Nonlinear tensor product approximation Vladimir Temlyakov ICERM; - - PowerPoint PPT Presentation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Nonlinear tensor product approximation

Vladimir Temlyakov ICERM; October 3, 2014

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

1

Introduction

2

Best nonlinear approximation

3

Constructive nonlinear approximation

4

Lemmas for trigonometric polynomials

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best multilinear approximation

We are interested in approximation of a multivariate function f (x1, . . . , xd) by linear combinations of products u1(x1) · · · ud(xd)

• f univariate functions ui(xi), i = 1, . . . , d.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best multilinear approximation

We are interested in approximation of a multivariate function f (x1, . . . , xd) by linear combinations of products u1(x1) · · · ud(xd)

• f univariate functions ui(xi), i = 1, . . . , d.

Definition For a function f (x1, . . . , xd) denote ΘM(f )X := inf

{ui

j },j=1,...,M,i=1,...,d f (x1, . . . , xd) −

M

• j=1

d

• i=1

ui

j(xi)X

and for a function class F define ΘM(F)X := sup

f ∈F

ΘM(f )X.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best multilinear approximation

We are interested in approximation of a multivariate function f (x1, . . . , xd) by linear combinations of products u1(x1) · · · ud(xd)

• f univariate functions ui(xi), i = 1, . . . , d.

Definition For a function f (x1, . . . , xd) denote ΘM(f )X := inf

{ui

j },j=1,...,M,i=1,...,d f (x1, . . . , xd) −

M

• j=1

d

• i=1

ui

j(xi)X

and for a function class F define ΘM(F)X := sup

f ∈F

ΘM(f )X. In the case X = Lp we write p instead of Lp in the notation.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Multilinear dictionary

In other words we are interested in studying M-term approximations of functions with respect to the dictionary Πd := {g(x1, . . . , xd) : g(x1, . . . , xd) =

d

• i=1

ui(xi)} where ui(xi) are arbitrary univariate functions. We discuss the case

• f 2π-periodic functions of d variables and approximate them in

the Lp spaces. Denote by Πd

p the normalized in Lp dictionary Πd of

2π-periodic functions.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Mega problem

Problem Find a simple algorithm such that for any f ∈ Lp it provides after M ≤ mφ(m) iterations an M-term with respect to Πd approximant AM(f ) such that f − AM(f )p ≤ C1Θm(f )p.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Mega problem

Problem Find a simple algorithm such that for any f ∈ Lp it provides after M ≤ mφ(m) iterations an M-term with respect to Πd approximant AM(f ) such that f − AM(f )p ≤ C1Θm(f )p. Simple – incremental, greedy.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Mega problem

Problem Find a simple algorithm such that for any f ∈ Lp it provides after M ≤ mφ(m) iterations an M-term with respect to Πd approximant AM(f ) such that f − AM(f )p ≤ C1Θm(f )p. Simple – incremental, greedy. Ideally, φ(m) = C2. Otherwise, the slower growing φ(m) the better.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

What does give a hope?

In the case d = 2, p = 2 the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm (Orthogonal Matching Pursuit) solve the problem with φ(m) = 1, C1 = 1.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

What does give a hope?

In the case d = 2, p = 2 the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm (Orthogonal Matching Pursuit) solve the problem with φ(m) = 1, C1 = 1. For the trigonometric system we have. Theorem (T., 2014) Let D be the normalized in Lp, 2 ≤ p < ∞, real d-variate trigonometric system. Then for any f ∈ Lp the Weak Chebyshev Greedy Algorithm with weakness parameter t gives fC(t,p,d)m ln(m+1)p ≤ Cσm(f , D)p. (1)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Dictionary

Definition A set of functions D from a Banach space X is a dictionary if each g ∈ D has norm one (g := gX = 1) and the closure of spanD coincides with X.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Dictionary

Definition A set of functions D from a Banach space X is a dictionary if each g ∈ D has norm one (g := gX = 1) and the closure of spanD coincides with X. Definition For a nonzero element f ∈ X we denote by Ff a norming (peak) functional for f : Ff = 1, Ff (f ) = f .

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Dictionary

Definition A set of functions D from a Banach space X is a dictionary if each g ∈ D has norm one (g := gX = 1) and the closure of spanD coincides with X. Definition For a nonzero element f ∈ X we denote by Ff a norming (peak) functional for f : Ff = 1, Ff (f ) = f . The existence of such a functional is guaranteed by the Hahn-Banach theorem.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Chebyshev greedy algorithm

Weak Chebyshev Greedy Algorithm (WCGA)(t) Let t ∈ (0, 1]. For a given f0 we inductively define for each m ≥ 1 ϕm ∈ D is any satisfying |Ffm−1(ϕm)| ≥ t sup

g∈D

|Ffm−1(g)|.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Chebyshev greedy algorithm

Weak Chebyshev Greedy Algorithm (WCGA)(t) Let t ∈ (0, 1]. For a given f0 we inductively define for each m ≥ 1 ϕm ∈ D is any satisfying |Ffm−1(ϕm)| ≥ t sup

g∈D

|Ffm−1(g)|. Define Φm := span{ϕj}m

j=1,

and define Gm to be the best approximant to f0 from Φm.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Chebyshev greedy algorithm

Weak Chebyshev Greedy Algorithm (WCGA)(t) Let t ∈ (0, 1]. For a given f0 we inductively define for each m ≥ 1 ϕm ∈ D is any satisfying |Ffm−1(ϕm)| ≥ t sup

g∈D

|Ffm−1(g)|. Define Φm := span{ϕj}m

j=1,

and define Gm to be the best approximant to f0 from Φm. Denote fm := f − Gm.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best m-term approximation

Definition We let Σm(D) denote the collection of all functions (elements) in X which can be expressed as a linear combination of at most m elements of D. Thus each function f ∈ Σm(D) can be written in the form f =

• g∈Λ

cgg, Λ ⊂ D, #Λ ≤ m, where the cg are real or complex numbers.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best m-term approximation

Definition We let Σm(D) denote the collection of all functions (elements) in X which can be expressed as a linear combination of at most m elements of D. Thus each function f ∈ Σm(D) can be written in the form f =

• g∈Λ

cgg, Λ ⊂ D, #Λ ≤ m, where the cg are real or complex numbers. Definition For a function f ∈ X we define its best m-term approximation error σm(f ) := σm(f , D) := inf

a∈Σm(D) f − a.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Nice bases

There are two fundamental systems: the d-variate trigonometric system T d := {ei(k,x)} and the prototype of the wavelets the Ud

• system. We define the system U := {UI} in the univariate case.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Nice bases

There are two fundamental systems: the d-variate trigonometric system T d := {ei(k,x)} and the prototype of the wavelets the Ud

• system. We define the system U := {UI} in the univariate case.

Denote U+

n (x) := 2n−1

• k=0

eikx = ei2nx − 1 eix − 1 , n = 0, 1, 2, . . .;

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Nice bases

There are two fundamental systems: the d-variate trigonometric system T d := {ei(k,x)} and the prototype of the wavelets the Ud

• system. We define the system U := {UI} in the univariate case.

Denote U+

n (x) := 2n−1

• k=0

eikx = ei2nx − 1 eix − 1 , n = 0, 1, 2, . . .; U+

n,k(x) := ei2nxU+ n (x − 2πk2−n),

k = 0, 1, . . . , 2n − 1; U−

n,k(x) := e−i2nxU+ n (−x + 2πk2−n),

k = 0, 1, . . . , 2n − 1.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

System U

It will be more convenient for us to normalize in L2 the system of functions {U+

m,k, U− n,k} and enumerate it by dyadic intervals. We

write UI(x) := 2−n/2U+

n,k(x)

with I = [(k + 1/2)2−n, (k + 1)2−n); UI(x) := 2−n/2U−

n,k(x)

with I = [k2−n, (k + 1/2)2−n); and U[0,1)(x) := 1.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

System Ud

In the multivariate case of x = (x1, . . . , xd) we define the system Ud as the tensor product of the univariate systems U. Let I = I1 × · · · × Id, Ij ∈ D, j = 1, . . . , d, then UI(x) :=

d

• j=1

UIj(xj).

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

System Ud

In the multivariate case of x = (x1, . . . , xd) we define the system Ud as the tensor product of the univariate systems U. Let I = I1 × · · · × Id, Ij ∈ D, j = 1, . . . , d, then UI(x) :=

d

• j=1

UIj(xj). Both T d and Ud have two fundamental features: orthogonality and tensor product structure. Definition We say that a dictionary D has a tensor product structure if all its elements have a form of products u1(x1) · · · ud(xd) of univariate functions ui(xi), i = 1, . . . , d.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

System Ud

In the multivariate case of x = (x1, . . . , xd) we define the system Ud as the tensor product of the univariate systems U. Let I = I1 × · · · × Id, Ij ∈ D, j = 1, . . . , d, then UI(x) :=

d

• j=1

UIj(xj). Both T d and Ud have two fundamental features: orthogonality and tensor product structure. Definition We say that a dictionary D has a tensor product structure if all its elements have a form of products u1(x1) · · · ud(xd) of univariate functions ui(xi), i = 1, . . . , d. Any dictionary with tensor product structure is a subset of Πd.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Function class W r

q

Let Fr(t) := 1 + 2

∞

• k=1

k−r cos(kt − πr/2) be the univariate Bernoulli kernel and let Fr(x) := Fr(x1, . . . , xd) :=

d

• i=1

Fr(xi).

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Function class W r

q

Let Fr(t) := 1 + 2

∞

• k=1

k−r cos(kt − πr/2) be the univariate Bernoulli kernel and let Fr(x) := Fr(x1, . . . , xd) :=

d

• i=1

Fr(xi). Definition We define W r

q := {f : f = Fr ∗ ϕ,

ϕq ≤ 1}, where ∗ denotes the convolution.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best m-term approximation with respect to Ud

Theorem (T, 2000; Th1) For 1 < q, p < ∞ and big enough r we have σm(W r

q , Ud)p ≍ m−r(ln m)(d−1)r.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best m-term approximation with respect to Ud

Theorem (T, 2000; Th1) For 1 < q, p < ∞ and big enough r we have σm(W r

q , Ud)p ≍ m−r(ln m)(d−1)r.

Theorem (T, 2000; Th2) For any orthonormal system Ψ we have for 1 ≤ q < ∞ σm(W r

q , Ψ)2 ≫ m−r(ln m)(d−1)r.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best m-term approximation with respect to Ud

Theorem (T, 2000; Th1) For 1 < q, p < ∞ and big enough r we have σm(W r

q , Ud)p ≍ m−r(ln m)(d−1)r.

Theorem (T, 2000; Th2) For any orthonormal system Ψ we have for 1 ≤ q < ∞ σm(W r

q , Ψ)2 ≫ m−r(ln m)(d−1)r.

Thus, the Ud is an ideal orthogonal system.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Bilinear approximation

In the case d = 2 the multilinear approximation problem is a classical problem of bilinear approximation. There are known results on the rate of decay of errors of best bilinear approximation in Lp under different smoothness assumptions on f . We only mention some known results for classes of functions W r

q . The

problem of estimating ΘM(f )2 in case d = 2 (best M-term bilinear approximation in L2) is a classical one and was considered for the first time by E. Schmidt in 1907. For many function classes F an asymptotic behavior of ΘM(F)p is known. For instance Theorem (T, 1986) In the case d = 2 for r > 1 and 1 ≤ q ≤ p ≤ ∞ we have ΘM(W r

q )p ≍ M−2r+(1/q−max(1/2,1/p))+

(2)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Multilinear approximation

In the case d > 2 almost nothing is known. There is an upper estimate in the case q = p = 2 Theorem (T, 1988) For r > 0 we have ΘM(W r

2 )2 ≪ M−rd/(d−1).

(3)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Multilinear approximation

In the case d > 2 almost nothing is known. There is an upper estimate in the case q = p = 2 Theorem (T, 1988) For r > 0 we have ΘM(W r

2 )2 ≪ M−rd/(d−1).

(3) The above theorems show that the rate M−r(log M)(d−1)r of best M-term approximation with respect to the basis Ud, which has a tensor structure, is not as good as best M-term approximation with respect to Πd (we have exponent r for Ud instead of

rd d−1 for Πd).

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Known lower bounds

New results of this talk are around the bound (3). First of all we discuss the lower bound matching the upper bound (3). In the case d = 2 the lower bound ΘM(W r

p )p ≫ M−2r,

1 ≤ p ≤ ∞. (4) follows from more general results in [T, 1986] (see (2) above).

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Known lower bounds

New results of this talk are around the bound (3). First of all we discuss the lower bound matching the upper bound (3). In the case d = 2 the lower bound ΘM(W r

p )p ≫ M−2r,

1 ≤ p ≤ ∞. (4) follows from more general results in [T, 1986] (see (2) above). A stronger result ΘM(W r

∞)1 ≫ M−2r

(5) follows from Theorem 1.1 in [T, 1992].

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

New lower bounds

We could not prove the lower bound matching the upper bound (3) for d > 2. Instead, we prove a weaker lower bound.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

New lower bounds

We could not prove the lower bound matching the upper bound (3) for d > 2. Instead, we prove a weaker lower bound. For a function f (x1, . . . , xd) denote Θb

M(f )X :=

inf

{ui

j },ui j X ≤bf 1/d X

f (x1, . . . , xd) −

M

• j=1

d

• i=1

ui

j(xi)X

and for a function class F define Θb

M(F)X := supf ∈F Θb M(f )X.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

New lower bounds

We could not prove the lower bound matching the upper bound (3) for d > 2. Instead, we prove a weaker lower bound. For a function f (x1, . . . , xd) denote Θb

M(f )X :=

inf

{ui

j },ui j X ≤bf 1/d X

f (x1, . . . , xd) −

M

• j=1

d

• i=1

ui

j(xi)X

and for a function class F define Θb

M(F)X := supf ∈F Θb M(f )X.

Theorem (Bazarkhanov and T., 2014) Θb

M(W r ∞)1 ≫ (M ln M)− rd

d−1 . Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Upper bounds

Secondly, we discuss some upper bounds which extend the bound (3). The relation (2) shows that for 2 ≤ p ≤ ∞ in the case d = 2

• ne has

ΘM(W r

2 )p ≪ M−2r.

(6) We extended (6) for d > 2.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Upper bounds

Secondly, we discuss some upper bounds which extend the bound (3). The relation (2) shows that for 2 ≤ p ≤ ∞ in the case d = 2

• ne has

ΘM(W r

2 )p ≪ M−2r.

(6) We extended (6) for d > 2. Theorem (Bazarkhanov and T., 2014) Let 2 ≤ p < ∞ and r > (d − 1)/d. Then ΘM(W r

2 )p ≪

• M

(log M)d−1 − rd

d−1

.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Upper bounds

Secondly, we discuss some upper bounds which extend the bound (3). The relation (2) shows that for 2 ≤ p ≤ ∞ in the case d = 2

• ne has

ΘM(W r

2 )p ≪ M−2r.

(6) We extended (6) for d > 2. Theorem (Bazarkhanov and T., 2014) Let 2 ≤ p < ∞ and r > (d − 1)/d. Then ΘM(W r

2 )p ≪

• M

(log M)d−1 − rd

d−1

. The proof of the above theorem is not constructive. It goes by induction and uses a nonconstructive bound in the case d = 2.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

TGA

We define a well-known Thresholding Greedy Algorithm with respect to a basis. It is convenient for us to enumerate the basis functions by dyadic intervals. Assume a given system Ψ of functions ψI indexed by dyadic intervals can be enumerated in such a way that {ψI j}∞

j=1 is a basis for Lp. Then we define the greedy

algorithm G p(·, Ψ) as follows. Let f =

∞

• j=1

cI j(f , Ψ)ψI j, cI(f , p, Ψ) := cI(f , Ψ)ψIp. Then cI(f , p, Ψ) → 0 as |I| → 0.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

TGA continue

Denote Λm a set of m dyadic intervals I such that min

I∈Λm cI(f , p, Ψ) ≥ max J / ∈Λm

cJ(f , p, Ψ). We define G p(·, Ψ) by formula G p

m(f , Ψ) :=

• I∈Λm

cI(f , Ψ)ψI.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

TGA is good

It is proved in [T., 2000] that for 1 < q, p < ∞ and big enough r sup

f ∈W r

q

f − G p

M(f , Ud)p ≍ σM(W r q , Ud)p

≍ M−r(log M)(d−1)r. (7)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

TGA is good

It is proved in [T., 2000] that for 1 < q, p < ∞ and big enough r sup

f ∈W r

q

f − G p

M(f , Ud)p ≍ σM(W r q , Ud)p

≍ M−r(log M)(d−1)r. (7) The above relation (7) illustrates two phenomena: (I) for the class W r

q the simple Thresholding Greedy Algorithm provides near best

M-term approximation; (II) the rate M−r(log M)(d−1)r of best M-term approximation with respect to the basis Ud, which has a tensor structure, is not as good as best M-term approximation with respect to Πd (we have exponent r for Ud instead of

rd d−1 for Πd).

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Constructive multilinear approximation

We use two very different greedy-type algorithms to provide a constructive multilinear approximant. Surprisingly, these two algorithms gave the same error bound. Theorem (Bazarkhanov and T., 2014) For big enough r the following constructive upper bound for 2 ≤ p < ∞ holds ΘM(W r

2 )p ≪

• M

(ln M)d−1 − rd

d−1 + β d−1

.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Constructive multilinear approximation

We use two very different greedy-type algorithms to provide a constructive multilinear approximant. Surprisingly, these two algorithms gave the same error bound. Theorem (Bazarkhanov and T., 2014) For big enough r the following constructive upper bound for 2 ≤ p < ∞ holds ΘM(W r

2 )p ≪

• M

(ln M)d−1 − rd

d−1 + β d−1

. This constructive upper bound has an extra term

β d−1 in the

exponent compared to the best M-term approximation. It would be interesting to find a constructive way to obtain the near best approximation in this case.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Trigonometric system. Some history

De la Vallee Poussin (1908) and Bernstein (1912) proved En(| sin x|)∞ ≍ n−1.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Trigonometric system. Some history

De la Vallee Poussin (1908) and Bernstein (1912) proved En(| sin x|)∞ ≍ n−1. Ismagilov (1974) and Maiorov (1986) proved σm(| sin x|, T )∞ ≍ n−3/2.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Trigonometric system. Some history

De la Vallee Poussin (1908) and Bernstein (1912) proved En(| sin x|)∞ ≍ n−1. Ismagilov (1974) and Maiorov (1986) proved σm(| sin x|, T )∞ ≍ n−3/2. Key: constructive, uses Gaussian sums to prove an inequality for trigonometric polynomials σm(f , T )∞ ≤ CN3/2m−1f 1, f ∈ T (N). (T1)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

More history

It is easy to see that (T1) follows from σm(f , T )∞ ≤ CN1/2m−1f A, f ∈ T (N). (T2) where f A :=

• k

|ˆ f (k)|.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

More history

It is easy to see that (T1) follows from σm(f , T )∞ ≤ CN1/2m−1f A, f ∈ T (N). (T2) where f A :=

• k

|ˆ f (k)|. Devore and T. (1995) σm(f , T )∞ ≤ Cm−1/2(ln(1 + N/m))1/2f A, f ∈ T (N). (T3) In a certain sense (T3) is much stronger than (T2). However, the proof was nonconstructive.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

History continues

Dilworth, Kutzarova, and T. (2002) σm(f , T )p ≤ C(p)m−1/2f A, 2 ≤ p < ∞, f ∈ T (N). (T4)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

History continues

Dilworth, Kutzarova, and T. (2002) σm(f , T )p ≤ C(p)m−1/2f A, 2 ≤ p < ∞, f ∈ T (N). (T4)

• T. (2005) gave a constructive proof of (T3). This proof and the

proof of (T4) use the Weak Chebyshev Greedy Algorithm.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

History continues

Dilworth, Kutzarova, and T. (2002) σm(f , T )p ≤ C(p)m−1/2f A, 2 ≤ p < ∞, f ∈ T (N). (T4)

• T. (2005) gave a constructive proof of (T3). This proof and the

proof of (T4) use the Weak Chebyshev Greedy Algorithm. In particular, (T4) implies σm(f , T )p ≤ C(p)N1/2m−1/2f 2, 2 ≤ p < ∞, f ∈ T (N). (T5)

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

Best multilinear approximation

Lemma (Bazarkhanov and T., 2014) Let f ∈ T(N). Denote v(N) := d

j=1 ¯

• Nj. Then for 2 ≤ p < ∞
• ne has

ΘM(f )p ≪ v(N)1− 1

d ( ¯

M)−1f 2, ¯ M = max(M, 1).

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

TGA-type algorithm

Lemma (Bazarkhanov and T., 2014) Suppose that f ∈ T(N). Denote v(N) := d

j=1 ¯

• Nj. Then for

1 ≤ q ≤ p ≤ ∞ Θm(f )p ≪ v(N)β( ¯ m)−βf q, β := 1 q − 1 p, ¯ m := max(1, m). (8) The bound (8) is realized by a simple greedy-type algorithm.

Vladimir Temlyakov Nonlinear tensor product approximation

Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials

General greedy-type algorithms

Lemma (Bazarkhanov and T., 2014) Let f ∈ T(N). Then for 2 ≤ p < ∞ Θm(f )p ≪ v(N)

1 2 − 1 pd ( ¯

m)−1/2f 2. The above bound is realized by the Weak Chebyshev Greedy Algorithm with weakness parameter t.

Vladimir Temlyakov Nonlinear tensor product approximation