DSpace at VNU: Multivariate approximation by translates of the Korobov function on Smolyak grids

Journal of Complexity 29 (2013) 424–437

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Journal of Complexity
journal homepage: www.elsevier.com/locate/jco

Multivariate approximation by translates of the
Korobov function on Smolyak grids
˜ a,∗ , Charles A. Micchelli b
Dinh Dung
a

Vietnam National University, Information Technology Institute, 144 Xuan Thuy, Hanoi, Viet Nam

b

Department of Mathematics and Statistics, SUNY Albany, Albany, 12222, USA

article

abstract

info

For a set W ⊂ Lp (Td ), 1 < p < ∞, of multivariate periodic
functions on the torus Td and a given function ϕ ∈ Lp (Td ), we
study the approximation in the Lp (Td )-norm of functions f ∈ W by
arbitrary linear combinations of n translates of ϕ . For W = Upr (Td )
and ϕ = κr ,d , we prove upper bounds of the worst case error of
this approximation where Upr (Td ) is the unit ball in the Korobov

Article history:
Received 17 January 2013
Accepted 6 June 2013
Available online 18 June 2013
Keywords:
Korobov space
Translates of the Korobov function
Reproducing kernel Hilbert space
Smolyak grids

space Kpr (Td ) and κr ,d is the associated Korobov function. To obtain
the upper bounds, we construct approximation methods based on
sparse Smolyak grids. The case p = 2, r > 1/2, is especially
important since K2r (Td ) is a reproducing kernel Hilbert space,
whose reproducing kernel is a translation kernel determined by
κr ,d . We also provide lower bounds of the optimal approximation
on the best choice of ϕ .
© 2013 Elsevier Inc. All rights reserved.

1. Introduction
The d-dimensional torus denoted by Td is the cross product of d copies of the interval [0, 2π ] with
the identification of the end points. When d = 1, we merely denote the d-torus by T. Functions on
Td are identified with functions on Rd which are 2π periodic in each variable. We shall denote by
Lp (Td ), 1 ≤ p < ∞, the space of integrable functions on Td equipped with the norm
−d/p

∥f ∥p := (2π )



Td

1/p

|f (x)| dx
p

.

(1.1)

We will consider only real valued functions on Td . However, all the results in this paper are true for
the complex setting. Also, we will use the Fourier series of a real valued function in complex form and

∗

Corresponding author.
˜
E-mail address: (D. Dung).

0885-064X/$ – see front matter © 2013 Elsevier Inc. All rights reserved.
/>

D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

425

somewhere estimate its Lp (Td )-norm via the Lp (Td )-norm of its complex valued components which
is defined as in (1.1).


For vectors x := (xl : l ∈ N [d]) and y := (yl : l ∈ N [d]) in Td we use (x, y) :=
l∈N [d] xl yl for the
inner product of x with y. Here, we use the notation N [m] for the set {1, 2, . . . , m} and later we will
use Z [m] for the set {0, 1, . . . , m − 1}. Also, for notational convenience we allow N [0] and Z [0] to stand
for the empty set. Given any integrable function f on Td and any lattice vector j = (jl : l ∈ N [d]) ∈ Zd ,
we let fˆ (j) denote the j-th Fourier coefficient of f defined by
fˆ (j) := (2π )−d


Td

f (x) χ−j (x) dx,

where we define the exponential function χj at x ∈ Td to be χj (x) = ei(j,x) . Frequently, we use the
superscript notation Bd to denote the cross product of a given set B.
The convolution of two functions f1 and f2 on Td , denoted by f1 ∗ f2 , is defined at x ∈ Td by equation

(f1 ∗ f2 )(x) := (2π )−d


Td

f1 (x) f2 (x − y) dy,

whenever the integrand is in L1 (Td ). We are interested in approximations of functions from the
Korobov space Kpr (Td ) by arbitrary linear combinations of n arbitrary shifts of the Korobov function

κr ,d defined below. The case p = 2 and r > 1/2 is especially important, since K2r (Td ) is a reproducing
kernel Hilbert space.

In order to formulate the setting for our problem, we establish some necessary definitions and
notation. For a given r > 0 and a lattice vector j := (jl : l ∈ N [d]) ∈ Zd we define the scalar λj by the
equation


λj :=

λjl ,

l∈N [d]

where for j ∈ Z,

λj :=



|j|r ,
1,

j ∈ Z \ {0},
otherwise.

Definition 1.1. The Korobov function κr ,d is defined at x ∈ Td by the equation

κr ,d (x) :=



1

λ−
χj (x)
j

j∈Zd

and the corresponding Korobov space is
Kpr (Td ) := {f : f = κr ,d ∗ g , g ∈ Lp (Td )}
with norm

∥f ∥Kpr (Td ) := ∥g ∥p .
Remark 1.2. The univariate Korobov function κr ,1 shall always be denoted simply by κr and therefore
κr ,d has at x = (xl : l ∈ N [d]) the alternate tensor product representation

κr ,d (x) =



κr (xl ).

l∈N [d]

Remark 1.3. For 1 ≤ p ≤ ∞ and r > 1/p, we have the embedding Kpr (Td ) ↩→ C (Td ), i.e., we can

consider Kpr (Td ) as a subset of C (Td ). Indeed, for d = 1, it follows from the embeddings
−1/p
Kpr (T) ↩→ Brp,∞ (T) ↩→ Br∞,∞
(T) ↩→ C (T),

where Brp,∞ (T) is the Nikol’skii–Besov space. See the proof of the embedding Kpr (T) ↩→ Brp,∞ (T) in

[26, Theorem I.3.1, Corollary 2 of Theorem I.3.4, (I.3.19)]. Corresponding relations for Kpr (Td ) can be
found in [26, III.3].


426

D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

Remark 1.4. Since κˆ r ,d (j) ̸= 0 for any j ∈ Zd it readily follows that ∥ · ∥Kpr (Td ) is a norm. Moreover, we
point out that the univariate Korobov function is related to the one-periodic extension of Bernoulli
polynomials. Specifically, if we denote the one-periodic extension of the Bernoulli polynomial as B¯ n
then for t ∈ T, we have that
B¯ 2m (t ) =

2m!
(1 − κ2m (2π t )).
(2π i)2m

When p = 2 and r > 1/2 the kernel K defined at x and y in Td as K (x, y) := κ2r ,d (x − y) is the
reproducing kernel for the Hilbert space K2r (Td ). This means, for every function f ∈ K2r (Td ) and x ∈ Td ,
we have that
f (x) = (f , K (·, x))K r (Td ) ,
2

where (·, ·)K r (Td ) denotes the inner product on the Hilbert space K2r (Td ). For a definitive treatment of
2
reproducing kernels, see, for example, [1].
Korobov spaces Kpr (Td ) are important for the study of smooth multivariate periodic functions. They
are sometimes called periodic Sobolev spaces of dominating mixed smoothness and are useful for the

study of multivariate approximation and integration; see, for example, the books [26,21].
The linear span of the set of functions {κr ,d (· − y) : y ∈ Td } is dense in the Hilbert space K2r (Td ).
In the language of Machine Learning, this means that the reproducing kernel for the Hilbert space
is universal. The concept of universal reproducing kernel has significant statistical consequences in
Machine Learning. In [20], a complete characterization of universal kernels is given in terms of its
feature space representation. However, no information is provided about the degree of approximation.
This unresolved question is the main motivation of this paper and we begin to address it in the context
of the Korobov space K2r (Td ). Specifically, we study approximations in the L2 (Td ) norm of functions
in K2r (Td ) when r > 1/2 by linear combinations of n translates of the reproducing kernel, namely,
κr ,d (· − yl ), yl ∈ Td , l ∈ N [n]. We shall also study this problem in the space Lp (Td ), 1 < p < ∞
for r > 1, because the linear span of the set of functions {κr ,d (· − y) : y ∈ Td }, is also dense in the
Korobov space Kpr (Td ).
For our purpose in this paper, the following concept is essential. Let W ⊂ Lp (Td ) and ϕ ∈ Lp (Td ) be
a given function. We are interested in the approximation in the Lp (Td )-norm of all functions f ∈ W by
arbitrary linear combinations of n translates of the function ϕ , that is, the functions ϕ(· − yl ), yl ∈ Td
and measure the error in terms of the quantity
















Mn (W, ϕ)p := sup inf f −
cl ϕ(· − yl ) : cl ∈ R, yl ∈ Td : f ∈ W .
 



l∈N [n]
p

The aim of the present paper is to investigate the convergence rate, when n → ∞, of Mn (Upr (Td ), κr ,d )p

where Upr (Td ) is the unit ball in Kpr (Td ). We shall also obtain a lower bound for the convergence rate
as n → ∞ of the quantity
Mn (U2r (Td ))2 := inf{Mn (U2r (Td ), ϕ)2 : ϕ ∈ L2 (Td )}
which gives information about the best choice of ϕ .
The paper [17] is directly related to the questions we address in this paper, and we rely upon
some results from [17] to obtain lower bounds for the quantity of Mn (Upr (Td ))p . Related material

can be found in [16,18]. Here, we shall provide upper bounds for Mn (Upr (Td ), κr ,d )p for 1 < p <

∞, r > 1, p ̸= 2 and r > 1/2 for p = 2, as well as lower bounds for Mn (U2r (Td ))2 . To obtain

our upper bound, we construct approximation methods based on sparse Smolyak grids. Although
these grids have a significantly smaller number of points than the corresponding tensor product grids,
the approximation error remains the same. Smolyak grids [25] and the related notion of hyperbolic
cross introduced by Babenko [2], are useful for high dimensional approximation problems; see,


D. D˜

ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

427

for example, [13,15]. For recent results on approximations and sampling on Smolyak grids see, for
example, [4,12,22,24].
To describe the main results of our paper, we recall the following notation. Given two sequences
{al : l ∈ N} and {bl : l ∈ N}, we write al ≪ bl provided there is a positive constant c such that for all
l ∈ N, we have that al ≤ cbl . When we say that al ≍ bl we mean that both al ≪ bl and bl ≪ al hold.
The main theorem of this paper is the following fact.
Theorem 1.5. If 1 < p < ∞, p ̸= 2, r > 1 or p = 2, r > 1/2, then
Mn (Upr (Td ), κr ,d )p ≪ n−r (log n)r (d−1) ,

(1.2)

while for r > 1/2, we have that
n−r (log n)r (d−2) ≪ Mn (U2r (Td ))2 ≪ n−r (log n)r (d−1) .

(1.3)

This paper is organized in the following manner. In Section 2, we give the necessary background
from Fourier analysis, construct methods for approximation of functions from the univariate Korobov
space Kpr (T) by linear combinations of translates of the Korobov function κr and prove an upper
bound for the approximation error. In Section 3, we extend the method of approximation developed
in Section 2 to the multivariate case and provide an upper bound for the approximation error. Finally,
in Section 4, we provide the proof of Theorem 1.5.
2. Univariate approximation
We begin this section by introducing the m-th Dirichlet function, denoted by Dm , and defined at
t ∈ T as
Dm (t ) :=



χl (t ) =

sin((m + 1/2)t )

|l|∈Z [m+1]

sin(t /2)

and corresponding m-th Fourier projection of f ∈ Lp (T), denoted by Sm (f ), and given as Sm (f ) :=
Dm ∗ f . The following lemma is a basic result.
Lemma 2.1. If 1 < p < ∞ and r > 0, then there exists a positive constant c such that for any
m ∈ N, f ∈ Kpr (T) and g ∈ Lp (Td ) we have

∥f − Sm (f )∥p ≤ c m−r ∥f ∥Kpr (T)

(2.1)

∥Sm (g )∥p ≤ c ∥g ∥p .

(2.2)

and

Remark 2.2. The proof of inequality (2.1) is easily verified while inequality (2.2) is given in Theorem
1, p. 137, of [3].
The main purpose of this section is to introduce a linear operator, denoted as Qm , which is constructed
from the m-th Fourier projection and prescribed translate of the Korobov function κr , needed for the

proof of Theorem 1.5. Specifically, for f ∈ Kpr (T) we define Qm (f ), where f is represented as f = κr ∗ g
for g ∈ Lp (T), to be
Qm (f ) := (2m + 1)−1


l∈Z [2m+1]

Sm ( g )



2π l
2m + 1




κr · −

2π l
2m + 1



.

Our main observation in this section is to establish that the operator Qm enjoys the same error
bound which is valid for Sm . We state this fact in the theorem below.

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D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

Theorem 2.3. If 1 < p < ∞ and r > 1, then there is a positive constant c such that for all m ∈ N and
f ∈ Kpr (T), we have that

∥f − Qm (f )∥p ≤ c m−r ∥f ∥Kpr (T)
and

∥Qm (f )∥p ≤ c ∥f ∥Kpr (T) .

(2.3)

The idea in the proof of Theorem 2.3 is to use Lemma 2.1 and study the function defined as
Fm := Qm (f ) − Sm (f ).
Clearly, the triangular inequality tells us that

∥f − Qm (f )∥p ≤ ∥f − Sm (f )∥p + ∥Fm ∥p .
Therefore, the proof of Theorem 2.3 hinges on obtaining an estimate for the Lp (T)-norm of the function
Fm . To this end, we recall some useful facts about trigonometric polynomials and Fourier series.
We denote by Tm the space of univariate trigonometric polynomials of degree at most m. That is,
we have that Tm := span{χl : |l| ∈ Z [m + 1]}. We require a readily verified quadrature formula which
says, for any f ∈ Ts , that
fˆ (0) =



1 

s l∈Z [s]

f

2π l



s

.

Using these facts leads to a formula from [9] which we state in the next lemma.
Lemma 2.4. If m, n, s ∈ N, such that m + n < s then for any f1 ∈ Tm and f2 ∈ Tn there holds the
following identity
2π l





f 1 ∗ f 2 = s −1

f1

 

s

l∈Z [s]

·−

f2

2π l



s

.

Lemma 2.4 is especially useful to us as it gives a convenient representation for the function Fm . In
fact, it readily follows, for f = κr ∗ g, that
Fm =

1



2m + 1 l∈Z [m+1]

Sm (g )

2π l





2m + 1



θm · −

2π l



2m + 1

,

(2.4)

where the function θm is defined as θm := κr − Sm (κr ). The proof of formula (2.4) may be based on the
equation
Sm (κr ∗ g ) =

1



2m + 1 l∈Z [2m+1]

Sm ( g )

2π l



2m + 1





(Sm κr ) · −

2π l
2m + 1



.

(2.5)

For the confirmation of (2.5) we use the fact that Sm is a projection onto Tm , so that
Sm (κr ∗ g ) = Sm (κr ) ∗ Sm (g ).
Now, we use Lemma 2.4 with f1 = Sm (g ), f2 = Sm (κr ) and s = 2m + 1 to confirm both (2.4) and (2.5).
The next step in our analysis makes use of Eq. (2.4) to get the desired upper bound for ∥Fm ∥p .
For this purpose, we need to appeal to two well-known facts attributed to Marcinkiewicz; see, for
example, [28]. To describe these results, we introduce the following notation. For any subset A of Z
and a vector a := (al : l ∈ A) and 1 ≤ p ≤ ∞ we define the lp (A)-norm of a by

∥a∥p,A :=



 



1/p
p

|al |

,

1 ≤ p < ∞,

l∈A

sup{|al | : l ∈ A},

p = ∞.


D. D˜
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429

Also, we introduce the mapping Wm : Tm → R2m defined at f ∈ Tm as
Wm (f ) =

2π l

 
f



2m + 1



: l ∈ Z [2m + 1] .

Lemma 2.5. If 1 < p < ∞, then there exist positive constants c and c ′ such that for any m ∈ N and
f ∈ Tm there hold the inequalities
c ∥f ∥p ≤ (2m + 1)−1/p ∥Wm (f )∥p,Z [2m+1] ≤ c ′ ∥f ∥p .
Remark 2.6. Lemma 2.5 appears in [28, p. 28, Volume II] as Theorem 7.5. We also remark in the case
that p = 2 the constants appearing in Lemma 2.5 are both one. Indeed, we have for any f ∈ Tm the
equation

(2m + 1)−1/2 ∥Wm (f )∥2,Z [2m+1] = ∥f ∥2 .

(2.6)

Lemma 2.7. If 1 < p < ∞ and there is a positive constant c such that for any vector a = (aj : j ∈ Z)
which satisfies for some positive constant A and any s ∈ Z, the condition
s+1 −1
±2

|aj − aj−1 | ≤ A,

j=±2s

and also ∥a∥∞,Z ≤ A, then for any functions f ∈ Lp (T), the function
Ma (f ) :=



aj fˆ (j)χj

j∈Z

belongs to Lp (T) and, moreover, we have that

∥Ma (f )∥p ≤ c A∥f ∥p .
Remark 2.8. Lemma 2.7 appears in [28, p. 232, Volume II] as Theorem 4.14 and is sometimes referred
as the Marcinkiewicz multiplier theorem.
We are now ready to prove Theorem 2.3.
Proof. For each j ∈ Z we define
bj :=

1



2m + 1 l∈Z [2m+1]

Sm (g )



2π l
2m + 1


e

2π ilj
− 2m
+1

(2.7)

and observe from Eq. (2.4) that
Fm =



bj |j|−r χj ,

(2.8)

j∈Z¯ [m]

where Z¯ [m] := {j ∈ Z : |j| > m}. Moreover, according to Eq. (2.7), we have for every j ∈ Z that
bj+2m+1 = bj .

(2.9)

2π l

Notice that Sm (g ) 2m
: l ∈ Z [2m + 1] is the discrete Fourier transform of bj : j ∈ Z [2m + 1]
+1
and therefore, we get for all l ∈ Z [2m + 1] that



Sm (g )





2π l
2m + 1




=




j∈Z [2m+1]

2π ilj

bj e 2m+1 .






430

D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

On the other hand, by definition we have
Sm (g )

2π l





2m + 1

2π ilj



=

gˆ (j)e 2m+1 .

|j|∈Z [m+1]

Hence,


bj =

gˆ (j),
gˆ (j − 2m − 1),

0 ≤ j ≤ m,
m + 1 ≤ j ≤ 2m.

(2.10)

We decompose the set Z¯ [m] as a disjoint union of finite sets each containing 2m + 1 integers.
Specifically, for each j ∈ Z we define the set
Im,j := {l : l ∈ N, j(2m + 1) − m ≤ l ≤ j(2m + 1) + m}
and observe that Z¯ [m] is a disjoint union of these sets. Therefore, using Eqs. (2.8) and (2.9) we can
compute
Fm =

 

|l|−r bl χl =



Gm,j χj(2m+1)−m

j∈N

j∈Z¯ [0] l∈Im,j

where



Gm,j :=

|l + j(2m + 1) − m|−r bl χl .

l∈Z [2m+1]

Hence, by the triangle inequality we conclude that

∥Fm ∥p ≤



∥Gm,j ∥p .

(2.11)

j∈Z¯ [0]

By using (2.9) and (2.10) we split the function Gm,j into two functions as follows
−
Gm,j = G+
m,j + χ2m+1 Gm,j ,

(2.12)

where
G+
m,j :=

m


|l + j(2m + 1) − m|−r gˆ (l)χl ,

l =0

G−
m,j :=

−1


|l + (j + 1)(2m + 1) − m|−r gˆ (l)χl .

l=−m

−
Now, we shall use Lemma 2.7 to estimate ∥G+
m,j ∥p and ∥Gm,j ∥p . For this purpose, we define for each
j ∈ N the components of a vector a = (al : l ∈ Z) as


al :=

|l + j(2m + 1) − m|−r ,
0,

l ∈ Z [m + 1],
otherwise,

we may conclude that Lemma 2.7 is applicable when a value of A is specified. For simplicity let us
consider the case j > 0, the other case can be treated in a similar way. For a fixed value of j and m,
we observe that the components of the vector a are decreasing with regard to |l| and moreover, it is
readily seen that a0 ≤ |jm|−r . Therefore, we may choose A = |jm|−r and apply Lemma 2.7 to conclude
−r
r
that ∥G+
m,j ∥p ≤ ρ|jm| ∥f ∥Kp (T) where ρ is a constant which is independent of j and m. The same
inequality can be obtained for ∥G−
m,j ∥p . Consequently, by (2.12) and the triangle inequality we have

∥Gm,j ∥p ≤ 2ρ|jm|−r ∥f ∥Kpr (T) .


D. D˜
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431

We combine this inequality with inequalities (2.11) and r > 1 to conclude, that there is positive
constant c, independent of m, such that

∥Fm ∥p ≤ c m−r ∥f ∥Kpr (T) .

We now turn our attention to the proof of inequality (2.3). Since κr is continuous on T, the proof of
(2.3) is transparent. Indeed, we successively use the Hölder inequality, the upper bound in Lemma 2.5
applied to the function Sm (g ) and the inequality (2.2) to obtain the desired result.
Remark 2.9. The restrictions 1 < p < ∞ and r > 1 in Theorem 2.3 are necessary for applying
the Marcinkiewicz multiplier theorem (Lemma 2.7) and processing the upper bound of ∥Fm ∥p . It is
interesting to consider this theorem for the case 0 < p ≤ ∞ and r > 0. However, this would go
beyond the scope of this paper.
We end this section by providing an improvement of Theorem 2.3 when p = 2.
Theorem 2.10. If r > 1/2, then there is a positive constant c such that for all m ∈ N and f ∈ K2r (T), we
have that

∥f − Qm (f )∥2 ≤ c m−r ∥f ∥K2r (T) .
Proof. This proof parallels that given for Theorem 2.3 but, in fact, is simpler. From the definition of
the function Fm we conclude that

∥Fm ∥22 =



|j|−2r |bj |2 =

 

|k|−2r |bk |2 .

j∈N k∈Im,j

j∈Z¯ [m]

We now use Eq. (2.9) to obtain that

∥Fm ∥22 =





|k + j(2m + 1) − m|−2r |bk |2

j∈Z¯ [m] k∈Z [2m+1]

≤ m−2r





|j|−2r

j∈Z¯ [0]

|bk |2 ≪ m−2r

k∈Z [2m+1]



|bk |2 .

k∈Z [2m+1]

Hence, appealing to Parseval’s
for discrete Fourier transforms applied to the pair


 identity
(bk : k ∈ Z [2m + 1]) and Sm (g ) 2m2π+l 1 : l ∈ Z [2m + 1] , and (2.6) we finally get that

∥Fm ∥22 ≪ m−2r ∥g ∥22 = m−2r ∥f ∥2Kpr (T)
which completes the proof.
3. Multivariate approximation
Our goal in this section is to make use of our univariate operators and create multivariate
operators from them which economize on the number of translates of Kr ,d used to approximate while
maintaining as high an order of approximation. To this end, we apply, in the present context, the
techniques of Boolean sum approximation. These ideas go back to Gordon [14] for surface design and
also Delvos and Posdorf [6] in the 1970s. Later, they appeared, for example, in [27,19,5] and because
of their importance continue to attract interest and applications. We also employ hyperbolic cross
and sparse grid techniques which date back to Babenko [2] and Smolyak [25] to construct methods of
multivariate approximation. These techniques then were widely used in numerous papers of Soviet
mathematicians (see surveys in [8,10,26] and bibliography there) and have been developed in [11–13,
22–24] for hyperbolic cross approximations and sparse grid sampling recoveries. Our construction of
approximation methods is a modification of those given in [10,12] (cf. [22–24]).


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ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

For m ∈ Zd+ , let the multivariate operator Qm in Kpr (Td ) be defined by

d


Qm :=

Qmj ,

(3.1)

j =1

where the univariate operator Qmj is applied to the univariate function f by considering f as a function

of variable xj with the other variables held fixed, Zd+ := {k ∈ Zd : kj ≥ 0, j ∈ N [d]} and kj denotes
the jth coordinate of k.
Set Zd−1 := {k ∈ Zd : kj ≥ −1, j ∈ N [d]}. For k ∈ Z−1 , we define the univariate operator Tk in
r
Kp (Td ) by
Tk := I − Q2k ,

k ≥ 0,

T−1 := I ,

where I is the identity operator. If k ∈ Zd−1 , we define the mixed operator Tk in Kpr (Td ) in the manner
of the definition of (3.1) as
Tk :=

d


Tki .

i=1

Set |k| :=



j∈N [d]

|kj | for k ∈ Zd−1 .

Lemma 3.1. Let 1 < p < ∞, p ̸= 2, r > 1 or p = 2, r > 1/2. Then we have for any f ∈ Kpr (Td ) and
k ∈ Zd−1 ,

∥Tk (f )∥p ≤ C 2−r |k| ∥f ∥Kpr (Td )
with some constant C independent of f and k.
Proof. We prove the lemma by induction on d. For d = 1 it follows from Theorems 2.3 and 2.10.
Assume that the lemma is true for d − 1. Set x′ := {xj : j ∈ N [d − 1]} and x = (x′ , xd ) for x ∈ Rd .
We temporarily denote by ∥f ∥p,x′ and ∥f ∥Kpr (Td−1 ),x′ or ∥f ∥p,xd and ∥f ∥Kpr (T),xd the norms applied to
the function f by considering f as a function of variable x′ or xd with the other variable held fixed,
respectively. For k = (k′ , kd ) ∈ Zd−1 , we get by Theorems 2.3 and 2.10 and the induction assumption
′

∥Tk (f )∥p = ∥ ∥Tk′ Tkd (f ) ∥p,x′ ∥p,xd ≪ ∥2−r |k | ∥Tkd (f ) ∥Kpr (Td−1 ),x′ ∥p,xd
′

′

= 2−r |k | ∥ ∥Tkd (f ) ∥p,xd ∥Kpr (Td−1 ),x′ ≪ 2−r |k | ∥2−rkd ∥f ∥Kpr (T),xd ∥Kpr (Td−1 ),x′
= 2−r |k| ∥f ∥Kpr (Td ) .
Let the univariate operator qk be defined for k ∈ Z+ , by
qk := Q2k − Q2k−1 ,

k > 0,

q0 := Q1 ,

and in the manner of the definition of (3.1), the multivariate operator qk for k ∈ Zd+ , by
qk :=

d


qkj .

j =1

For k ∈ Zd+ , we write k → ∞ if kj → ∞ for each j ∈ N [d].
Theorem 3.2. Let 1 < p < ∞, p ̸= 2, r > 1 or p = 2, r > 1/2. Then every f ∈ Kpr (Td ) can be
represented as the series
f =


k∈Zd+

qk (f )

(3.2)

D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

433

converging in the Lp -norm, and we have for k ∈ Zd+ ,

∥qk (f )∥p ≤ C 2−r |k| ∥f ∥Kpr (Td )

(3.3)

with some constant C independent of f and k.
Proof. Let f ∈ Kpr (Td ). In a way similar to the proof of Lemma 3.1, we can show that

∥f − Q2k (f )∥p ≪ max 2−rkj ∥f ∥Kpr (Td ) ,
j∈N [d]

and therefore,

∥f − Q2k (f )∥p → 0,

k → ∞,

where 2k = (2kj : j ∈ N [d]). On the other hand,



Q2k =

qs (f ).

sj ≤kj , j∈N [d]

This proves (3.2). To prove (3.3) we notice that from the definition it follows that
qk =



(−1)|e| Tke ,

e⊂N [d]

where ke is defined by kei = ki if i ∈ e, and kei = ki − 1 if i ̸∈ e. Hence, by Lemma 3.1

∥qk (f )∥p ≤



∥Tke (f )∥p ≪



e
2−r |k | ∥f ∥Kpr (Td ) ≪ 2−r |k| ∥f ∥Kpr (Td ) .

e⊂N [d]

e⊂N [d]

For approximation of f ∈ Kpr (Td ), we introduce the linear operator Pm , m ∈ Z+ , by
Pm (f ) :=



qk (f ).

(3.4)

|k|≤m

In the next section, we will see that Pm defines a method of approximation by a certain linear
combination of translates of the Korobov function κr ,d on a Smolyak grid. The following theorem gives
an upper bound for the error of the approximation of functions f ∈ Kpr (Td ) by the operator Pm .
Theorem 3.3. Let 1 < p < ∞, p ̸= 2, r > 1 or p = 2, r > 1/2. Then, we have for every m ∈ Z+ and
f ∈ Kpr (Td ),

∥f − Pm (f )∥p ≤ C 2−rm md−1 ∥f ∥Kpr (Td )
with some constant C independent of f and m.
Proof. From Theorem 3.2 we deduce that








∥f − Pm (f )∥p = 

qk (f ) ≤
∥qk (f )∥p
|k|>m

|k|>m
p


≪
2−r |k| ∥f ∥Kpr (Td ) ≪ ∥f ∥Kpr (Td )
2−r |k|
|k|>m
−rm

≪2

|k|>m
d−1

m

∥f ∥Kpr (Td ) .

4. Convergence rate and optimality
We choose a positive integer m ∈ N, a lattice vector k ∈ Zd+ with |k| ≤ m and another lattice
vector s = (sj : j ∈ N [d]) ∈ ⊗j∈N [d] Z [2kj +1 + 1] to define the vector yk,s =



2π sj

k +1
2 j +1


: j ∈ N [d] . The


434

D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

Smolyak grid on Td consists of all such vectors and is given as
Gd (m) := {yk,s : |k| ≤ m, s ∈ ⊗j∈N [d] Z [2kj +1 + 1]}.
A simple computation confirms, for m → ∞ that

|Gd (m)| =

 

(2kj +1 + 1) ≍ 2d md−1 ,

|k|≤m j∈N [d]

so, Gd (m) is a sparse subset of a full grid of cardinality 2dm . Moreover, by the definition of the linear
operator Pm given in Eq. (3.4) we see that the range of Pm is contained in the subspace
span{κr ,d (· − y) : y ∈ Gd (m)}.
In other words, Pm defines a multivariate method of approximation by translates of the Korobov
function κr ,d on the sparse Smolyak grid Gd (m). An upper bound for the error of this approximation
of functions from Kpr (Td ) is given in Theorem 3.3.

Now, we are ready to prove the next theorem, thereby establishing inequality (1.2) in Theorem 1.5.
Theorem 4.1. If 1 < p < ∞, p ̸= 2, r > 1 or p = 2, r > 1/2, then
Mn (Upr , κr ,d )p ≪ n−r (log n)r (d−1) .
Proof. If n ∈ N and m is the largest positive integer such that |Gd (m)| ≤ n, then n ≍ 2m md−1 and by
Theorem 3.3 we have that
Mn (Upr (Td ), κr ,d ) ≤ sup{∥f − Pm (f )∥Lp (Td ) : f ∈ Upr (Td )} ≪ 2−rm md−1 ≍ n−r (log n)r (d−1) .
Next, we prepare for the proof of the lower bound of (1.3) in Theorem 1.5. To this end, let Pq (Rl ) be
the set of algebraic polynomials of total degree at most q on Rl , and Em the subset of Rm of all vectors
t = (tj : j ∈ N [m]) with components in absolute value one. That is, for every j ∈ N [m] we demand
that |tj | = 1. We choose a polynomial vector field p : Rl → Rm such that each component of the
vector field p is in Pq (Rl ). Corresponding to this polynomial vector field, we introduce the polynomial
manifold in Rm defined as Mm,l,q := p(Rl ). That is, we have that

Mm,l,q := {(pj (u) : j ∈ N [m]) : pj ∈ Pq (Rl ), j ∈ N [m], u ∈ Rl }.
We denote the euclidean norm of a vector x in Rm as ∥x∥2 . For a proof of the following lemma see [17].
Lemma 4.2 (V. Maiorov). If m, l, q ∈ N satisfy the inequality l log(
t ∈ Em and a positive constant c such that

4emq
l

)≤

m
,
4

then there are a vector

inf{∥t − x∥2 : x ∈ Mm,l,q } ≥ c m1/2 .

Remark 4.3. If we denote the euclidean distance of t ∈ Rm to the manifold Mm,l,q by dist2 (t, Mm,l,q ),
then the lemma of V. Maiorov above says that
sup{dist2 (y, Mm,l,q ) : y ∈ Em } ≥ cm−1/2 .
Let us now prove inequality (1.3) in Theorem 1.5.
Theorem 4.4. If r > 1/2, then we have that
n−r (log n)r (d−2) ≪ Mn (U2r )2 ≪ n−r (log n)r (d−1) .

(4.1)

Proof. The upper bound of (4.1) was proved in Theorem 4.1, and so we only need to prove the lower
bound by borrowing a technique used in the proof of [17, Theorem 1.1]. For every positive number a


D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

435

we define a subset H(a) of lattice vectors given by




H(a) :=

k : k = (kj : j ∈ N [d]) ∈ Z ,
d



|kj | ≤ a .

j∈N [d]

Recall that, for a → ∞, we have that |H(a)| ≍ a(log a)d−1 ; see, for example, [7]. To apply Lemma 4.2,
we choose for any n ∈ N, q = ⌊n(log n)−d+2 ⌋ + 1, m = 5(2d + 1)⌊n log n⌋ and l = (2d + 1)n. With
these choices we observe that

|H(q)| ≍ m

(4.2)

q ≍ m(log m)−d+1

(4.3)

and

as n → ∞. Also, we readily confirm that



l

lim

log

m

n→∞

4emq


=

l

1
5

and so the hypothesis of Lemma 4.2 is satisfied for n → ∞.
Now, there remains the task of specifying the polynomial manifold Mm,l,q . To this end, we introduce
the positive constant ζ := q−r m−1/2 and let Y be the set of trigonometric polynomials on Td , defined
by






ζ

Y :=

|H(q)|

tk χk : t = (tk : k ∈ H(q)) ∈ E

.

k∈H(q)

If f ∈ Y is written in the form
f =ζ



tk χk ,

k∈H(q)

then f = κr ,d ∗ g for some trigonometric polynomial g such that

∥g ∥2L (Td ) ≤ ζ 2
2



|λk |2 ,

k∈H(q)

where λk was defined earlier before Definition 1.1. Since

ζ2



|λk |2 ≤ ζ 2 q2r |H(q)| = m−1 |H(q)|,

k∈H(q)

we see from (4.2) that there is a positive constant c such that ∥g ∥L2 (Td ) ≤ c for all n ∈ N. So, we can
either adjust functions in Y by dividing them by c or we can assume without loss of generality that
c = 1. We choose the latter possibility so that Y ⊆ U2r (Td ).
We are now ready to obtain a lower bound for Mn (U2r (Td ))2 . We choose any ϕ ∈ L2 (Td ) and let v
be any function formed as a linear combination of n translates of the function ϕ . Thus, for some real
constants cj ∈ R and vectors yj ∈ Td , j ∈ N [n] we have that

v=



cj ϕ(· − yj ).

j∈N [n]

By the Bessel inequality we readily conclude for
f =ζ


k∈H(q)

tk χk ∈ Y


436

D. D˜
ung, C.A. Micchelli / Journal of Complexity 29 (2013) 424–437

that

∥f − v∥


2
 
ϕ(
ˆ k)  i(yj ,k) 
≥ζ
cj e
tk −
 .


ζ j∈N [n]
k∈H(q)

2
L2 (Td )

2

(4.4)

We now introduce a polynomial manifold so that we can use Lemma 4.2 to get a lower bound for
the expressions on the left hand side of inequality (4.4). To this end, we define the vector c = (cj : j ∈

N [n]) ∈ Rn and for each j ∈ N [n], let zj = (zj,l : l ∈ N [d]) be a vector in Cd and then concatenate these
vectors to form the vector z = (zj : j ∈ N [n]) ∈ Cnd . We employ the standard multivariate notation
zkj =



k

zj,ll

l∈N [d]

and require vectors w = (c, z) ∈ Rn × Cnd and u = (c, Re z, ℑz) ∈ Rl written in concatenate form.
Now, we introduce for each k ∈ H(q) the polynomial qk defined at w as
qk (w) :=

ϕ(
ˆ k)  k
cj z .
ζ j∈H(q)

We only need to consider the real part of qk , namely, pk = Re qk since we have that



2


 
ϕ(

ˆ k)  i(yj ,k) 

cj e
inf
 : cj ∈ R, yj ∈ Td
tk −


k∈H(q) 
ζ j∈N [n]



|tk − pk (u)|2 : u ∈ Rl .
≥ inf
k∈H(q)

Therefore, by Lemma 4.2 and (4.3) we conclude that there are a vector t0 = (tk0 : k ∈ H(q)) ∈ Ehq
and the corresponding function
f0 = ζ



tk0 χk ∈ Y

k∈H(q)

for which there is a positive constant c such that for every v of the form



v=

cj ϕ(· − yj ),

j∈N [n]

we have that
1

∥f 0 − v∥L2 (Td ) ≥ c ζ m 2 = q−r ≍ n−r (log n)r (d−2)
which proves the lower bound of (4.1).
Acknowledgments
˜
Dinh Dung’s
research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under Grant no. 102.01-2012.15. Charles A. Micchelli’s research is
supported by US National Science Foundation Grant DMS-1115523 and US Air Force Office of
Scientific Research under Grant FA9550-09-1-0511. The authors would like to thank the referees for
a critical reading of the manuscript and for several valuable suggestions which helped to improve its
presentation.
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