We will show later that there are quasi-greedy bases which are not unconditional. Actually there are Banach spaces which do not contain any unconditional basic sequence, but in which every normalized weakly null sequence contains a quasi-greedy subsequence.

8 2 CHAPTER. THE THRESHOLD ALGORITHM Proof of Proposition..7. Let A N be finite and ( n : n 2 A) {±}. Then if we let 2 (0, ) and put m =#{j 2 A : j =+} we obtain n2a, n=+ e n = G T m apple C n2a, n=+ n2a, n=+ e n + e n + n2a, n= n2a, n= ( )e n ( )e n. By taking >0 to be arbitrarily small, we obtain that e n apple C e n. n2a, n=+ n2a Similarly we have and thus, e n apple C e n, n2a, n= n2a ne n apple 2C e n. n2a n2a We now present a characterization of greedy bases obtained by Konyagin and Temliakov. We need the following notation. Definition..8. We call a a normalized basic sequence democratic if there is a constant C so that for all finite E,F N, with#e =#F it follows that (.3) j2e e j apple C j2f In that case we call the smallest constant, so that (.3) holds, the Constant of Democracy of (e i ) and denote it by C d. Theorem..9. [KT] Anormalizedbasis(e n ) is greedy if and only it is unconditional and democratic. In this case (.4) max(c s,c d ) apple C g apple C d C s C 2 u + C u, where C u is the unconditional constant and C s is the suppression constant. Remark. The proof will show that the first inequality is sharp. Recently it was shown in [DOSZ] that the second inequality is also sharp. e j

9 .. GREEDY AND QUASI GREEDY BASES 3 Proof of Theorem..9. ( Let x = P e i (x)e i 2, n 2 N and let >0. Choose x = P i2 n a ie i so that # n = n which is up to the best n term approximation to x (since we allow a i to be 0, we can assume that # is exactly n), i.e. (.5) kx xk apple n (x)+. Let n be a set of n coordinates for which b := min i2 n e i (x) We need to show that max e i (x) and G T n (x) = e i (x)e i. i2n\ n i2 n kx G T n (x)k apple(c d C s C 2 u + C u )( n (x)+ ). Then x G T n (x) = e i (x)e i = i2n\ n e i (x)e i + e i (x)e i. i2 n \ n i2n\( n [ n) But we also have (.6) e i (x)e i i2 n \ n apple bc u e i (By Proposition 4..) i2 n \ n apple bc u C d e i i2 \ n n [Note that #( n \ n)=#( n \ n )] apple CuC 2 d e i (x)e i i2 n\ n [Note that e i (x) b if i 2 n \ n ] apple C s CuC 2 d (e i (x) a i )e i + e i (x)e i i2 n i2n\ n = C s C 2 uc d kx xk applec s C 2 uc d ( n (x)+ ) and (.7) (.8) e i (x)e i apple C s (e i (x) a i )e i + e i (x)e i i2n\( n[ n) i2 n i2n\ n = C s kx xk applec s ( n (x)+ ). This shows that (e i ) is greedy and, since > 0 is arbitrary, we deduce that C g apple C s C 2 uc d + C s.

10 4 CHAPTER. THE THRESHOLD ALGORITHM ) Assume that (e i ) is greedy. In order to show that (e i ) is democratic let, 2 N with # =# 2. Let >0 and put m =#( 2 \ ) and x = e i +(+ ) e i. i2 i2 2 \ Then it follows i2 e i = kx G T m(x)k apple C g apple C g m (x) (since(e i )isc g -greedy) x apple C g e i +(+ ) e i. i2 \ 2 i2 2 \ i2 \ 2 e i Since >0 can be taken arbitrary, we deduce that e i apple C g e i. i2 i2 2 Thus, it follows that (e i ) is democratic and C d apple C g. In order to show that (e i ) is unconditional let x = P e i (x)e i 2 have finite support S. Let S and put y = i2 e i (x)e i + b i2s\ with b>max i2s e i (x). For n =#(S \ ) it follows that G T n (y) =b e i, i2s\ and since (e i ) is greedy we deduce that (note that #supp(y x) =n) e i (x)e i = ky G T n (y)k applec g n (y) apple C g ky (y x)k = C g kxk, i2 which implies that (e i ) is unconditional with C s apple C g..2 The Haar basis is greedy in L p [0, ] and L p (R) Theorem.2.. For <p< there are two constants c p apple C p, depending only on p, sothatforalln2n and all A T with #A = n c p n /p apple t2a e i, h (p) t apple C p n /p. In particular (h (p) t ) t2t is democratic in L p [0, ].