Kolmogorov nWidths of Function Classes Defined by a NonDegenerate Differential Operator

Let P(D) be the differential operator generated by a polynomial P, and let U P 2 be the class of multivariate periodic functions f such that kP(D)(f)k2 6 1. The problem of computing the asymptotic order of the Kolmogorov nwidths dn(U P 2 , L2) in the general case when U P 2 is compactly embedded into L2 has been open for a long time. In the present paper, we solve it in the case when P(D) is nondegenerate

Kolmogorov n-Widths of Function Classes Defined by a Non-Degenerate Differential Operator

Patrick L Combettes1 and Dinh D˜ung2∗

1 Sorbonne Universit´ es – UPMC Univ Paris 06 UMR 7598, Laboratoire Jacques-Louis Lions

F-75005 Paris, France

plc@math.jussieu.fr

2 Information Technology Institute Vietnam National University

144 Xuan Thuy, Cau Giay Hanoi, Vietnam

dinhzung@gmail.com

August 20, 2014 version 3.0

Abstract

Let P (D) be the differential operator generated by a polynomial P , and let U2[P ]be the class

of multivariate periodic functions f such that kP (D)(f )k 2 6 1 The problem of computing the asymptotic order of the Kolmogorov n-widths d n (U2[P ], L 2 ) in the general case when U2[P ] is compactly embedded into L 2 has been open for a long time In the present paper, we solve it

in the case when P (D) is non-degenerate.

Keywords Kolmogorov n-widths · Non-degenerate differential operator ·

Mathematics Subject Classifications (2010) 41A10; 41A50; 41A63

∗ Corresponding author Email: dinhzung@gmail.com.

1 Introduction

The aim of the present paper is to study Kolmogorov n-widths of classes of multivariate periodic functions given by a differential operator In order to describe the exact setting of the problem let

us introduce some notation

We first recall the notion of Kolmogorov n-widths [12,19] Let X be a normed space, let F be

a nonempty subset of X such that F = −F , and let Gnbe the class of all vector subspaces of X of dimension at most n The Kolmogorov n-width, dn(F, X ), of F in X is given by

dn(F, X ) = inf

G∈G n

sup

f ∈F

inf

This notion quantifies the error of the best approximation to the elements of F by elements in a vector subspace in X of dimension at most n [19,26,27] Recently, there has been strong interest

in applications of Kolmogorov n-width and its dual Gelfand n-widths to compressive sensing [3,9,

10,20], and in Kolmogorov n-width and its inverse ε-dimension of classes mixed smoothness in high-dimensional approximations [4,8] ε-dimension and more general information complexity is

a tool for study of tractability of high-dimensional approximation problems; see [4,8,16,17,18] for details and references

We consider functions on Rd which are 2π-periodic in each variable as functions defined on the d-dimensional torus Td = [−π, π]d Denote by L2(Td) the Hilbert space of functions on Td

equipped with the standard scalar product, i.e.,

(∀f ∈ L2(Td))(∀g ∈ L2(Td)) hf | gi = 1

(2π)d

Z

Td

and by S0(Td)the space of distributions on Td The norm of f ∈ L2(Td)is kf k2 =phf | fi and, given k ∈ Zd, the kth Fourier coefficient of f ∈ L2(Td)is ˆf (k) = ihk|·i Every f ∈ S0

(Td)can

be identified with the formal Fourier series

f = X

k∈Z d

ˆ

where the sequence ( ˆf (k))k∈Zd forms a tempered sequence [23,27] By Parseval’s identity, L2(Td)

is the subset of S0(Td)of all distributions f for which

X

k∈Z d

Let α = (α1, , αd) ∈ Ndand let f ∈ S0(Td) We set

Zd0(α) =(k1, , kd) ∈ Zd(∀j ∈ {1, , d}) αj 6= 0 ⇒ kj 6= 0 (1.5)

As usual, we set |α| =Pd

j=1αjand, given z = (z1, , zd) ∈ Cd, zα =Qd

j=1zαj

j The αth derivative

of f ∈ S0(Td)is the distribution f(α) ∈ S0(Td)given through the identification

f(α) = X

k∈Zd0 (α)

The differential operator Dα on S0(Td) is defined by Dα: f 7→ (−i)|α|f(α) Now let A ⊂ Nd be a nonempty finite set, let (cα)α∈Abe nonzero real numbers, and define a polynomial by

P : x 7→X

α∈A

The differential operator P (D) on S0(Td)generated by P is

P (D) =X

α∈A

Set

W2[P ] =f ∈ S0

(Td)P (D)(f ) ∈ L2(Td) , (1.9)

denote the seminorm of f ∈ W2[P ]by

kf k

and let

U2[P ]=

n

f ∈ W2[P ] kf k

W2[P ]6 1

o

The problem of computing asymptotic orders of dn(U2[P ], L2(Td)) in the general case when W2[P ]

is compactly embedded into L2(Td)has been open for a long time; see, e.g., [25, Chapter III] for details Our main contribution is to solve it for a non-degenerate differential operator P (D) by establishing the asymptotic order

dn U2[P ], L2(Td)
 n−rlogνrn, (1.12) where r and ν depend only on P

The first exact values of n-widths of univariate Sobolev classes were obtained by Kol-mogorov [12] (see also [13, pp 186–189]) The problem of computing the asymptotic order

of dn(U2[P ], L2(Td))is directly related to hyperbolic crosses trigonometric approximations and to n-widths of classes multivariate periodic functions with a bounded mixed smoothness This line of work was initiated by Babenko in [1,2]; in particular, the asymptotic orders of n-widths in L2(Td)

of these classes were established in [1] Further work on asymptotic orders and hyperbolic cross approximation can be found in [6,7,25] and recent developments in [14,22,24,28] In [5], the strong asymptotic order of dn(U2A, L2(Td))was computed in the case when UA

2 is the closed unit ball of the space WA

2 of functions with several bounded mixed derivatives (see Subsection4.4for

a precise definition) Recently, Kolmogorov n-widths in the classical isotropic Sobolev space Hsof classes of multivariate periodic functions with anisotropic smoothness have been investigated in high-dimensional settings [4,8], where although the dimension n of the approximating subspace

is the main parameter in the study of convergence rates with respect to n going to infinity, the parameter d may seriously affect this rate when d is large

The paper is organized as follows In Section 2, we provide as auxiliary results Jackson-type and Bernstein-type inequalities for trigonometric approximations of functions from W2[P ] We also characterize the compactness of U2[P ]in L2(Td)and of non-degenerateness of P (D) In Section3,

we present the main result of the paper, namely the asymptotic order of dn(U2[P ], L2(Td))in the case when P (D) is non-degenerate In Section4, we derive norm equivalences relative to k · kW[P ]

2

and, based on them, we provide examples of n-widths dn(U2[P ], L2(Td))for non-degenerate differential operators

2 Preliminaries

2.1 Notation, standing assumption, and definitions

Let Θ be an abstract set, and let Φ and Ψ be functions from Θ to R Then we write

if there exist constants C1 and C2 in ]0, +∞[ such that (∀θ ∈ Θ) C1Φ(θ) 6 Ψ(θ) 6 C2Φ(θ) The unit vectors of Rdare denoted by (uj)16j6d

Definition 2.1 Let A be a nonempty finite subset of Rd

+ The polyhedron spanned by A is the convex hull conv(A) of A,

∆(A) =nα ∈ Aλα

λ ∈ [1, +∞[ ∩ conv(A) = {α}o, (2.2) and E(A) the set of vertices of ∆(A) In addition,

∀x ∈ Rd+

mA(x) = max

and

(∀t ∈ [0, +∞[) ΩA(t) =k ∈ Nd

Throughout the paper, we make the following standing assumption

Assumption 2.2 A is a nonempty finite subset of Ndand (cα)α∈A are nonzero real numbers We set

P : x 7→X

α∈A

cαxα, M : x 7→ max

α∈E(A)

|xα|, and τ = inf

k∈Z d|P (k)| (2.5) Moreover, for every t ∈ [0, +∞[,

K(t) =k ∈ Zd

|P (k)| 6 t

and V (t) =



f ∈ S0(Td)

f = X

k∈K(t)

ˆ

f (k)eihk|·i

 (2.6)

Remark 2.3 Suppose that t ∈ ]τ, +∞[ Then K(t) 6= ∅ and dimV (t) = |K(t)|, where |K(t)|

denotes the cardinality of K(t) In addition, if |K(t)| < +∞, then V (t) is the space of trigonometric polynomials with frequencies in K(t)

Definition 2.4 The Newton diagram of P is ∆(A) and the Newton polyhedron of P is Γ(P ) =

conv(A) The intersection of Γ(P ) with a supporting hyperplane of Γ(P ) is called a face of Γ(P ) The dimension of a face ranges from 0 to d − 1 A vertex is a 0-dimensional face The set of

vertices of Γ(P ) is ϑ(P ) and the set of faces of Γ(P ) is Σ(P ) The differential operator P (D) is

non-degenerate if P and, for every σ ∈ Σ(P ), Pσ: Rd→ R : x 7→P

α∈σcαxα do not vanish outside the coordinate planes of Rd, i.e.,

∀x ∈ Rd

d

Y

j=1

xj 6= 0 ⇒ ∀σ ∈ Σ(P )

P (x)Pσ(x) 6= 0 (2.7)

2.2 Trigonometric approximations

We first prove a Jackson-type inequality

Lemma 2.5 Let t ∈ ]0, +∞[ and define the linear operator St: S0(Td) → S0(Td)by

∀f ∈ S0(Td)
St(f ) = X

k∈K(t)

ˆ

Let f ∈ W2[P ]and suppose that t > τ Then the distribution f − St(f )represents a function in L2(Td)

and

kf − St(f )k26 t−1kf k

Proof Set g = f − St(f ) Then g ∈ S0(Td) On the other hand, Parseval’s identity yields

kf k2

W2[P ]= X

k∈Z d

Hence,

X

k∈Z d

|ˆg(k)|2= X

k∈Z d \K(t)

| ˆf (k)|2

6 sup

k∈Z d \K(t)



|P (k)|−2 X

k∈Z d \K(t)

|P (k)|2| ˆf (k)|2



6 t−2kf k2

which means that f − St(f )represents a function in L2(Td)for which (2.9) holds

Corollary 2.6 Let t ∈ ]τ, +∞[ Then

sup

f ∈U2[P ]

inf

g∈V (t)

f −g∈L 2 (T d )

Next, we prove a Bernstein-type inequality

Lemma 2.7 Let f ∈ V (t) ∩ L2(Td)and let t ∈ ]τ, +∞[ Then

kf k

Proof By (2.10), we have

kf k2

W2[P ]= X

k∈K(t)

|P (k)|2| ˆf (k)|2 6

 sup

k∈K(t)

|P (k)|2

 X

k∈K(t)

| ˆf (k)|2 6 t2kf k2

2, (2.14)

which provides the announced inequality

2.3 Compactness and non-degenerateness

We start with a characterization of the compactness of the unit ball defined in (1.11)

Lemma 2.8 The set U2[P ]is a compact subset of L2(Td)if and only if the following hold:

(i) For every t ∈ ]τ, +∞[, K(t) is finite.

(ii) τ > 0.

Proof To prove sufficiency, suppose that (i) and (ii)hold, and fix t ∈ ]τ, +∞[ By (i), V (t) is a set of trigonometric polynomials and, consequently, a subset of L2(Td) In particular, using the notation (2.8), (∀f ∈ S0(Td)) St(f ) ∈ L2(Td) Hence, by Lemma2.5,

∀f ∈ W2[P ]

f = (f − St(f )) + St(f ) ∈ L2(Td) (2.15) Thus, W2[P ] ⊂ L2(Td) On the other hand, (2.10) implies that U2[P ] is a closed subset of L2(Td) Therefore, it is compact in L2(Td)if, for every ε ∈ ]0, +∞[, there exists a finite ε-net in L2(Td)for

U2[P ]or, equivalently, if the following following two conditions are satisfied:

(iii) For every ε ∈ ]0, +∞[, there exists a finite dimensional vector subspace Gε of L2(Td) such that

sup

f ∈U2[P ]

inf

g∈G ε

(iv) U2[P ]is bounded in L2(Td).

It follows from (2.10) that (ii)⇔(iv) On the other hand, since dim V (t) = |K(t)|, Corollary2.6

yields(i)⇒(iii) To prove necessity, suppose that(i)does not hold Then dim V (˜t) = |K(˜t)| = +∞ for some ˜t ∈ ]0, +∞[ By Lemma 2.7, eU = f ∈ V (˜t) ∩ L2(Td)  kf k2 6 1/˜t is a subset of U2[P ] which is not compact in L2(Td) If (ii) does not hold, then U2[P ]∩ L2(Td) is unbounded and, consequently, not compact in L2(Td)

The following lemma characterizes the non-degenerateness of P (D)

Lemma 2.9 Then P (D) is non-degenerate if and only if

(∃ C ∈ ]0, +∞[)(∀x ∈ Rd) |P (x)| > C max

α∈ϑ(P )

Proof As proved in [11,15], P (D) is non-degenerate if and only if

(∃ C1∈ ]0, +∞[)(∀x ∈ Rd) |P (x)| > C1

X

α∈ϑ(P )

Hence, since there exist constants C2 and C3 in ]0, +∞[ such that

(∀x ∈ Rd) C2 max

α∈ϑ(P )

|xα| 6 X

α∈ϑ(P )

|xα| 6 C3 max

α∈ϑ(P )

the proof is complete

Lemma 2.10 Let B be a nonempty finite subset of Rd

+and let t ∈ [0, +∞[ Then ΩB(t)is finite if and only

(∀j ∈ {1, , d})(∃ aj ∈ ]0, +∞[) B ∩span(uj) = {ajuj} (2.20)

Proof. If (2.20) holds, then ΩB(t) ⊂ Td

j=1x ∈ Rd

+

xj 6 t1/aj and, consequently, ΩB(t)

is bounded Conversely, if (2.20) does not hold, there exists j ∈ {1, , d} such that

muj

m ∈ N ⊂ ΩB(t), which shows that ΩB(t)is unbounded

Theorem 2.11 Suppose that P (D) is non-degenerate Then U2[P ]is a compact subset of L2(Td)if and only if (2.20) is satisfied and 0 ∈ A.

Proof It follows from Young’s inequality that there exists C1 ∈ ]0, +∞[ such that

(∀x ∈ Rd) |P (x)| 6 C1 max

α∈ϑ(P )

Hence, by Lemma2.9, there exist C2∈ ]0, +∞[ such that

(∀x ∈ Rd) C2 max

α∈ϑ(P )|xα| 6 |P (x)| 6 C1 max

Consequently, by Lemma2.8, U2[P ]is a compact set in L2(Td)if and only if, for every t ∈ [0, +∞[,

ΩA(t)is finite and

inf

By Lemma2.10, the first condition is equivalent to (2.20), and the second to 0 ∈ A

3 Main result

The following facts will be necessary to prove our main result

Lemma 3.1 Let B be a nonempty finite subset of Rd

+and let x ∈ Rd

+ Then mB(x) = mE(B)(x).

Proof It is clear that mE(B)(x) 6 mB(x) Conversely, let α ∈ B r E(B) On the one hand, there exists ρ ∈ ]1, +∞[ such that α0 = ρα ∈ ∆(B) One the other hand, by Carath´eodory’s theorem [21, Theorem 17.1], α0is a convex combination of points (αj)16j6d+1in E(B), say

α0 =

d+1

X

j=1

λjαj, where {λj}16j6d+1⊂ [0, +∞[ and

d+1

X

j=1

λj = 1 (3.1)

Hence, by Young’s inequality

xα< xα0 =

d+1

Y

j=1

(xαj)λj 6

d+1

X

j=1

λjxαj 6 max

16j6d+1xαj 6 mE(B)(x) (3.2)

Corollary 3.2 Suppose that P (D) is non-degenerate Then there exist C1 ∈ ]0, +∞[ and C2 ∈ ]0, +∞[such that

∀x ∈ Rd

C1M (x) 6 |P (x)| 6 C2M (x) (3.3)

Proof Combine (2.22), Lemma2.9, and Lemma3.1

For a given finite set B ⊂ Rd

+, consider the following convex problem in Rd

max

x∈B ◦

d

X

j=1

where

B◦ =x ∈ Rd

is the polar of B Denote by µ(B) the (maximal) value of Problem (3.4) and by ν(B) the dimension

of its set of solutions We put also

One can verify that

r(B) = maxρ

and ν(B) is the dimension of the minimal face of the polyhedron conv(B) which contains the point r(B)1as a relative interior point, where 1 = (1, , 1) ∈ Rd Notice also that 0 6 ν(B) 6 d − 1

Lemma 3.3 Let B meet all the axes at a point ajuj for some aj > 0 Then

(∀t ∈ [2, +∞[) |ΩB(t)|  tµ(B)logν(B)t (3.8)

Proof Fix t ∈ [2, +∞[ and set ΛB(t) =x ∈ Rd

+

mB(x) 6 t Then, as in the proof of Lemma2.10, one can see that ΛB(t)is a bounded subset in Rd

+ If we denote by vol ΛB(t)the volume of ΛB(t), then it follows from [5, Theorem 1] that

Furthermore, proceeding as in the proof of [5, Theorem 2], one shows that

These asymptotic relations prove the lemma

In computational mathematics, the so-called ε-dimension nε = nε(W, X) is used to quantify the computational complexity It is defined by

nε(W, X) := inf

(

n : ∃ Ln: sup

f ∈W

inf

g∈L n

kf − gkX 6 ε

) ,

where Ln is a linear subspace in X of dimension 6 n This approximation characteristic is the inverse of dn(W, X) In other words, the quantity nε(W, X)is the minimal number nε such that the approximation of W by a suitably chosen approximant nε-dimensional subspace L in X gives the approximation error 6 ε

Our main result can now be stated and proved

Theorem 3.4 Suppose that P (D) is non-degenerate, that (2.20) is satisfied, and that 0 ∈ A Then,

for n sufficiently large,

dn U2[P ], L2(Td)
 n−r(ϑ(A))logν(ϑ(A))r(ϑ(A))n, (3.11)

or equivalently,

nε U2[P ], L2(Td)
 ε−1/r(ϑ(A))| log ε|ν(ϑ(A)) (3.12)

Proof Set ¯t = max{2, τ } It follows from Corollary?? that

(∀t ∈ [¯t, +∞[) |Ωϑ(A)(t)|  |K(t)| (3.13) Since A satisfies (2.20), so does ϑ(A) Hence applying Lemma3.3to ϑ(A), we have

(∀t ∈ [¯t, +∞[) |K(t)|  tµlogνt, (3.14) where µ = µ(ϑ(A)) and ν = ν(ϑ(A)) In turn, for every m ∈ N, there exists C1 ∈ ]0, +∞[ such that

For n ∈ N large enough, there exist m ∈ N such that

C1m1/rlogνm 6 n < C1(m + 1)1/rlogν(m + 1) 6 C2m1/rlogνm, (3.16) where C2 ∈ ]0, +∞[ is independent from n and m It follows from (3.15), (3.16), and Corollary2.6

that

dn(U2[P ], L2(Td)) 6 m−1 n−rlogνrn (3.17) The upper bound of (3.11) is proven To establish the lower bound, let us recall from [26] that, for every n + 1-dimensional subspace Ln+1of X and every ρ ∈ ]0, +∞[, we have

dn(Bn+1(ρ), X ) = ρ, where Bn+1(ρ) = {f ∈ Ln+1| kf kX 6 ρ} (3.18) Similarly to (3.15) and (3.16), for n ∈ N sufficiently large, there exists m ∈ N such that

dim V (m) > C3m1/rlogνm > n > C4m1/rlogνm, (3.19) where C3 ∈ ]0, +∞[ and C4∈ ]0, +∞[ are independent from n and m Consider the set

U (m) =f ∈ V (m)

By Lemma2.7, U (m) ⊂ U2[P ]and consequently, it follows from (3.18) and (3.19) that

dn U2[P ], L2(Td)
> dn(U (m), L2(Td)) > m−1 n−rlogνrn, (3.21) which concludes the proof

Remark 3.5 We have actually proven a bit more than Theorem3.4 Namely, suppose that P (D) satisfies the conditions of compactness for U2[P ]stated in Lemma2.8and for every n ∈ N, let m(n)

be the maximal number such that |K(m(n))| 6 n Then, for n sufficiently large, we have

dn U2[P ], L2(Td)
 1

The paper is organized as follows... in the case when UA< /small>

2 is the closed unit ball of the space WA< /small>

2 of functions with several bounded mixed derivatives (see... constants C1 and C2 in ]0, +∞[ such that (∀θ ∈ Θ) C1Φ(θ) Ψ(θ) C2Φ(θ) The unit vectors of Rdare denoted by (uj)16j6d