Kolmogorov nWidths of Function Classes Defined by a NonDegenerate Differential Operator

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

2

Information Technology Institute
Vietnam National University
144 Xuan Thuy, Cau Giay
Hanoi, Vietnam


August 20, 2014 -- version 3.0

Abstract
[P ]

Let P (D) be the differential operator generated by a polynomial P , and let U2 be the class
of multivariate periodic functions f such that P (D)(f ) 2 1. The problem of computing the
[P ]
[P ]
asymptotic order of the Kolmogorov n-widths dn (U2 , L2 ) in the general case when U2 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 non-degenerate.
Keywords. Kolmogorov n-widths · Non-degenerate differential operator ·

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

∗

Corresponding author. Email:

1


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 Gn be 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 sup inf

G∈Gn f ∈F g∈G

f −g

(1.1)

X.

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 ))

f |g =

1
(2π)d

(1.2)

f (x)g(x)dx,
Td

and by S (Td ) the space of distributions on Td . The norm of f ∈ L2 (Td ) is f 2 =
f | f and,
d
d
i
k|·
ˆ
given k ∈ Z , the kth Fourier coefficient of f ∈ L2 (T ) is f (k) = f | e
. Every f ∈ S (Td ) can

be identified with the formal Fourier series
fˆ(k)ei

f=

k|·

(1.3)

,

k∈Zd

where the sequence (fˆ(k))k∈Zd forms a tempered sequence [23, 27]. By Parseval’s identity, L2 (Td )
is the subset of S (Td ) of all distributions f for which
|fˆ(k)|2 < +∞.

(1.4)

k∈Zd

Let α = (α1 , . . . , αd ) ∈ Nd and let f ∈ S (Td ). We set
Zd0 (α) = (k1 , . . . , kd ) ∈ Zd (∀j ∈ {1, . . . , d}) αj = 0 ⇒ kj = 0 .

(1.5)
α

As usual, we set |α| = dj=1 αj and, given z = (z1 , . . . , zd ) ∈ Cd , z α = dj=1 zj j . The αth derivative
of f ∈ S (Td ) is the distribution f (α) ∈ S (Td ) given through the identification
(ik)α fˆ(k)ei

f (α) =

k|·

(1.6)

.

k∈Zd0 (α)

2


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

P: x→

(1.7)

α∈A

The differential operator P (D) on S (Td ) generated by P is
cα Dα .

P (D) =

(1.8)

α∈A

Set
[P ]

W2

= f ∈ S (Td ) P (D)(f ) ∈ L2 (Td ) ,
[P ]

denote the seminorm of f ∈ W2
f

[P ]

W2

(1.9)

by
(1.10)

= P (D)(f ) 2 ,

and let
[P ]

U2

[P ]

= f ∈ W2

f

[P ]

W2

(1.11)

1 .
[P ]

[P ]

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

dn U2 , L2 (Td )

n−r logνr n,

(1.12)

where r and ν depend only on P .

The first exact values of n-widths of univariate Sobolev classes were obtained by Kolmogorov [12] (see also [13, pp. 186–189]). The problem of computing the asymptotic order
[P ]
of dn (U2 , 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 U2A is the closed unit
ball of the space W2A of functions with several bounded mixed derivatives (see Subsection 4.4 for
a precise definition). Recently, Kolmogorov n-widths in the classical isotropic Sobolev space H s of
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.
3


The paper is organized as follows. In Section 2, we provide as auxiliary results Jackson-type
[P ]
and Bernstein-type inequalities for trigonometric approximations of functions from W2 . We also
[P ]
characterize the compactness of U2 in L2 (Td ) and of non-degenerateness of P (D). In Section 3,
[P ]
we present the main result of the paper, namely the asymptotic order of dn (U2 , L2 (Td )) in the case
when P (D) is non-degenerate. In Section 4, we derive norm equivalences relative to · W [P ] and,
2

[P ]
dn (U2 , L2 (Td ))

based on them, we provide examples of n-widths
operators.

2

for non-degenerate differential

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
(∀θ ∈ Θ)

Φ(θ)

(2.1)

Ψ(θ)

if there exist constants C1 and C2 in ]0, +∞[ such that (∀θ ∈ Θ) C1 Φ(θ)
unit vectors of Rd are denoted by (uj )1 j d .

C2 Φ(θ). The

Ψ(θ)

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) = α ∈ A

λα λ ∈ [1, +∞[ ∩ conv(A) = {α} ,

(2.2)

and E(A) the set of vertices of ∆(A). In addition,
∀x ∈ Rd+

mA (x) = max xα

(2.3)

α∈A

and
(∀t ∈ [0, +∞[)

ΩA (t) = k ∈ Nd mA (k)

(2.4)

t .

Throughout the paper, we make the following standing assumption.
Assumption 2.2 A is a nonempty finite subset of Nd and (cα )α∈A are nonzero real numbers. We
set
cα xα ,

P: x→

M : x → max |xα |,
α∈E(A)

α∈A

and τ = inf |P (k)|.

(2.5)

k∈Zd

Moreover, for every t ∈ [0, +∞[,
K(t) = k ∈ Zd |P (k)|

t

and V (t) =

fˆ(k)ei

f ∈ S (Td ) f =
k∈K(t)

4

k|·

.

(2.6)


Remark 2.3 Suppose that t ∈ ]τ, +∞[. Then K(t) = ∅ 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 → α∈σ cα xα do not vanish outside
the coordinate planes of Rd , i.e.,
d

∀x ∈ R

d

⇒

xj = 0

∀σ ∈ Σ(P )

P (x)Pσ (x) = 0.

(2.7)

j=1

2.2

Trigonometric approximations

We first prove a Jackson-type inequality.
Lemma 2.5 Let t ∈ ]0, +∞[ and define the linear operator St : S (Td ) → S (Td ) by
∀f ∈ S (Td )

fˆ(k)ei

St (f ) =

k|·

(2.8)

.

k∈K(t)
[P ]

Let f ∈ W2
and

and suppose that t > τ . Then the distribution f − St (f ) represents a function in L2 (Td )

f − St (f )

t−1 f

2

[P ]

W2

(2.9)

.

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

2
[P ]
W2

|P (k)|2 |fˆ(k)|2 .

=

(2.10)

k∈Zd

Hence,
|fˆ(k)|2

|ˆ

g (k)|2 =
k∈Zd

k∈Zd \K(t)

|P (k)|−2

sup
k∈Zd \K(t)

t−2 f

|P (k)|2 |fˆ(k)|2
k∈Zd \K(t)

2
[P ] ,
W2

(2.11)

which means that f − St (f ) represents a function in L2 (Td ) for which (2.9) holds.
5


Corollary 2.6 Let t ∈ ]τ, +∞[. Then
f −g

sup

inf

[P ]
f ∈U2

g∈V (t)
f −g∈L2 (Td )

2

t−1 .

(2.12)

Next, we prove a Bernstein-type inequality.
Lemma 2.7 Let f ∈ V (t) ∩ L2 (Td ) and let t ∈ ]τ, +∞[. Then
f

t f

[P ]

W2

(2.13)

2.

Proof. By (2.10), we have
f

2
[P ]
W2

|P (k)|2 |fˆ(k)|2

=
k∈K(t)

|fˆ(k)|2

sup |P (k)|2
k∈K(t)

t2 f

2
2,

(2.14)

k∈K(t)

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).
[P ]

Lemma 2.8 The set U2

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 ∈ S (Td )) St (f ) ∈ L2 (Td ). Hence, by Lemma 2.5,
[P ]

∀f ∈ W2

f = (f − St (f )) + St (f ) ∈ L2 (Td ).

[P ]

(2.15)
[P ]

Thus, W2 ⊂ L2 (Td ). On the other hand, (2.10) implies that U2 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
[P ]
U2 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

[P ]

f ∈U2

inf

g∈Gε

f −g

2

(2.16)

ε.

6


[P ]

(iv) U2

is bounded in L2 (Td ).

It follows from (2.10) that (ii)⇔(iv). On the other hand, since dim V (t) = |K(t)|, Corollary 2.6
yields (i)⇒(iii). To prove necessity, suppose that (i) does not hold. Then dim V (t˜) = |K(t˜)| = +∞
[P ]
for some t˜ ∈ ]0, +∞[. By Lemma 2.7, U = f ∈ V (t˜) ∩ L2 (Td )
f 2 1/t˜ is a subset of U2

[P ]

which is not compact in L2 (Td ). If (ii) does not hold, then U2
consequently, not compact in L2 (Td ).

∩ L2 (Td ) is unbounded and,

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 )

C max |xα |.

|P (x)|

(2.17)

α∈ϑ(P )

Proof. As proved in [11, 15], P (D) is non-degenerate if and only if
(∃ C1 ∈ ]0, +∞[)(∀x ∈ Rd )

|P (x)|

|xα |.

C1

(2.18)

α∈ϑ(P )

Hence, since there exist constants C2 and C3 in ]0, +∞[ such that
(∀x ∈ Rd )

C2 max |xα |
α∈ϑ(P )

|xα |

C3 max |xα |,

(2.19)

α∈ϑ(P )

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 ) = {aj uj }.

(2.20)

Proof. If (2.20) holds, then ΩB (t) ⊂ dj=1 x ∈ Rd+ xj 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.
[P ]

Theorem 2.11 Suppose that P (D) is non-degenerate. Then U2
only if (2.20) is satisfied and 0 ∈ A.

is a compact subset of L2 (Td ) if and

Proof. It follows from Young’s inequality that there exists C1 ∈ ]0, +∞[ such that
(∀x ∈ Rd )

|P (x)|

C1 max |xα |.

(2.21)

α∈ϑ(P )

Hence, by Lemma 2.9, there exist C2 ∈ ]0, +∞[ such that
(∀x ∈ Rd )

C2 max |xα |
α∈ϑ(P )

|P (x)|

C1 max |xα |.
α∈ϑ(P )

7

(2.22)

[P ]

Consequently, by Lemma 2.8, U2
ΩA (t) is finite and

is a compact set in L2 (Td ) if and only if, for every t ∈ [0, +∞[,
(2.23)

inf mA (x) > 0.

x∈Nd

By Lemma 2.10, 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) mB (x). Conversely, let α ∈ B E(B). On the one hand, there
exists ρ ∈ ]1, +∞[ such that α = ρα ∈ ∆(B). One the other hand, by Carath´eodory’s theorem [21,
Theorem 17.1], α is a convex combination of points (αj )1 j d+1 in E(B), say
d+1

d+1

λj α j ,

α =

where

{λj }1

j d+1

⊂ [0, +∞[

j=1

and

λj = 1.

(3.1)

j=1

Hence, by Young’s inequality
d+1
α

α

x
d+1

α j λj

=

λ j xα

(x )
j=1

j

max xα

1 j d+1

j=1

j

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

C1 M (x)

|P (x)|

(3.3)

C2 M (x).

Proof. Combine (2.22), Lemma 2.9, and Lemma 3.1.
For a given finite set B ⊂ Rd+ , consider the following convex problem in Rd
d

max

x∈B ◦

(3.4)

xj ,
j=1

where
B ◦ = x ∈ Rd (∀α ∈ B) α | x

(3.5)

1
8


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
(3.6)

r(B) = 1/µ(B).
One can verify that
r(B) = max ρ ρ1 ∈ conv(B) ,

(3.7)

and ν(B) is the dimension of the minimal face of the polyhedron conv(B) which contains the point
r(B)1 as a relative interior point, where 1 = (1, . . . , 1) ∈ Rd . Notice also that 0 ν(B) d − 1.
Lemma 3.3 Let B meet all the axes at a point aj uj 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) t . Then, as in the proof of Lemma 2.10,
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
volΛB (t)

tµ(B) logν(B) t.

(3.9)

Furthermore, proceeding as in the proof of [5, Theorem 2], one shows that
|ΩB (t)|

vol ΛB (t).

(3.10)

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 inf

f ∈W g∈Ln

f −g

X

ε ,

where Ln is a linear subspace in X of dimension
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 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,
[P ]

dn U2 , L2 (Td )

n−r(ϑ(A)) logν(ϑ(A))r(ϑ(A)) n,

(3.11)

ε−1/r(ϑ(A)) | log ε|ν(ϑ(A)) .

(3.12)

or equivalently,
[P ]

nε U2 , L2 (Td )

9


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 Lemma 3.3 to ϑ(A), we have
tµ logν t,

(∀t ∈ [t¯, +∞[) |K(t)|

(3.14)

where µ = µ(ϑ(A)) and ν = ν(ϑ(A)). In turn, for every m ∈ N, there exists C1 ∈ ]0, +∞[ such that

dim V (m)

C1 m1/r logν m.

(3.15)

For n ∈ N large enough, there exist m ∈ N such that
C1 m1/r logν m

n < C1 (m + 1)1/r logν (m + 1)

C2 m1/r logν m,

(3.16)

where C2 ∈ ]0, +∞[ is independent from n and m. It follows from (3.15), (3.16), and Corollary 2.6
that
[P ]

dn (U2 , L2 (Td ))

m−1

n−r logνr n.

(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+1 of X and every ρ ∈ ]0, +∞[, we have
dn (Bn+1 (ρ), X ) = ρ,

where

Bn+1 (ρ) = {f ∈ Ln+1 | f

X

ρ}.

(3.18)

Similarly to (3.15) and (3.16), for n ∈ N sufficiently large, there exists m ∈ N such that
dim V (m)

C3 m1/r logν m > n

C4 m1/r logν m,

(3.19)

where C3 ∈ ]0, +∞[ and C4 ∈ ]0, +∞[ are independent from n and m. Consider the set
U (m) = f ∈ V (m)

f
[P ]

By Lemma 2.7, U (m) ⊂ U2
[P ]

dn U2 , L2 (Td )

2

m−1 .

(3.20)

and consequently, it follows from (3.18) and (3.19) that

dn (U (m), L2 (Td ))

m−1

n−r logνr n,

(3.21)

which concludes the proof.
Remark 3.5 We have actually proven a bit more than Theorem 3.4. Namely, suppose that P (D)
[P ]
satisfies the conditions of compactness for U2 stated in Lemma 2.8 and for every n ∈ N, let m(n)
be the maximal number such that |K(m(n))| n. Then, for n sufficiently large, we have
[P ]

dn U2 , L2 (Td )

1
.
m(n)

(3.22)

10


4

Examples

First, we establish norm equivalences and, based on them, we provide some examples of
[P ]
dn (U2 , L2 (Td )) for non-degenerate and degenerate differential operators.
Theorem 4.1 Suppose that P (D) is non-degenerate and set
xα .

Q: x →

(4.1)

α∈E(A)

Then
[P ]

∀f ∈ W2

f

2
[P ]

W2

f

Dα f

2
[Q]
W2

2
2

α∈E(A)

max

Dα f

α∈E(A)

2
2.

(4.2)

Moreover, the semi-norms in (4.2) are a norm if and only if 0 ∈ A.
[P ]

Proof. Let f ∈ W2 . It is clear that

Dα f

2
2

Dα f

max
α∈E(A)

α∈E(A)

2
2.

(4.3)

Parseval’s identity and Corollary 3.2 yield
Dα f

max

2
2

α∈E(A)

|k|2α |fˆ(k)|2

max

α∈E(A)

k∈Zd

|M (k)|2 |fˆ(k)|2 .

(4.4)

k∈Zd

Let us decompose Zd into the subsets Zd (α), α ∈ E(A), such that
Zd =

Zd (α), Zd (α) ∩ Zd (α ) = ∅, α = α,

(4.5)

α∈E(A)

and
M (k) = |k α |,

k ∈ Zd (α).

(4.6)

(Such a decomposition is easily constructed). Then we have
max
α∈E(A)

Dα f

2
2

|k 2α ||fˆ(k)|2

= max
α∈E(A)

α ∈E(A) k∈Zd (α )

|k 2α | |fˆ(k)|2
α ∈E(A) k∈Zd (α )

|M (k)|2 |fˆ(k)|2 .

=
k∈Zd

11

(4.7)


Thus, we have proven the following equation
max

Dα f

α∈E(A)

2
2

|M (k)|2 |fˆ(k)|2 .

=

(4.8)

k∈Zd

Hence, by Corollary 3.2 and (2.10) we obtain
max

Dα f

2
2

f

2
[P ] .
W2

(4.9)

Dα f

2
2

f

2
[Q] .
W2

(4.10)

α∈E(A)

The relation
max
α∈E(A)

follows from the last semi-norm equivalence and the equation E(Q) = E(A). If follows from (4.2)
that the semi-norms in (4.2) are a norm if and only if 0 ∈ A.

4.1

Isotropic Sobolev classes

Let s ∈ N∗ . The isotropic Sobolev space H s is the Hilbert space of functions f ∈ L2 (Td ) equipped
with the norm · H s which is defined by
f

2

Hs

=

f

2
2

+

f (α) 22 .

(4.11)

xα ,

(4.12)

|α|=s

Consider
xα =

P: x→1+

α∈A

|α|=s

where A = 0 ∪ {α : |α| = s}. If s is even, the differential operator P (D) is non-degenerate and
consequently, by Theorem 4.1 the norm f H s is equivalent to one of the norms in (4.2) with
E(A) = 0 ∪ α = suj 1 j d and
d

xsj .

Q(x) = 1 +

(4.13)

j=1

Moreover, we have r(A) = s and ν(a) = 0, and therefore, for the unit ball U s in H s from Theorem 3.4 we again retrieve the well known result
dn U s , L2 (Td )

n−s .

(4.14)

12


4.2

Anisotropic Sobolev classes

For β = (β1 , . . . , βd ) ∈ N∗d , the anisotropic Sobolev space H β is the Hilbert space of functions
f ∈ L2 equipped with the norm f H β which is defined by
d

f

2
Hβ

= f

2
2

j)

f (βj u

+

2
2.

(4.15)

j=1

Consider
d
β

xα ,

xj j =

P: x→1+
j=1

(4.16)

α∈A

where A = {0} ∪ {α = βj uj | j = 1, . . . , d}. If the coordinates of β are even, the differential
operator P (D) is non-degenerate and consequently, by Theorem 4.1 the norm f H β is equivalent
to one of the norms in (4.2) with E(A) = A and
(4.17)

Q(x) = P (x).
We have


−1

d

r = r(A) = 

(4.18)

1/βj 
j=1

and ν(A) = 0, and therefore, for the unit ball U r in H β from Theorem 3.4 we again retrieve the

well-known result
n−r .

dn U β , L2 (Td )

4.3

(4.19)

Classes of functions with a bounded mixed derivative

Let α = (α1 , . . . , αd ) ∈ Nd with 0 < α1 = · · · = αν+1 < αν+2 = · · · = αd for some 0 ν d − 1.
For a set e ⊂ {1, . . . , d}, let the vector α(e) ∈ Nd be defined by α(e)j = αj if j ∈ e, and α(e)j = 0
otherwise (in particular, α(∅) = 0 and α({1, . . . , d}) = α). The space W2α is the Hilbert space of
functions f ∈ L2 equipped with the norm · W2α which is defined by
f

2
W2α

f (α(e)) 22 .

=

(4.20)

e⊂{1,...,d}

Consider
xα(e) =

P: x→
e⊂{1,...,d}

xα ,

(4.21)

α∈A

13


where A = α(e) e ⊂ {1, . . . , d} . If the coordinates of α are even, the differential operator P (D)
is non-degenerate and consequently, by Theorem 4.1 the norm · W2α is equivalent to one of the
norms in (4.2) with E(A) = A and
(4.22)

Q(x) = P (x).

We have r(A) = α1 and ν(A) = ν, and therefore, for the unit ball U2α in W2α from Theorem 3.4 we
again retrieve the result proven in [1], namely that for n sufficiently large
n−α1 logνα1 n.

dn U2α , L2 (Td )

(4.23)

In the particular case α = r1, we have
n−r log(d−1)r n.

dn U2r1 , L2 (Td )

4.4

(4.24)

Classes of functions with several bounded mixed derivatives

Suppose that (2.20) is satisfied and that 0 ∈ A. The space W2A is the Hilbert space of functions
f ∈ L2 (Td ) equipped with the norm · W A which is defined by
2

f

2
W2A

f (α) 22 .

=

(4.25)

α∈A

Notice that spaces H s , H r , and W2α are a particular cases of W2A . Now consider
xα .

P: x→

(4.26)

α∈A

If the coordinates of every α ∈ E(A) are even, the differential operator P (D) is non-degenerate
and consequently, by Theorem 4.1, the norm · W A is equivalent to one of the norms in (4.2). If
2
r = r(E(A)) and ν = ν(E(A)), for the unit ball U2A in W2A from Theorem 3.4 we again retrieve
the result proven in [5], namely that for n sufficiently large
n−r logνr n.

dn U2A , L2 (Td )

4.5

(4.27)

Classes of functions given by a differential operator
[P ]

We give two examples of space W2
Consider the following polynomials

with non-degenerate differential operator P (D) for d = 2.

P1 : x → = 8x41 − 4x31 − 3x31 x2 − 2x21 x2 − 4x1 x2 + 6x22 − 4x1 − 3x2 + 13,
(4.28)
P2 : x → =

6x61

+

x41 x22

−

6x51

−

x31 x22

+

5x42

−

14

4x32

+ 3.


We have
A1 = {(4, 0), (3, 0), (2, 1), (2, 0), (1, 1), (0, 2), (1, 0), (0, 1), (0, 0)},
E(A1 ) = {(4, 0), (0, 2), (0, 0)},

(4.29)
A2 = {(6, 0), (4, 2), (5, 0), (3, 2), (0, 4), (0, 3), (0, 0)},
E(A2 ) = {(6, 0), (4, 2), (0, 4), (0, 0)}.
It is easy to verify that for i = 1, 2, Pj (D) is non-degenerate, (2.20) holds, and 0 ∈ Ai . Moreover, r(E(A1 )) = 4/3, ν(E(A1 )) = 0 and r(E(A2 )) = 8/3, ν(E(A2 )) = 1. From Theorem 3.4 we
have
dn U [P1 ] , L2 (T2 )

n−4/3 ,

(4.30)

dn U [P2 ] , L2 (T2 )

n−8/3 log8/3 n.

(4.31)

and

Let us give an example of degenerate differential operator. For
P3 (x) = x41 − 2x31 x2 + x21 x22 + x21 + x22 + 1,

(4.32)

the differential operator P3 (D) is degenerate, although P3 (x)
1 for every x ∈ R2 , and U [P3 ] is
2
[P
]
a compact set in L2 (T ). Therefore, we cannot compute dn (U 3 , L2 (T2 )) by using Theorem 3.4.

However, by a direct computation we get |K(t)| t1/2 log t. Hence, by (3.22) we have
dn U [P3 ] , L2 (T2 )

4.6

n−2 log2 n.

(4.33)

A conjecture
[P ]

Suppose that U2

(i) For every t

is compact in L2 (Td ). In view of Lemma 2.8, this is equivalent to the conditions:
0, K(t) is finite.

(ii) τ > 0.
As mentioned in (3.22), for n ∈ N sufficiently large, if m(n) is the maximal number such that
|K(m(n))| n, then
[P ]

dn U2 , L2 (Td )

1
.
m(n)

(4.34)
[P ]

This means that the problem of computing the asymptotic order of dn (U2 , L2 (Td )) is equivalent
to the problem of computing that of |K(t)| when t → +∞. Let us formulate it as the following
conjecture.
15


Conjecture 4.2 Suppose that, for every t ∈ [0, +∞[, K(t) is finite (the condition τ > 0 is not
essential). Then there exist integers α, β, and ν such that 0 < α β 0 ν < d, and, for t large
enough,
|K(t)|

tα/β logν t.

(4.35)

In view of (3.14), we know that the conjecture is true when P satisfies conditions (2.7) and
(2.20).
Acknowledgment. Dinh Dung’s research work is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under Grant No. 102.01-2014.02, and a part
of it was done when Dinh Dung was working as a research professor at and Patrick Combettes was
visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). Both authors thank
the VIASM for providing fruitful research environment and working condition.

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17