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 (ei). Given x ∈ X, we want to approximate x by a linear combination of the ei. 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: σm(x) = inf

• x −
• i∈A

aiei

• : A ⊂ N, |A| ≤ m, (ai)i∈A ⊂ R
• .

The greedy algorithm and greedy bases

Let x ∈ X and write x = xiei ∈ X. We fix a permutation ρ = ρx of N such that |xρ(1)| ≥ |xρ(2)| ≥ . . . . We then define the mth greedy approximant to x by Gm(x) =

m

• i=1

xρ(i)eρ(i) . We say (ei) is a greedy basis for X if there exists C > 0 (C-greedy) such that x − Gm(x) ≤ Cσm(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 (ei) is normalized: ei = 1 for all i ∈ N.

Greedy characterization

A basis (ei) is said to be unconditional if there is a constant K such that

• aiei
• ≤ K ·
• biei
• whenever |ai| ≤ |bi| for all i ∈ N .

We also say (ei) is K-unconditional. Can always renorm so that K = 1 works. A basis (ei) is said to be democratic if there is a constant ∆ such that

• i∈A

ei

• ≤ ∆
• i∈B

ei

• whenever |A| ≤ |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 (ei) is 1-unconditional, then ∆ ≤ C ≤ 1 + ∆.

Examples

• 1. The unit vector basis of ℓp (1 ≤ p < ∞) or c0 is 1-greedy.
• 2. Any orthonormal basis of a separable Hilbert space is 1-greedy.
• 3. The Haar basis of Lp[0, 1] (1 < p < ∞) is greedy [V. N. Temlyakov, ’98].
• 4. The Haar system in one-dimensional dyadic Hardy space Hp(R), 0 < p < ∞.

[P. Wojtaszczyk, ’00]. 5.

• ⊕∞

n=1 ℓn p

• ℓq has a greedy basis whenever 1 ≤ p ≤ ∞ and 1 < q < ∞

[S. J. Dilworth, D. Freeman, E. Odell and Th. Schlumprecht, ’11].

• 6. None of the space
• ∞
• n=1

ℓp

• ℓq, 1 ≤ p = q < ∞,
• ∞
• n=1

ℓp

• c0, 1 ≤ p < ∞, and
• ∞
• n=1

c0

• ℓq, 1 ≤ q < ∞, have greedy bases [G. Schechtman, ’14].

Questions

Let X be a Banach space with a greedy basis (ei).

• Q1. Can X be renormed so that (ei) 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 (ei) is normalized and 1-unconditional. Recall: ∆ ≤ C ≤ 1 + ∆.

• Q3. Can X be renormed so that (ei) 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 H1 [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 (ei) of a Banach space X is defined by ϕ(n) = sup

|A|≤n

• i∈A

ei

• .

E.g., for ℓp or Lp (p < ∞) we have ϕ(n) ∼ n1/p. Note: ϕ(n) is increasing and that (ϕ(n)/n) is decreasing. The dual fundamental function ϕ∗ of (ei) is the fundamental function of (e∗

i ).

Note that n = n

i=1 ei, n i=1 e∗ i

• ≤ ϕ(n)ϕ∗(n).

We say that (ei) is bidemocratic if there is a constant ∆ ≥ 1 (∆-bidemocratic) such that ϕ(n)ϕ∗(n) ≤ ∆n for all n ∈ N. If (ei) is bidemocratic with constant ∆, then both (ei) and (e∗

i ) are democratic

with constant ∆ [DKKT, ’03].

Bidemocratic case

Suppose that (ei) is a greedy and bidemocratic basis for a Banach space X. Theorem 1 [DOSZ, ’11] There is an equivalent norm on X in which (ei) is normalized, 1-unconditional and 1-bidemocratic. In particular, (ei) 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 (ei) is normalized, 1-unconditional, 1-bidemocratic and (1 + ε)-greedy. Notation: a) x, x∗ = x∗(x) for x ∈ X, x∗ ∈ X ∗ b) 1A will denote either

i∈A ei or i∈A e∗ i . E.g., 1A =

• i∈A ei
• and

1A∗ =

• i∈A e∗

i

• .

c) Let x = xiei ∈ X. We set |x| = |xi|ei, and write x ≥ 0 if xi ≥ 0 for all i.

Proof of Theorem 1

WLOG (ei) is normalized and 1-unconditional. Let ∆ be the bidemocracy constant. Define |||x||| = x ∨ sup

• |x|, ϕ(n)

n 1A

• : n ∈ N, A ⊂ N, n = |A|
• .
• ϕ(n)

n 1A

• ∗ ≤ ϕ(n)

n ϕ∗(n) ≤ ∆. So x ≤ |||x||| ≤ ∆x.

Let |E| = n. Then |||1E||| ≥

• 1E, ϕ(n)

n 1E

• = ϕ(n). For |A| = m, we have
• 1E, ϕ(m)

m 1A

• = ϕ(m)

m |E ∩ A| ≤ ϕ(|E∩A|) |E∩A| |E ∩ A| ≤ ϕ(n) .

So |||1E||| = ϕ(n) and

• ϕ(n)

n 1E

• ∗ = 1. QED

Remark: Assume, instead of bidemocracy, that for all q ∈ (0, 1) there exists C > 0 such that A =

• A ⊂ N : n = |A| < ∞,
• ϕ(n)

n 1A

• ∗ ≤ C
• satisfies: ∀ finite E ⊂ N there exists A ∈ A such that A ⊂ E and |A| ≥ q|E|.

Characterizing C-greedy bases

Given vectors x = xiei and y = yiei 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 (ei) has Property (A) with constant C if for all x, y we have y ≤ Cx whenever y is a greedy rearrangement of x. Theorem [Albiac, Wojtaszczyk, ’06] If (ei) is 1-unconditional, then (ei) is C-greedy if and only if it satisfies Property (A) with constant C.

Proof of AW-characterization

Assume (ei) is 1-unconditional and has Property (A) with constant C. Fix x = xiei and m ∈ N. Write Gm(x) =

i∈A xiei. Let

s = min{|xi| : i ∈ A}, and note that |xi| ≤ s ≤ |xj| for i / ∈ A, j ∈ A. Let b =

i∈B biei be an arbitrary m-term approximation.

x − b =

• i∈A\B

xiei +

• i∈B

(xi − bi)ei +

• i /

∈A∪B

xiei

• ≥
• i∈A\B

sei +

• i /

∈A∪B

xiei

• ≥ 1

C

• i∈B\A

sei +

• i /

∈A∪B

xiei

• ≥ 1

C

• i∈B\A

xiei +

• i /

∈A∪B

xiei

• = 1

C

• x − Gm(x)
• .

QED.

Proof of Theorem 2

Let (ei) 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 (ei) is normalized, 1-unconditional, 1-bidemocratic and (1 + ε)-greedy. WLOG (ei) is normalized, 1-unconditional and 1-bidemocratic (by Theorem 1). Define |||x||| = sup

• |x|, x∗ +

1 ϕ∗(n)1A

• : x∗ ∈ εBX ∗, n ∈ N, A ⊂ N, |A| = n
• .

Calculation shows that (ei) 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 (ei) has the URP if there exists 0 < β < 1 and C > 0 such that ϕ(n) ≤ C n

m

βϕ(m) for all m ≤ n . This was introduced in [DKKT, ’03]. They showed that a greedy basis with the URP is bidemocratic, and that a greedy basis (ei) 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

• n Lp[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 (ei) be a normalized, 1-unconditional, ∆-democratic basis of a Banach space X. Given 0 < q < 1, fix C >

∆ q(1−q) and set

A =

• A ⊂ N : n = |A| < ∞,
• ϕ(n)

n 1A

• ∗ ≤ C
• .

Then ∀ finite E ⊂ N there exists A ∈ A such that A ⊂ E and |A| ≥ q|E|. Corollary [DKOSZ] Let (ei) be a greedy basis of a Banach space X. For any ε > 0 there is an equivalent norm on X with respect to which (ei) 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 H1 [DOSZ, ’11]. Problem: Can we replace (2 + ε)-greedy by (1 + ε)-greedy?

Proof of Lemma

Fix τ > 0 such that C >

(1+τ)∆ τq(1−q). Let E ⊂ N and n = |E|. Set F1 = E.

Pick z(1) ∈ BX ∗ with 1F1, z(1) = 1F1. WLOG z(1) ≥ 0 and supp(z(1)) ⊂ F1. Let E1 = {i ∈ F1 : z(1)

i

≥ τ}, and F2 = F1 \ E1 If |F2| < (1 − q)n, then stop, else go to next step. Pick z(2) ∈ BX ∗ with 1F2, z(2) = 1F2. WLOG z(2) ≥ 0 and supp(z(2)) ⊂ F2. Let E2 = {i ∈ F2 : z(1)

i

+ z(2)

i

≥ τ}, and F3 = F2 \ E2 If |F3| < (1 − q)n, then stop, else go to next step. Continue inductively. After m steps we end up with disjoint subsets E1, . . . , Em

• f E, and sets E = F1 ⊃ · · · ⊃ Fm+1 where Fk = E \ k−1

i=1 Ei for

1 ≤ k ≤ m + 1, and functionals z(1), . . . , z(m) ∈ B+

X ∗ such that supp(z(k)) ⊂ Fk

and Ek = {i ∈ Fk : z(1)

i

+ · · · + z(k)

i

≥ τ} for k = 1, . . . , m .

Proof of Lemma (contd.)

• 1E, z(1) + · · · + z(m)

=

m

• k=1
• 1Ek , z(1) + · · · + z(m)

+

• 1Fm+1, z(1) + · · · + z(m)

< (1 + τ)n .

• 1E, z(1) + · · · + z(m)

=

m

• k=1
• 1Fk , z(k)

≥ m ϕ

• (1 − q)n
• ∆

≥ m(1 − q)ϕ(n) ∆ . So m ≤ (1 + τ)∆ (1 − q) · n ϕ(n). Set A = m

k=1 Ek. Then A ⊂ E and |A| ≥ qn.

Finally, τ1A∗ ≤ z(1) + · · · + z(m)∗ ≤ m, and hence

• ϕ(|A|)

|A| 1A

• ∗ ≤ mϕ(|A|)

τ|A|

≤ · · · ≤ C . QED.

The general case

Corollary [DKOSZ] Let (ei) be a greedy basis of a Banach space X. For any ε > 0 there is an equivalent norm on X with respect to which (ei) 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 H1 [DOSZ, ’11]. Problem: Can we replace (2 + ε)-greedy by (1 + ε)-greedy?

The δ-parameter

Let ϕ be a fundamental function ϕ, i.e., ϕ: N → R+ is increasing and ϕ(n)

n

is

• decreasing. Define

δϕ(m) = lim inf

n→∞

ϕ(mn) mϕ(n) , and δ(ϕ) = lim

m→∞ δϕ(m) .

Properties of ϕ imply that δ(ϕ) ∈ [0, 1]. If (ei) has the URP, then δ(ϕ) = 0. Indeed, ϕ(mn) ≤ C mn

n

βϕ(n) , and so δϕ(m) ≤ Cmβ−1 . We are interested in the case δ(ϕ) > 0. E.g., if ϕ(n) ∼ n (Tsirelson space T, dyadic Hardy space H1), or if ϕ(n) ∼

n log n (Schlumprecht space S).

General greedy renorming

Theorem [DKOSZ] Let (ei) be a greedy basis of a Banach space X with fundamental function ϕ. Assume that δ(ϕ) > 0. Then for all ε > 0 there is an equivalent norm on X with respect to which (ei) is normalized, 1-unconditional and (1 + ε)-greedy. Lemma [DKOSZ] Let ϕ be a fundamental function with δ(ϕ) > 0. Then for all ε > 0 and for all m ∈ N there exists a fundamental function ψ ∼ ϕ such that δψ(m) >

1 1+ε.

Corollary [DKOSZ] Let X be either H1 with the Haar basis or T with the unit vector basis. Then for all ε > 0 there is an equivalent norm on X such that the basis is normalized, 1-unconditional and (1 + ε)-greedy.

Open Problems

Problem 1 Let (ei) be a bidemocratic basis of a Banach space X. Does there exist an equivalent norm on X with respect to which (ei) is 1-greedy? Problem 2 [AW, ’06, Problem 6.2] Let 1 < p < ∞. Does there exist an equivalent norm on Lp[0, 1] with respect to which the Haar basis is 1-greedy? Problem 3 Let (ei) be a greedy basis of a Banach space X. Does there exist for any ε > 0 an equivalent norm on X with respect to which the basis is (1 + ε)-greedy?