Renorming Banach spaces with greedy basis. Andr as Zs ak - PowerPoint PPT Presentation

Renorming Banach spaces with greedy basis. Andr´ as Zs´ ak Peterhouse, Cambridge (Joint work with S. J. Dilworth, D. Kutzarova, E. Odell, Th. Schlumprecht.) Aleksander Pe� lczy´ nski Memorial Conference, July 2014, Bedlewo, Poland

Approximating signals We are given a Banach space X with a basis ( e i ). Given x ∈ X , we want to approximate x by a linear combination of the e i . We also seek an algorithm that produces for any m ∈ N , a good m -term approximant. We measure the efficiency of any such algorithm against the smallest theoretical error: �� � � � σ m ( x ) = inf � x − a i e i � : A ⊂ N , | A | ≤ m , ( a i ) i ∈ A ⊂ R . � � i ∈ A

The greedy algorithm and greedy bases Let x ∈ X and write x = � x i e i ∈ X . We fix a permutation ρ = ρ x of N such that | x ρ (1) | ≥ | x ρ (2) | ≥ . . . . We then define the m th greedy approximant to x by m � G m ( x ) = x ρ ( i ) e ρ ( i ) . i =1 We say ( e i ) is a greedy basis for X if there exists C > 0 ( C-greedy ) such that � x − G m ( x ) � ≤ C σ m ( x ) for all x ∈ X and for all m ∈ N . The smallest C is the greedy constant of the basis. We shall often assume that ( e i ) is normalized: � e i � = 1 for all i ∈ N .

Greedy characterization A basis ( e i ) is said to be unconditional if there is a constant K such that � � � � � � a i e i � ≤ K · b i e i whenever | a i | ≤ | b i | for all i ∈ N . � � � � � � � We also say ( e i ) is K-unconditional . Can always renorm so that K = 1 works. A basis ( e i ) is said to be democratic if there is a constant ∆ such that � � � � � � e i � ≤ ∆ e i whenever | A | ≤ | B | . � � � � � � � i ∈ A i ∈ B We will use the term ∆ -democratic . Theorem [S. V. Konyagin, V. N. Temlyakov, ’99] A basis is greedy if and only if it is unconditional and democratic. If ( e i ) is 1-unconditional, then ∆ ≤ C ≤ 1 + ∆.

Examples 1. The unit vector basis of ℓ p (1 ≤ p < ∞ ) or c 0 is 1-greedy. 2. Any orthonormal basis of a separable Hilbert space is 1-greedy. 3. The Haar basis of L p [0 , 1] (1 < p < ∞ ) is greedy [V. N. Temlyakov, ’98]. 4. The Haar system in one-dimensional dyadic Hardy space H p ( R ), 0 < p < ∞ . [P. Wojtaszczyk, ’00]. ⊕ ∞ n =1 ℓ n � � 5. ℓ q has a greedy basis whenever 1 ≤ p ≤ ∞ and 1 < q < ∞ p [S. J. Dilworth, D. Freeman, E. Odell and Th. Schlumprecht, ’11]. ∞ ∞ � � � � � � 6. None of the space ℓ p ℓ q , 1 ≤ p � = q < ∞ , ℓ p c 0 , 1 ≤ p < ∞ , and n =1 n =1 ∞ � � � ℓ q , 1 ≤ q < ∞ , have greedy bases [G. Schechtman, ’14]. c 0 n =1

Questions Let X be a Banach space with a greedy basis ( e i ). Q1. Can X be renormed so that ( e i ) is C -greedy in the new norm, where C is a universal constant? YES with C = 2 + ε . Q2. Can we take C = 1 in Q1? NO in general. Can take C = 1 + ε for certain bases. Note: WLOG ( e i ) is normalized and 1-unconditional. Recall: ∆ ≤ C ≤ 1 + ∆. Q3. Can X be renormed so that ( e i ) is ∆-democratic in the new norm, where ∆ is a universal constant? YES with ∆ = 1 + ε . Q4. Can we take ∆ = 1 in Q3? NO in general: e.g., the unit vector basis of Tsirelson space T or the Haar basis of dyadic Hardy space H 1 [Dilworth, Odell, Schlumprecht, Z, 11]. YES for certain bases.

Bidemocratic bases. Introduced by S. J. Dilworth, N. J. Kalton, D. Kutzarova and V. N. Temlyakov. The fundamental function ϕ of a basis ( e i ) of a Banach space X is defined by � � � ϕ ( n ) = sup e i � . � � � | A |≤ n i ∈ A E.g., for ℓ p or L p ( p < ∞ ) we have ϕ ( n ) ∼ n 1 / p . Note: ϕ ( n ) is increasing and that ( ϕ ( n ) / n ) is decreasing. The dual fundamental function ϕ ∗ of ( e i ) is the fundamental function of ( e ∗ i ). � � n i =1 e i , � n i =1 e ∗ ≤ ϕ ( n ) ϕ ∗ ( n ). Note that n = � i We say that ( e i ) is bidemocratic if there is a constant ∆ ≥ 1 (∆ -bidemocratic ) such that ϕ ( n ) ϕ ∗ ( n ) ≤ ∆ n for all n ∈ N . If ( e i ) is bidemocratic with constant ∆, then both ( e i ) and ( e ∗ i ) are democratic with constant ∆ [DKKT, ’03].

Bidemocratic case Suppose that ( e i ) is a greedy and bidemocratic basis for a Banach space X . Theorem 1 [DOSZ, ’11] There is an equivalent norm on X in which ( e i ) is normalized, 1-unconditional and 1-bidemocratic. In particular, ( e i ) and ( e ∗ i ) are 1-democratic and 2-greedy. Remark: The implication “1-democratic ⇒ 2-greedy” is sharp. Theorem 2 [DKOSZ] For all ε > 0 there is an equivalent norm on X in which ( e i ) is normalized, 1-unconditional, 1-bidemocratic and (1 + ε )-greedy. Notation: a) � x , x ∗ � = x ∗ ( x ) for x ∈ X , x ∗ ∈ X ∗ � and i ∈ A e ∗ � � b) 1 A will denote either � i ∈ A e i or � i . E.g., � 1 A � = � � i ∈ A e i � 1 A � ∗ = i ∈ A e ∗ � � � � � . i c) Let x = � x i e i ∈ X . We set | x | = � | x i | e i , and write x ≥ 0 if x i ≥ 0 for all i .

Proof of Theorem 1 WLOG ( e i ) is normalized and 1-unconditional. Let ∆ be the bidemocracy constant. �� � | x | , ϕ ( n ) Define ||| x ||| = � x � ∨ sup n 1 A � : n ∈ N , A ⊂ N , n = | A | . � ∗ ≤ ϕ ( n ) � ϕ ( n ) n ϕ ∗ ( n ) ≤ ∆. So � x � ≤ ||| x ||| ≤ ∆ � x � . � � n 1 A 1 E , ϕ ( n ) � � Let | E | = n . Then ||| 1 E ||| ≥ n 1 E = ϕ ( n ). For | A | = m , we have 1 E , ϕ ( m ) = ϕ ( m ) m | E ∩ A | ≤ ϕ ( | E ∩ A | ) � � m 1 A | E ∩ A | | E ∩ A | ≤ ϕ ( n ) . � ∗ = 1. QED � ϕ ( n ) � �� �� � �� �� So ||| 1 E ||| = ϕ ( n ) and n 1 E Remark: Assume, instead of bidemocracy, that for all q ∈ (0 , 1) there exists C > 0 such that � ∗ ≤ C � � � � ϕ ( n ) � A = A ⊂ N : n = | A | < ∞ , n 1 A satisfies: ∀ finite E ⊂ N there exists A ∈ A such that A ⊂ E and | A | ≥ q | E | .

Characterizing C -greedy bases Given vectors x = � x i e i and y = � y i e i in X , we say y is a greedy rearrangement of x if it is obtained from x by rearranging and possibly changing the sign of some of the coefficients of x of maximum modulus. x = ( − 2 , 0 , 3 , 3 , 1 , 0 , 0 , − 1 , 2 , − 3 , 0 , 0 , . . . ) and y = ( − 2 , − 3 , 0 , 3 , 1 , 0 , 3 , − 1 , 2 , 0 , 0 , 0 , . . . ). We say ( e i ) has Property (A) with constant C if for all x , y we have � y � ≤ C � x � whenever y is a greedy rearrangement of x . Theorem [Albiac, Wojtaszczyk, ’06] If ( e i ) is 1-unconditional, then ( e i ) is C -greedy if and only if it satisfies Property (A) with constant C .

Proof of AW-characterization Assume ( e i ) is 1-unconditional and has Property (A) with constant C . Fix x = � x i e i and m ∈ N . Write G m ( x ) = � i ∈ A x i e i . Let s = min {| x i | : i ∈ A } , and note that | x i | ≤ s ≤ | x j | for i / ∈ A , j ∈ A . Let b = � i ∈ B b i e i be an arbitrary m -term approximation. � � � � � � x − b � = x i e i + ( x i − b i ) e i + x i e i � � � � i ∈ A \ B i ∈ B i / ∈ A ∪ B � ≥ 1 � � � � � � � � ≥ se i + x i e i se i + x i e i � � � � C � � � i ∈ A \ B i / ∈ A ∪ B i ∈ B \ A i / ∈ A ∪ B ≥ 1 � = 1 � � � � � . � � x i e i + x i e i � x − G m ( x ) QED. � � C C � i ∈ B \ A i / ∈ A ∪ B

Proof of Theorem 2 Let ( e i ) be a greedy and bidemocratic basis of a Banach space X . Theorem 2 [DKOSZ] For all ε > 0 there is an equivalent norm on X in which ( e i ) is normalized, 1-unconditional, 1-bidemocratic and (1 + ε )-greedy. WLOG ( e i ) is normalized, 1-unconditional and 1-bidemocratic (by Theorem 1). Define �� | x | , x ∗ + : x ∗ ∈ ε B X ∗ , n ∈ N , A ⊂ N , | A | = n � 1 � ||| x ||| = sup ϕ ∗ ( n ) 1 A . Calculation shows that ( e i ) has Property (A) with constant 1 + ε , and hence it is (1 + ε )-greedy. A little more work . . . QED.

The Upper Regularity Property (URP) We say that ( e i ) has the URP if there exists 0 < β < 1 and C > 0 such that � n � β ϕ ( m ) ϕ ( n ) ≤ C for all m ≤ n . m This was introduced in [DKKT, ’03]. They showed that a greedy basis with the URP is bidemocratic, and that a greedy basis ( e i ) of a Banach space X with non-trivial type has the URP. Corollary [DKOSZ] Let 1 < p < ∞ . For all ε > 0 there is an equivalent norm on L p [0 , 1] in which the Haar basis is normalized, 1-unconditional, 1-bidemocratic and (1 + ε )-greedy. Problem: Can one make the Haar basis 1-greedy? Problem: Can one make a bidemocratic, greedy basis 1-greedy?

The general case Lemma [DKOSZ] Let ( e i ) be a normalized, 1-unconditional, ∆-democratic ∆ basis of a Banach space X . Given 0 < q < 1, fix C > q (1 − q ) and set � ∗ ≤ C � � � ϕ ( n ) � � A = A ⊂ N : n = | A | < ∞ , n 1 A . Then ∀ finite E ⊂ N there exists A ∈ A such that A ⊂ E and | A | ≥ q | E | . Corollary [DKOSZ] Let ( e i ) be a greedy basis of a Banach space X . For any ε > 0 there is an equivalent norm on X with respect to which ( e i ) is normalized, 1-unconditional and (1 + ε )-democratic, and hence (2 + ε )-greedy. Remark: We cannot replace (1 + ε )-democratic by 1-democratic. E.g., Tsirelson space or dyadic Hardy space H 1 [DOSZ, ’11]. Problem: Can we replace (2 + ε )-greedy by (1 + ε )-greedy?