the work of nigel kalton on greedy algorithms in...

The work of Nigel Kalton on greedyalgorithms in Banach spaces

July, 2011

References

GK1 M. Ganichev and N. J. Kalton, Convergence of the weakdual greedy algorithm in Lp spaces, J. Approx. Theory 124(2003), 89-95.

GK2 M. Ganichev and N. J. Kalton, Convergence of the DualGreedy Algorithm in Banach Spaces, New York J. Math. 15(2009), 73-95.

Notation

X is a separable infinite-dimensional Banach space and D is anormalized dictionary for X , i.e:

(i) ‖ϕ‖ = 1 for all ϕ ∈ D.(ii) D is a fundamental system for X , i.e. its linear span is

dense in X .(iii) D is symmetric, i.e. ϕ ∈ D ⇒ −ϕ ∈ D.

Hilbert Space: Pure Greedy Algorithm

For x ∈ H (Hilbert space), define (xn)n>0 ⊂ H and (ϕn)n>1 ⊂ Dinductively. Set x0 := x .

I 1. Choose ϕn ∈ D such that

〈xn−1, ϕn〉 = maxϕ∈D

〈xn−1, ϕ〉 > 0.

I 2. Setxn := xn−1 − 〈xn−1, ϕn〉ϕn.

RemarkCan replace 1) by:

〈xn−1, ϕn〉 > τ supϕ∈D

〈xn−1, ϕ〉.

where τ ∈ (0, 1) is a weakness parameter.

〈xn−1, ϕ〉 > 0.

〈xn−1, ϕ〉.

〈xn−1, ϕ〉 > 0.

〈xn−1, ϕ〉.

Set Gn(x) := x − xn =∑n

i=1〈xi−1, ϕi〉ϕi

Theorem (L. Jones, ’85)The PGA converges, i.e.

x =∞∑

〈xn−1, ϕn〉ϕn = limn→∞

Gn(x).

Set Gn(x) := x − xn =∑n

i=1〈xi−1, ϕi〉ϕi

Theorem (L. Jones, ’85)The PGA converges, i.e.

x =∞∑

〈xn−1, ϕn〉ϕn = limn→∞

Gn(x).

Smoothness in Banach spaces

I Recall for 0 6= x ∈ X a (Hahn-Banach) norming functionalFx ∈ X ∗ satisfies

Fx(x) = ‖x‖ and ‖Fx‖ = 1.

I X is smooth if Fx is unique (x ∈ X ).I If X is smooth then its norm is Gateaux differentiable, i.e.

for all 0 6= x , y ∈ X

Fx(y) = limt→0

‖x + ty‖ − ‖x‖t

exists.

I The modulus of smoothness ρX (τ) is defined for τ > 0 by

ρX (τ) := sup‖x‖=‖y‖=1

‖x + τy‖+ ‖x − τy‖2

− 1.

Fx(x) = ‖x‖ and ‖Fx‖ = 1.

Fx(y) = limt→0

‖x + ty‖ − ‖x‖t

exists.

ρX (τ) := sup‖x‖=‖y‖=1

‖x + τy‖+ ‖x − τy‖2

− 1.

Fx(x) = ‖x‖ and ‖Fx‖ = 1.

Fx(y) = limt→0

‖x + ty‖ − ‖x‖t

exists.

ρX (τ) := sup‖x‖=‖y‖=1

‖x + τy‖+ ‖x − τy‖2

− 1.

Fx(x) = ‖x‖ and ‖Fx‖ = 1.

Fx(y) = limt→0

‖x + ty‖ − ‖x‖t

exists.

ρX (τ) := sup‖x‖=‖y‖=1

‖x + τy‖+ ‖x − τy‖2

− 1.

I X is uniformly smooth if

limτ↓0

ρX (τ)

τ= 0.

I X is uniformly convex if given ε > 0 there exists δ > 0 suchthat whenever x , y ∈ X satisfy ‖x‖ 6 1, ‖y‖ 6 1, and‖x − y‖ > ε then

‖x + y‖2

6 1− δ

Example`p and Lp[0, 1] are uniformly smooth and uniformly convex for1 < p < ∞.

limτ↓0

ρX (τ)

τ= 0.

‖x + y‖2

6 1− δ

limτ↓0

ρX (τ)

τ= 0.

‖x + y‖2

6 1− δ

4. Weak Dual Greedy Algorithm (DGA) in Banachspaces (introduced by V. Temlyakov)

We extend the PGA to Banach spaces by replacing the Hilbertspace inner product by norming functionals from the dual spaceX ∗.

Let 0 < τ < 1 be a weakness parameter. Set x0 := x .For each n > 1 define ϕn ∈ D and xn ∈ X :

I 1. ϕn satisfies

Fxn−1(ϕn) > τ supϕ∈D

Fxn−1(ϕ).

I 2. Define an ∈ R:

‖xn−1 − anφn‖ = mina∈R

‖xn−1 − aφn‖.

I 3. Setxn := xn−1 − anφn.

I 1. ϕn satisfies

Fxn−1(ϕ).

‖xn−1 − aφn‖.

I 1. ϕn satisfies

Fxn−1(ϕ).

‖xn−1 − aφn‖.

I 1. ϕn satisfies

Fxn−1(ϕ).

‖xn−1 − aφn‖.

The WDGA converges if ‖xn‖ → 0. Then

x =∑

anϕn.

RemarkIn [GK2] the WDGA is generalized to find the minimizer of asmooth f : X → R. The greedy step is to choose ϕn ∈ Dsatisfying

〈ϕn,∇f (xn−1)〉 > τ supϕ∈D

〈ϕ,∇f (xn−1)〉.

Here f (y) = ‖y‖ and x0 = x is the usual WDGA.

The WDGA converges if ‖xn‖ → 0. Then

x =∑

anϕn.

RemarkIn [GK2] the WDGA is generalized to find the minimizer of asmooth f : X → R. The greedy step is to choose ϕn ∈ Dsatisfying

〈ϕn,∇f (xn−1)〉 > τ supϕ∈D

〈ϕ,∇f (xn−1)〉.

Here f (y) = ‖y‖ and x0 = x is the usual WDGA.

Property Γ (Ganichev-Kalton, 03)

DefinitionX has property Γ if there exists γ > 0 such that for all x , y ∈ X ,with Fx(y) = 0, we have

‖x + y‖ > ‖x‖+ γFx+y (y).

TheoremIf X has property Γ then the WDGA converges (for everydictionary D and every initial vector x0).

TheoremFor 1 < p < ∞, Lp[0, 1] with its usual norm

‖f‖p = (

0|f (x)|p dx)1/p

has property Γ (and hence the WDGA converges).

‖x + y‖ > ‖x‖+ γFx+y (y).

‖f‖p = (

0|f (x)|p dx)1/p

‖x + y‖ > ‖x‖+ γFx+y (y).

‖f‖p = (

0|f (x)|p dx)1/p

Property Γ and Tame Functions (Ganichev-Kalton ’09)DefinitionA convex function f : X → R is tame if there exists a constantγ > 0 such that ∀x , y ∈ X

f (x + 2y) + f (x − 2y)− 2f (x)

6 γ(f (x + y) + f (x − y) −2f (x)).

Examplef (t) = |t |p is tame iff p > 1

TheoremLet f : X → R be a continuous convex function. Then f is tameiff f is Gateaux differentiable and there exists λ < ∞ (called theindex of f such that

〈y − x ,∇f (y)−∇f (x)〉6 λ(f (y)− 〈∇f (x), y − x〉 − f (x))

Property Γ and Tame Functions (Ganichev-Kalton ’09)DefinitionA convex function f : X → R is tame if there exists a constantγ > 0 such that ∀x , y ∈ X

f (x + 2y) + f (x − 2y)− 2f (x)

6 γ(f (x + y) + f (x − y) −2f (x)).

Examplef (t) = |t |p is tame iff p > 1

TheoremLet f : X → R be a continuous convex function. Then f is tameiff f is Gateaux differentiable and there exists λ < ∞ (called theindex of f such that

〈y − x ,∇f (y)−∇f (x)〉6 λ(f (y)− 〈∇f (x), y − x〉 − f (x))

TheoremLet f : X → R be a continuous tame convex function with indexλ.

I f is continuously Fréchet differentiable and f 7→ ∇f islocally Hölder continuous.

I If f is proper, i.e. f (x) →∞ as ‖x‖ → ∞ then f has aunique minimum at some point a ∈ X and there exist c > 0such that

f (x) > f (a) + c min(‖x − a‖λ, ‖x − a‖λ′),

where λ′ = λ/(λ− 1).

TheoremLet f : X → R be a proper tame continuous convex function.Then for any dictionary and any initial point the WDGAproduces a sequence (an) converging to the minimizer a of f .

f (x) > f (a) + c min(‖x − a‖λ, ‖x − a‖λ′),

Property Γ revisited

TheoremX has property Γ if and only if x 7→ ‖x‖r is tame for some(equiv. all) r > 1

I Property Γ passes to subspaces and quotient spaces.I Property Γ ⇒ uniformly smooth and uniformly convex.I X has Property Γ ⇔ X ∗ has property Γ.I Property Γ is preserved by real and complex interpolation.

Banach Spaces with Property Γ

I The Orlicz function space LF has property Γ for the Orliczand the Luxemberg norms ⇔ x 7→ F (|x |) is a tame functionon R.

I Let X be a Banach lattice which is p-convex with constant1 and q-concave with constant 1, where 1 < p < q < ∞.Then X has property Γ.

I If X has property Γ then Lr (X ) has property Γ for all1 < r < ∞.

I The Schatten ideals Sp have property Γ for 1 < p < ∞.

The X -greedy algorithm (XGA)(V. Temlyakov)

The XGA is a natural extension of the PGA to Banach spaces:simply make (almost) the best approximation at each step.

Set x0 := x . For each n > 1 define ϕn ∈ D and xn ∈ X :I 1. Choose ϕn ∈ D and an ∈ R such that

‖xn−1‖−‖xn−1 − anϕn ‖> τ(‖xn−1‖ − min

ϕ∈D,a∈R‖xn−1 − aϕ‖)

I 2. Setxn := xn−1 − anϕn.

Theorem (E. Livshits)The XGA can fail to converge in smooth Banach spaces.

‖xn−1‖−‖xn−1 − anϕn ‖> τ(‖xn−1‖ − min

An easy exampleLet (ei) be the unit vector basis of `p:

‖∑

aiei‖p = (∑

|ai |p)1/p.

Consider the dictionary D = {±ei : i > 1}.I Let x0 =

∑∞i=1 aiei .

I By symmetry we may assume

|a1| > |a2| > |a3| . . .

I Clearly,

x1 =∞∑

aiei , x2 =∞∑

aiei , . . . , xn =∞∑

aiei .

I So xn → 0.

‖∑

aiei‖p = (∑

|ai |p)1/p.

∑∞i=1 aiei .

|a1| > |a2| > |a3| . . .

I Clearly,

x1 =∞∑

aiei , x2 =∞∑

aiei , . . . , xn =∞∑

aiei .

I So xn → 0.

‖∑

aiei‖p = (∑

|ai |p)1/p.

∑∞i=1 aiei .

|a1| > |a2| > |a3| . . .

I Clearly,

x1 =∞∑

aiei , x2 =∞∑

aiei , . . . , xn =∞∑

aiei .

I So xn → 0.

‖∑

aiei‖p = (∑

|ai |p)1/p.

∑∞i=1 aiei .

|a1| > |a2| > |a3| . . .

I Clearly,

x1 =∞∑

aiei , x2 =∞∑

aiei , . . . , xn =∞∑

aiei .

I So xn → 0.

RemarkCrucial to this argument is that selected coefficients are alwaysset to zero because (ei) is a 1-unconditional basis for `p.

I A basis (ei) for a Banach space X is 1-unconditional if forall scalars (ai)

‖∑

±aiei‖ = ‖∑

aiei‖.

I The argument breaks down for the Haar basis in Lp[0, 1]which is not 1-unconditional.

The `p example easily generalizes:

PropositionSuppose (ei) is a 1-unconditional basis for a uniformly convexBanach space X . Then the XGA converges for the dictionaryD = {±ei : i > 1}.

‖∑

±aiei‖ = ‖∑

aiei‖.

‖∑

±aiei‖ = ‖∑

aiei‖.

‖∑

±aiei‖ = ‖∑

aiei‖.

Weak Convergence (SJD, D. Kutzarova, K. Shuman,V. Temlyakov, and P. Wojtaszczyk, 2008)

Recall that a sequence (xn) ⊂ X is weakly null if

limn→∞

F (xn) = 0 (F ∈ X ∗).

DefinitionX has the WN Property if whenever (xn) ⊂ X with ‖xn‖ = 1(n > 1) is such that (Fxn) is weakly null in X ∗, then (xn) isweakly null in X .

TheoremIf X is uniformly smooth and has the WN property then (forevery dictionary D and x0) the XGA (and also the DGA)converge weakly, i.e. (xn) is weakly null.

limn→∞

F (xn) = 0 (F ∈ X ∗).

limn→∞

F (xn) = 0 (F ∈ X ∗).

Spaces with the WN property

I If X is uniformly convex and has a 1-unconditional basisthen X has the WN property.

I A large class of Orlicz sequence spaces including `p

I Lp[0, 1] with the square function norm:

|||∞∑

aihi ||| = ‖(∑

a2i h2

i )1/2‖p.

This norm is equivalent to ‖ · ‖p. Moreover, this space isisometrically isomorphic to a closed subspace of Lp with itsusual norm.

I If X is reflexive and has the uniform Opial property then Xhas the WN property.

I If X is reflexive and has a uniformly reverse monotonebasis then X has the WN property.

|||∞∑

aihi ||| = ‖(∑

a2i h2

i )1/2‖p.

|||∞∑

aihi ||| = ‖(∑

a2i h2

i )1/2‖p.

|||∞∑

aihi ||| = ‖(∑

a2i h2

i )1/2‖p.

|||∞∑

aihi ||| = ‖(∑

a2i h2

i )1/2‖p.

I Lp[0, 1] with its usual norm does NOT have the WNproperty unless p = 2.

CorollaryThe XGA converges weakly in `p.

PropositionIf Xn is a smooth finite-dimensional normed space for eachn > 1 then the XGA converges weakly in the space

X = (∞∑

Xn)`p .

Hence uniform smoothness is not necessary for weakconvergence of the XGA.

ProblemDoes the XGA converge (weakly) in Lp[0, 1] with its usualnorm?

X = (∞∑

Xn)`p .

X = (∞∑

Xn)`p .

The Chebyshev XGA

‖xn−1 − anϕn‖ = minϕ∈D,a∈R

‖xn−1 − aϕ‖

I 2. DefineΦn := Span {ϕj}n

and define Gn to be the best approximant to x from Φn.I 3. Set

xn := x −Gn.

This generalizes the Orthogonal Greedy Algorithm.

The Chebyshev XGA

‖xn−1 − aϕ‖

xn := x −Gn.

The Chebyshev XGA

‖xn−1 − aϕ‖

xn := x −Gn.

The Chebyshev XGA

‖xn−1 − aϕ‖

xn := x −Gn.

Recall that X has the Kadets-Klee property if xn → x whenever

x = w − lim xn and ‖x‖ = lim ‖xn‖.

Theorem (SJD-Kutzarova-Shuman, 2006)Let X be a separable smooth reflexive Banach space with theKadets-Klee property. Then the CXGA converges for everydictionary D and every initial vector x0.

CorollaryEvery separable reflexive Banach space admits an equivalentnorm for which the CXGA always converges.Weak-∗ compactness yields results for special dictionaries innon-reflexive spaces.

TheoremThe CXGA converges in the Hardy spaces H1(Um) andBergman spaces B1(Um) for the dictionary of monomials

D = {Πmi=1zni

i : ni ∈ N}.

D = {Πmi=1zni

i : ni ∈ N}.

D = {Πmi=1zni

i : ni ∈ N}.

D = {Πmi=1zni

i : ni ∈ N}.

the work of nigel kalton on greedy algorithms in...

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