Multivariate approximation by translates of tensor product kernel on Smolyak grids

Multivariate approximation by translates of tensor product kernel on
Smolyak grids
Dinh D˜
unga∗, Charles A. Micchellib , Vu Nhat Huyc
a

Vietnam National University, Information Technology Institute
144 Xuan Thuy, Hanoi, Vietnam
b

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

College of Science, Vietnam National University
334 Nguyen Trai, Thanh Xuan, Ha Noi
August 9, 2014 -- Version 0.9

Abstract
Keywords: Korobov space; Translates of the Korobov function; Reproducing kernel Hilbert space;
Smolyak grids.
Mathematics Subject Classifications: (2010) 41A46; 41A63; 42A99.

1

Introduction

The purpose of this paper is to improve and extend the ideas in the recent paper [14] on sparse
approximation by translates of the multivariate Korobov function. The motivation for the results
given in [14], and those presented here come from Machine Learning, since certain cases of our results
here relate to approximation of a function by sections of a reproducing kernel corresponding to specific

Hilbert space of functions. This relationship to ML is described in the paper [14] and is not reviewed
in detail here.
We shall begin our discussion here by establishing notation used throughout the paper. In this
regard, we merely follow closely the presentation in [14]. 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
1/p

f

p

:= (2π)−d/p

|f (x)|p dx
Td

∗

Corresponding author. Email:

1

.

(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.
Here, we use the notation Nm for the set {1, 2, . . . , m} and later for r, s ∈ Z we will use Zr,s
for the set {r, r + 1, . . . , s}. For vectors x := (xl : l ∈ Nd ) and y := (yl : l ∈ Nd ) in Td we use
(x, y) := l∈Nd xl yl for the inner product of x with y. Also, for notational convenience we allow
N0 and Z0 to stand for the empty set. Given any integrable function f on Td and any lattice vector
j = (jl : l ∈ Nd ) ∈ Zd , we let fˆ(j) denote the j-th Fourier coefficient of f defined by the equation
fˆ(j) := (2π)−d

f (x) ei(j,x) dx.
Td

Frequently, we use the superscript notation Bd to denote the cross product of d copies of a given set
B in Rd .
Let S (Td ) be the space of distribution on Td . Every f ∈ S (Td ) can be identified with the formal
Fourier series
f=
fˆ(j)ei(j,.)
j∈Zd

where the sequence (fˆ(j) : j ∈ Zd ) forms a tempered sequence [Z,.].
Let λ := (λj : j ∈ Z) be a bounded sequence with nonzero components. With the univariate λ
we associate a multivariate tensor product sequence λ := (λj : j ∈ Zd ) defined on a lattice vectors
j := (jl : l ∈ Nd ) whose component are given by
d

λj :=

λjl .
l=1

Let us introduce the space Φλ,p (Td ) of all f ∈ S (Td ) such that
f=
j∈Zd

gˆ(j) i(j,.)
e
.
λ(j)

Let λ := (λj : j ∈ Zd ) be an absolutely summable vectors with nonzero components. We introduce the
function ϕλ,d , defined at x ∈ Td by the equation
i(j,x)
λ−1
.
j e

ϕλ,d (x) :=

(1.2)

j∈Zd

Moreover, in the case that d = 1 we merely write ϕλ for the univariate function ϕλ,1 . According to
our hypothesis on the univariate vector (λj : j ∈ Z) we see that the function ϕλ,d is continuous on Td .
The special cases that the univariate vector (λj : j ∈ Z) is given, for r ∈ (0, ∞) by the equation
λj =

|j|r
1

if j = 0
if j = 0

(1.3)

corresponds to the Korobov function which was the focus of study in [14]. In general, we introduce a
subspace of Lp (Td ) defined as
Φλ,p (Td ) := {f : f = ϕλ,d ∗ g, g ∈ Lp (Td )}
2


with norm
f

:= g

Φλ,p (Td )

p

where 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

f1 (x) f2 (x − y) dy,
Td

whenever the integrand is in L1 (Td ).
The case p = 2 is particularly interesting as it has an interpretation in ML which is described in

detail in the paper [14]. As in that paper we are concerned with the following concept. Let W ⊂ Lp (Td )
be a prescribed subset of Lp (Td ) and ψ ∈ Lp (Td ) be a given function on Td . We are interested in the
approximation in Lp (Td )-norm of all functions f ∈ W by arbitrary linear combinations of n translates
of the function ψ, that is, by the functions in the set {ψ(· − yl ) : yl ∈ Td , l ∈ Nn } and measure the
error in terms of the quantity
cl ψ(· − yl )

Mn (W, ψ)p := sup{ inf{ f −

p

: cl ∈ R, yl ∈ Td } : f ∈ W}.

l∈Nn

The aim of the present paper is to investigate the convergence rate, when n → ∞, of Mn (Uλ,p (Td ), ψ)p
where Uλ,p (Td ) is the unit ball in Φλ,p (Td ). We shall also obtain a lower bound for the convergence
rate as n → ∞ of the quantity
Mn (Uλ,2 (Td ))2 := inf{Mn (Uλ,2 (Td ), ψ)2 : ψ ∈ L2 (Td )}
which gives information about the best choice of ψ.
This paper is organized in the following manner.

2

The Univariate Case

In this section, we introduce a method of approximation induced by translates of the function defined
in equation (1.2) in the univariate case. We do this in some greater generality than described earlier.
To the end, we start with the functions ϕλ , ϕβ given in equation (1.2) and we consider a even, nonincreasing function h : R → [0, 1] defined on [0, ∞) such that
h(t) =

if t ∈ [− 21 , 21 ]
if t ∈ (−1, 1).

1,
0,

(2.4)

Corresponding to this function we introduce a trigonometric polynomial Hm ∈ Tm define at x ∈ T as
Hm (x) =

h(k/m)
k∈Z

βk ikx
e :=
λk

αk eikx .

(2.5)

k∈Z

p
We define lp,0 (Z) := (θk : k ∈ Z) :
(2m + 1)j − m ≤ k ≤
k∈Z |θk | < ∞ , Im,j = {k ∈ Z :
(2m + 1)j + m},

For a function f ∈ Φλ,p (T) represented as f = ϕλ ∗ g, g ∈ Lp (T), we define the operator

Qm (f ) :=

1
2m + 1

Vm (g)(δm l)ϕβ (· − δm l),
l∈Z2m+1

3

(2.6)


where δm := 2π/(2m + 1) and Vm (g) := Hm ∗ g. Our goal is to obtain an estimate for the error of
approximating a function f ∈ Φλ,p (T) by Qm (f ) the linear combinations of n translates of the function
ϕβ .

2.1

Results for L2 (T) spaces

Theorem 2.1 We put
εm = max{ sup |λ−1
k |,
|k|>m/2

Γ2m,k },
k∈Z

where γk = αk βk−1 and Γm,j = max{|γk | : k ∈ Im,j } for j ∈ Z, j = 0, Γm,0 = max{|γk | : k ∈
(m/2, m)}. Then there exists a positive constant c such that for all m ∈ N and f ∈ Φλ,2 (T) we have
that
f − Qm (f ) 2 ≤ cεm f Φλ,2 (T)
(2.7)
and
Qm (f )

2

≤c f

Φλ,2 (T) .

(2.8)

Proof. We define the kernel Pm (x, t) for x, t ∈ T as
Pm (x, t) :=

1
2m + 1

ϕβ (x − δm l)Hm (δm l − t)
l∈Z2m+1

and easily obtain from our definition (2.6) the equation
Qm (f )(x) =

Pm (x, t)g(t) dt.

(2.9)

T

We now use equation (1.2), the definition of the trigonometric polynomial Hm given in equation (2.5)
and the easily verified fact, for k, s ∈ Z, s ∈ [−m, m], that
1
2m + 1

ik(t−δm l) is(δm l−t)

e

e

=

l∈Z2m+1


0,

if

k−s
2m+1

∈Z

 i(k−k )t
e
,

if

k−s
2m+1

∈Z

to conclude that
γk ei(k−k

Pm (x, t) =

)x

.

(2.10)

k∈Z

We again use the formula for the function ϕλ given in equation (1.2) to get that
−ikt
eikx (γk e−ik t − λ−1
).
k e

Pm (x, t) − ϕλ (x − t) =

(2.11)

k∈Z

For k ∈ [−m/2, m/2] we have that k = k ≤ m and since the function h given in (2.4) is one on interval
[−1/2, 1/2] the above expression becomes
−ikt
eikx (γk e−ik t − λ−1
) =: Am (x) − Bm (x).
k e

Pm (x, t) − ϕλ (x − t) =
|k|>m/2

4

(2.12)


By the triangle inequality we have
Qm (f ) − f

≤

2

Am

2

+ Bm 2 .

(2.13)

Parseval’s identity gives
Am

2
2

|γk |2 |ˆ
g (k )|2

=
|k|>m/2

|γk |2 |ˆ
g (k)|2 +

=

|γk |2 |ˆ
g (k − (2m + 1)j)|2
j∈Z∗ k∈Im,j

[m/2]+1≤|k|
≤ Γ20 |ˆ

g (k)|2 +

span

|Γm,j |2 |ˆ
g (k − (2m + 1)j)|2 .
j∈Z\0 k∈Im,j

Hence, by the hypothesis of εm and the inequality
|ˆ
g (k − (2m + 1)j)|2 ≤ g 22 ,
k∈Im,j

we obtain
Am

2
2

≤ Γ2m,0 g

2
2

Γ2m,j g

+

2
2

≤ ε2m g 22 .

(2.14)

j∈Z\0

Next, by Parseval’s identity, we have
Bm

2
2

|λ−2
g (k)|2 ≤ sup |λ−1
k |
k ||ˆ

=

|k|>m/2

|k|>m/2

|ˆ
g (k)|2 ≤ ε2m g 22 .
|k|>m/2

This together with (2.13) and (2.14) proves the theorem.
Corollary 2.2 Let h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1] and

εm = max{ sup |λ−1
k |,
|k|>m

where Γm,j = max{|γk | :

Γ2m,k },
k∈Z\0

k ∈ Im,j } for j ∈ Z. Then for all m ∈ N and f ∈ Φλ,2 (T) we have that
f − Qm (f )

2

≤ cεm f

Φλ,2 (T)

and
Qm (f )

2

≤c f

Φλ,2 (T) .

Proof. From the definition of function h and the proof in above theorem we have
−ikt
eikx (γk e−ik t − λ−1

).
k e

Pm (x, t) − ϕλ (x − t) =
|k|>m

Similar the proof of above theorem we complete the proof.

5

(2.15)


Corollary 2.3 Let λk = βk = λ−k = β−k for all k ∈ Z and h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if
x ∈ [−1, 1]; the sequence {|λk |}k∈N is non decreasing. Then there exists a positive constant c such that
for all m ∈ N and f ∈ Φλ,2 (T) we have that
f − Qm (f )

2

≤ cεm f

Φλ,2 (T)

and
Qm (f )

2

≤c f

Φλ,2 (T) .

where
λ−2
mk },

εm = max{|λ−1
m |,
k∈N

−1
Proof. From the hypothesis we have γk = λk and sup|k|>m |λ−1
k | ≤ |λm | and Γm,k ≤ |λmk | for all
k ∈ N. From this and corollary 2.2, we complete the proof.
From the above corollary we have the following result

Corollary 2.4 Let λk = βk = λ−k = β−k for all k ∈ Z and h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if
x ∈ [−1, 1]; the sequence { |λkkr | }k∈N is non decreasing for some r > 12 . Then there exists a positive
constant c such that for all m ∈ N and f ∈ Φλ,2 (T) we have that
f − Qm (f )

2

≤ c|λ−1
m | f

Qm (f )

2

≤c f

Φλ,2 (T)

and
Φλ,2 (T) .

Proof. We see from the hypothesis that
|λm |
|λmk |
≥
(mk)r
mr
and then

|λmk |
≥ jr
|λm |

for all m, k ∈ N. Hence
−1
λ−2
mk ≤ |λm |
k∈N

Note that, since r >
the proof.

1

2

we have

k∈N

k −2r

k −2r .
k∈N

< ∞ and then by applying above corollary we complete

Corollary 2.5 Let βk = β−k = λ2k = λ2−k for all k ∈ Z and h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if
x ∈ [−1, 1]; the sequence {|λk |}k∈N is non decreasing. Then there exists a positive constant c such that
for all m ∈ N and f ∈ Φλ,2 (T) we have that
f − Qm (f )

2

≤ cεm f

Φλ,2 (T)

and
Qm (f )

2

≤c f

Φλ,2 (T) .

where
λ−2
mk },

εm = max{|λ−1
m |,
k∈N

6


−1
Proof. We see that |γk | = |λ−2
k λk | ≤ |λk | and then it follows from the sequence {|λk |}k∈N is non
decreasing that Γm,k ≤ |λ−1
mk | for all k ∈ N. From this and corollary 2.2, we complete the proof.

Corollary 2.6 Let h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1], βk = λ2k = β−k = λ2−k and the
sequence { |λkkr | }k∈N is non decreasing for some r > 21 . Then there exists a positive constant c such that
for all m ∈ N and f ∈ Φλ,2 (T) we have that
f − Qm (f )

2

≤ c|λ−1
m | f

Qm (f )

2

≤c f

Φλ,2 (T)

(2.16)

and
Φλ,2 (T) .

(2.17)

Proof. We see from the hypothesis that
|λm |
|λmk |
≥
r
(mk)
mr
and then

|λmk |
≥ kr
|λm |

for all m, k ∈ N. Hence
−1

λ−2
mk ≤ |λm |
k∈N

Note that, since r >
the proof.

1
2

we have

k∈N k

−2r

k −2r .
k∈N

< ∞ and then by applying above corollary we complete

Corollary 2.7 Let h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1], λk = βk = |k|1/2 ln |k| for all
k ∈ Z, k = 0. Then there exists a positive constant c such that for all m ∈ N and f ∈ Φλ,2 (T) we have
that
f − Qm (f ) 2 ≤ cm(ln m)2 f Φλ,2 (T)
and
Qm (f )

2

≤c f

Φλ,2 (T) .

Let χ be a normed space, let F be a nonempty subset of χ such that F = −F and let Gn be the
class of all vector subspaces of χ of dimension at most n. The Kolmogorov n-width, dn (F, χ), of F in
χ is given by
dn (F, χ) = inf sup inf f − g χ .
g∈Gn f ∈F g∈G

Theorem 2.8 Let λk = λ−k for all k ∈ Z and the sequence { |λkkr | }k∈N is non decreasing for some
r > 12 . Then
−1
C2 |λ−1
n | ≤ dn (Uλ (T), L2 ) ≤ C1 |λn |.
Proof. By using above corollary we have the estimate
dn (Uλ (T), L2 ) ≤ C1 |λ−1
n |.
The proof is complete.
7


2.2

Results for L2 (Td ) spaces

Definition 2.9 For k ∈ Rd we define
( dj=1 |kj |p )1/p
max1≤j≤d |kj |

|k|p =

if 1 ≤ p < ∞
if p = ∞.

(2.18)

Corresponding to the univariate function we introduce a trigonometric polynomial Hm define at
x ∈ Td as
βk
h(k1 /m)...h(kd /m) eikx :=
αk eikx .
(2.19)
Hm (x) =
λ
k
d
d
k∈Z

For a function f ∈ Φλ,2

(Td )

k∈Z

represented as f = ϕλ,d ∗ g, g ∈ L2 (Td ), we define the operator
1
(2m + 1)d

Qm (f ) :=

Vm (g)(δm l)ϕβ,d (· − δm l),

(2.20)

l∈Zd2m+1

where δm := 2π/(2m + 1) and Vm (g) := Hm ∗ g. Put
εm := max{

sup
k∈Zd \[−m/2,m/2]d

where γk = βk−1 αk , k ∈ [−m, m]d ,
Γm,j =

kj −kj
2m+1

|λ−1
k |;
j∈Zd

∈ Z for all j = 1, 2, .., d, and





max

k∈[−m,m]d \[−m/2,m/2]d




Γ2m,j }

|γk |

|γk+(2m+1)j |

max

k∈[−m,m]d

if j = 0
if j = 0.

Theorem 2.10 There exists a positive constant c such that for all f ∈ Φλ,2 (Td ) and m ∈ N, we have
that
f − Qm (f ) 2 ≤ cεm f Φλ,2 (Td )
(2.21)
and
Qm (f )

2

≤c f

Φλ,2 (Td ) .

(2.22)

Proof. We define the kernel Pm (x, t) for x, t ∈ Td as
Pm (x, t) :=

1
(2m + 1)d

ϕβ,d (x − δm l)Hm (δm l − t)
l∈Zd2m+1

and easily obtain from our definition (2.20) the equation
Qm (f )(x) =

Pm (x, t)g(t) dt.

(2.23)

Td

We now use equation (1.2), the definition of the trigonometric polynomial Hm given in equation (2.19)
and the easily verified fact, for k, s ∈ Zd , s ∈ [−m, m]d , that

kj −sj
if 2m+1
∈ Z for some j ∈ [1, d]
0,

1
ik(t−δm l) is(δm l−t)
e
e
=
 i(k−k )t
(2m + 1)d
kj −sj
e
,
if 2m+1
∈ Z for all j ∈ [1, d]
l∈Zd2m+1
8


to conclude that
γk ei(k−k )x .

Pm (x, t) =

(2.24)

k∈Zd

where γk := βk−1 αk , k ∈ [−m, m]d . We again use the formula for the function ϕλ,d given in equation
(1.2) to get that
−ikt
eikx (γk e−ik t − λ−1
).

k e

Pm (x, t) − ϕλ,d (x − t) =

(2.25)

k∈Zd

For k ∈ [−m/2, m/2]d we have that k = k and since the function h given in (2.4) is one on interval
[−1/2, 1/2] the above expression becomes
−ikt
eikx (γk e−ik t − λ−1
)
k e

Pm (x, t) − ϕλ,d (x − t) =
k∈Zd \[−m/2,m/2]d

span
ikx
λ−1
g
ˆ
(k)e
=:
A
(x)
−
B
(x).

m
m
k

γk gˆ(k )eikx −

=

k∈Zd \[−m/2,m/2]d

k∈Zd \[−m/2,m/2]d

(2.26)
By the triangle inequality we have
Qm (f ) − f

2

≤

Am

2

+ Bm 2 .

(2.27)

Parseval’s identity gives
Am

2
2

|γk |2 |ˆ
g (k )|2

=
k∈Zd \[−m/2,m/2]d

|γk+(2m+1)j |2 |ˆ
g (k)|2

|γk |2 |ˆ
g (k)|2 +

=

span

j∈Zd∗ k∈[−m,m]d

k∈[−m,m]d \[−m/2,m/2]d

≤ Γ2m,0 |ˆ
g (k)|2 +

|Γm,j |2 |ˆ
g (k)|2 .
j∈Zd \0 k∈[−m,m]d

Hence, by the hypothesis of εm and the inequality
|ˆ
g (k)|2 ≤ g 22 ,
k∈[−m,m]d

we obtain
Am

2
2

≤ Γ2m,0 g

2
2

Γ2m,j g

+

2
2

≤ ε2m g 22 .

(2.28)

j∈Zd \0

Next, by Parseval’s identity, we have
Bm

2
2

|λ−2
g (k)|2 ≤
k ||ˆ

=
k∈Zd \[−m/2,m/2]d

sup
k∈Zd \[−m/2,m/2]d

This together with (2.13) and (2.14) proves the theorem.

9

|λ−2
k |

|ˆ
g (k)|2 ≤ ε2m g 22 .
k∈Zd \[−m/2,m/2]d


Definition 2.11 The sequence {θk }k∈Zd will be called a non decreasing-type sequence if θk ≥ cθl for
all k, l ∈ Zd satisfies |kj | ≥ |lj |, j = 1, 2, .., d.

From the above corollary we have the following result
k|
Corollary 2.12 Let h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1]; the sequence { |k||βr |λ
} d is a
k | k∈Z

non decreasing-type sequence for some r > d2 . Then there exists a positive constant c such that for all
m ∈ N and f ∈ Φλ,2 (Td ) we have that
f − Qm (f )

2

≤c

sup
k∈Zd \[−m/2,m/2]d

|λ−1
k | f

Φλ,2 (Td )

and
Qm (f )

2

≤c f

Φλ,2 (Td ) .

Proof. We see from the hypothesis for all k ∈ [−m/2, m/2]d that
|βk+(2m+1)j |
|k + (2m + 1)j| |βk |
≥c
|λk+(2m+1)j |
|k|
|λk |
and then
|γk+(2m+1)j | = |λ−1
k+(2m+1)j |

|λk+(2m+1)j |βk |
r
≤ c1
sup
|λ−1
k ||j|
|βk+(2m+1)j |λk |
k∈Zd \[−m/2,m/2]d

where |j| = |j1 | + ... + |jd |. Hence
Γm,j ≤ c1

sup
k∈Zd \[−m/2,m/2]d

r
|λ−1
k ||j|

Similarly,
Γm,0 ≤ c1

sup
k∈Zd \[−m/2,m/2]d

|λ−1
k |.

Hence
∞
−2
γm,j
j∈Zd

≤ c1

sup
k∈Zd \[−m/2,m/2]d

Note that for 2r > d then

|λ−1
k |

−2r

|j|

sup
k∈Zd \[−m/2,m/2]d

j∈Zd

∞
d−1 j −2r
j=1 j

≤ c1

|λ−1
k |

j d−1 j −2r .
j=1

is convergent. The proof is complete.

Corollary 2.13 Let h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1], βk = λ2k for all k ∈ Zd ; the
d
k|
sequence { |λ
|k|r }k∈Zd is non decreasing-type for some r > 2 . Then there exists a positive constant c
such that for all m ∈ N and f ∈ Φλ,2 (Td ) we have that
f − Qm (f )

2

≤c

sup
k∈Zd \[−m/2,m/2]d

|λ−1
k | f

and
Qm (f )

2

≤c f

10

Φλ,2 (Td ) .

Φλ,2 (Td )


Corollary 2.14 Let h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1], βk = αk for all k ∈ Zd ; the
d
k|
sequence { |λ
|k|r }k∈Zd is non decreasing-type for some r > 2 . Then there exists a positive constant c
such that for all m ∈ N and f ∈ Φλ,2 (Td ) we have that
f − Qm (f )

2

≤c

sup
k∈Zd \[−m/2,m/2]d

|λ−1
k | f

Φλ,2 (Td )

and
Qm (f )

2

≤c f

Φλ,2 (Td ) .

Proof. We see from the hypothesis for all k ∈ [−m/2, m/2]d that
|βk+(2m+1)j |
|k + (2m + 1)j| |βk |
≥c
|λk+(2m+1)j |
|k|
|λk |
and then
|γk+(2m+1)j | = |λ−1
k+(2m+1)j | ≤ c1

sup
k∈Zd \[−m/2,m/2]d

r
|λ−1
k ||j|

where |j| = |j1 | + ... + |jd |. Hence
Γm,j ≤ c1

sup
k∈Zd \[−m/2,m/2]d

r
|λ−1
k ||j|

Similarly,
Γm,0 ≤ c1

sup
k∈Zd \[−m/2,m/2]d

|λ−1
k |.

Hence
∞
−2

γm,j
j∈Zd

≤ c1

sup
k∈Zd \[−m/2,m/2]d

Note that for 2r > d then

2.3

|λ−1
k |

−2r

|j|

sup
k∈Zd \[−m/2,m/2]d

j∈Zd

∞
d−1 j −2r
j=1 j

≤ c1

|λ−1
k |

j d−1 j −2r .
j=1

is convergent. The proof is complete.

Results for Lp (T) spaces

For this purpose, we define, for m ∈ N, the quantity
|∆λ−1
k |,

εm = max{
|k|>m/2

|∆γk | +
|k|>m/2

|γk(2m+1)+m |},

(2.29)

k∈Z

where γk := βk−1 αk , k the unique integer in Z−m,m such that the number (k − k )/(2m + 1) is an
integer and ∆γk := γk − γk+1 .
Now, we are ready to state the the following result.
Theorem 2.15 If 1 < p < ∞ then there exists a positive constant c such that for all f ∈ Φλ,p (T) and

m ∈ N, we have that
f − Qm (f ) p ≤ cεm f Φλ,p (T) .
(2.30)

11


Before we give the proof of the above theorem, we recall, for 1 < p < ∞, that there exists a positive
constant c such that for all f ∈ Lp (T) there holds the inequality
Sm (f )

p

≤c f

(2.31)

p

fˆ(j)eijx , see for

where Sm is the Fourier projection onto Tm given at x ∈ T as Sm (f )(x) =
j∈Z−m,m

example, Theorem 1, page 137, of [2].
For r, s ∈ Z, we introduce a functional which is central to the proof of Theorem 2.15 which is
defined at x ∈ T and g ∈ Lp (T) as
eikx gˆ(k).

Gr,s (x) :=

(2.32)

k∈Zr,s

From this definition , we easily obtain the following lemma.
Lemma 2.16 If 1 < p < ∞ then there exists a positive constant c such that for all f ∈ Lp (T) and
r, s ∈ Z we have
Gr,s (f ) p ≤ c f p .
(2.33)
Proof. The proof of the result is strong forward when s = r + 2m for some nonnegative integer m we
have that
Gr,s (x) = ei(r+m)x (Sm g1 )(x)
where g1 is defined at x ∈ T as

g1 (x) = ei(r+m)x g(x).

From this formula and inequality (2.31) we obtain inequality (2.33).
When r = r + 2m + 1 we have that
Gr,s (x) = ei(r+m+1)x (Sm+1 g1 )(x) − ei(r+2m+2)x (S0 g2 )(x)
when now g1 , g2 are defined at x ∈ T as g1 (x) = ei(r+m+1)x g(x) and g2 (x) = ei(r+2m+2)x g(x).
Now, we ready to present the proof of Theorem 2.15.
Proof. (Proof of Theorem 2.15) We define the kernel Pm (x, t) for x, t ∈ T as
Pm (x, t) :=

1
2m + 1

ϕβ (x − δm l)Hm (δm l − t)
l∈Z2m+1

and easily obtain from our definition (2.6) the equation
Qm (f )(x) =

Pm (x, t)g(t) dt.

(2.34)

T

We now use equation (1.2), the definition of the trigonometric polynomial Hm given in equation (2.5)
and the easily verified fact, for k, s ∈ Z, s ∈ [−m, m], that
1
2m + 1

e

ik(t−δm l) is(δm l−t)

e

l∈Z2m+1

12

=


0,

if

k−s
2m+1

∈Z

 i(k−k )t
e
,

if

k−s
2m+1

∈Z


to conclude that
γk ei(k−k

Pm (x, t) =

)x

.

(2.35)

k∈Z

We again use the formula for the function ϕλ given in equation (1.2) to get that
−ikt
eikx (γk e−ik t − λ−1
).
k e

Pm (x, t) − ϕλ (x − t) =

(2.36)

k∈Z

For k ∈ [−m/2, m/2] we have that k = k ≤ m and since the function h given in (2.4) is one on interval
[−1/2, 1/2] the above expression becomes
−ikt
eikx (γk e−ik t − λ−1
).
k e

Pm (x, t) − ϕλ (x − t) =

(2.37)

|k|>m/2

From this equation and the definition of f we finally deduce that
ikx
λ−1

gˆ(k).
k e

γk eikx gˆ(k ) −

Qm (f )(x) − f (x) =
|k|>m/2

(2.38)

|k|>m/2

The next step is to decompose each sum above into two parts. Specifically, we write the first sum of
above as
γk eikx gˆ(k ) +
γk eikx gˆ(k ).
(2.39)
k>m/2

k<−m/2

We call the the first sum in equation (2.39)
A+
m (x) in the form
A+
m (x) =

A+
m (x)

and the other A−
m (x). Now, we readily rewrite

γk eikx gˆ(k) +

γk eikx gˆ(k ),

(2.40)

j∈N k∈Im,j

[m/2]+1≤k
where Im,j := {l : l ∈ Z, j(2m + 1) − m ≤ l ≤ j(2m + 1) + m}.
We shall express the right hand side of equation (2.41) in an alternate form by using summation
by part. For this purpose, we introduce the modified difference operator defined on vectors as
Λγk =

γk − γk+1 , if j(2m + 1) − m ≤ k < j(2m + 1) + m
γk ,
if k = j(2m + 1) + m.

(2.41)

With this notation in hand and the fact that, for k ∈ Im,j we have k = k − j(2m + 1), we conclude
that
A+
m (x) =

γk eikx gˆ(k − j(2m + 1))

Λγk G[m/2]+1,k +
j∈N k∈Im,j

[m/2]+1≤k
span
=

Λγk G[m/2]+1,k +

Λγk e

ij(2m+1)x

G−m,k−j(2m+1) .

j∈N k∈Im,j

[m/2]+1≤k
Consequently, according the Lemma 2.16 and the H¨older inequality, there exists a positive constant
c1 such that
A+
|Λγk | +
|Λγk | g p .
(2.42)
m p ≤ c1
j∈N k∈Im,j

m/2+1≤k≤m

13


From this inequality and the definition of εm given in equation (2.29) we conclude that
A+
m
A bound on A−
m

p

≤ c1 εm g p .

p

(2.43)

follows by a similar argument and yields the inequality
A−
m

≤ c1 εm g p .

p

(2.44)

There still remains the task of bounding the second sum in equation (2.38). As before, we split it

into two parts
ikx
λ−1
gˆ(k)
(2.45)
λk−1 eikx gˆ(k) +
k e
k<−m/2

k>m/2

+ (x) and the second sum B − (x). As before, summation by parts yields
and call the first sum above Bm
m
the alternate form
+
∆λ−1
(2.46)
Bm
(x) =
k Gm/2+1,k
[m/2]+1≤k<∞

from which we deduce that
+
Bm
(x)

p

|∆λ−1
k | g p.

≤
[m/2]+1≤k<∞

Therefore, by (2.29) we obtain that
+
Bm
(x)

p

≤ c1 εm g

p

(2.47)

and in a similar way we prove that
−
Bm

≤ c1 εm g p .

p

(2.48)

So far, we have proved inequality (2.30). The proof is complete.

Now, we are ready to state the the following result.
Corollary 2.17 Let 1 < p < ∞, h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1] and {λk }∞
k=0 is the
decreasing sequence and λk = λ−k = βk = β−k for all k ∈ Z. Then there exists a positive constant c
such that for all f ∈ Φλ,p (T) and m ∈ N, we have that
f − Qm (f )

p

≤ cεm f

Φλ,p (T)

(2.49)

and
Qm (f )
where εm =

p

≤c f

Φλ,p (T) .

(2.50)

−1
k∈N λk[m/2] .

Note that for
λj =

|j|r
1

if j = 0
if j = 0

then Φλ,p become Korobov function and we have the following estimate which have known in [14]
f − Qm (f )

p

≤ cm−r f

14

Φλ,p (T) .


Theorem 2.18 Let 1 < p < ∞, h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1]. Then there exists a
positive constant c such that for all f ∈ Φλ,p (T) and m ∈ N, we have that
f − Qm (f )

p

≤ cεm f

(2.51)

Φλ,p (T)

and
Qm (f )

p

≤c f

Φλ,p (T) .

(2.52)

where
|∆λ−1
k |,

εm = max{

|∆γk | +
|k|>m

|k|>m

|γk(2m+1)+m |},
k∈Z

β

where γk := βk−1 λk , k the unique integer in Z−m,m such that the number (k − k )/(2m + 1) is an
k
integer.
Proof. We have known in the proof of Theorem 2.15 that
ikx
λ−1
gˆ(k).
k e

γk eikx gˆ(k ) −

Qm (f )(x) − f (x) =
|k|>m/2

|k|>m/2

Note that, from h(x) = 1 for x ∈ [−1, 1], we have γk = λ−1
k for all integer k ∈ [−m, m] and then
ikx
λ−1
gˆ(k).
k e

γk eikx gˆ(k ) −

Qm (f )(x) − f (x) =
|k|>m

|k|>m

Similar to the proof of Theorem 2.15, we also have
f − Qm (f )

p

≤ cεm f

Φλ,p (T) .

The proof is complete.
From this theorem we have the following corollary
Corollary 2.19 Let 1 < p < ∞, λk = βk = e−s|k| for all k ∈ Z where s > 0 and h(x) = 1
if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1]. Then there exists a positive constant c such that for all
f ∈ Φλ,p (T) and m ∈ N, we have that
f − Qm (f )

p

≤ ce−sm f

Qm (f )

p

≤c f

Φλ,p (T)

(2.53)

and
Φλ,p (T) .

Proof. From the hypothesis we have
−1
−s(m+1)
|∆λ−1
k | = λm+1 = e
|k|>m

and
|γk(2m+1)+m | = e−s(m+1) +

|∆γk | +
|k|>m

e−s|k(2m+1)+m| ≤ c(s)e−s(m+1) .
k∈Z

k∈Z

Hence, by using Theorem 2.18 we complete the proof.
15

(2.54)


Definition 2.20 Let β > 0. A function b : R → R will be called a mask of type β if b is an even, 2
times continuously differentiable such that for t > 0, b(t) = (1 + |t|)−β Fb (log |t|) for some Fb : R → R
(k)

such that |Fb (t)| ≤ c(b) for all t > 1, k = 0, 1. A sequence {bk }k∈Z will be called a sequence mask of
type β.
−1
Definition 2.21 We put λk := λ−1
k and β k := βk .

Theorem 2.22 Let 1 < p < ∞ and the sequence {λk }k∈Z = {β}k∈Z be a sequence mask of type r > 1
and h(x) = 1 if x ∈ [−1, 1], h(x) = 0 if x ∈ [−1, 1]. Then there exists a positive constant c such that
for all f ∈ Φλ,p (T) and m ∈ N,
f − Qm (f )

p

Qm (f )

p

≤ cm−r f

Φλ,p (T) .

and
≤c f

Φλ,p (T) .

Proof. According the hypothesis we have
f − Qm (f )

p

≤ cεm f

Φλ,p (T)

where
|∆λk | +

εm =

|λk(2m+1)+m |.

|k|>m/2

Note that for k ∈ N

k∈Z

|∆λ−1
k | = |λ (k) − λ (k + 1)| = |λ (c)|

where c ∈ (k, k + 1). We have that for k ∈ N
|λ (c)| = |(1 + c)−(r+1) (

−1
r+1
r+1
F (log c) + Fλ (log c))| ≤
c(λ)(1 + c)−(r+1) ≤ c(λ)
(1 + k)−(r+1) .

r λ
r
r

Therefore
|∆λ−1
k | ≤ 2c(λ)
|k|>m/2

r+1
r

r + 1 −r
m .
r

(2.55)

(1 + k(2m + 1) + m)−r ≤ c(λ)c1 m−r

(2.56)

(1 + k)−(r+1) ≤ c(λ)
k>m/2

We also have
|λ−1
k(2m+1)+m | ≤ c(λ)
k∈Z

where c1 =

1
k∈N kr .

k∈Z

From (2.55) and (2.56) we complete the proof.

Definition 2.23 A function b : R → R will be called a exponent - type if b is 2 times continuously
differentiable and there exists a positive constant s such that b(t) = e−s|t| Fb (|t|) for some decreasing
function Fb : [0, +∞) → [0, +∞). The sequence {bk }k∈Z be called a sequence exponent - type.
Theorem 2.24 Let 1 < p < ∞ and the sequence {λ}k∈Z be a sequence mash of type r > 1, the
sequence {β k }k∈Z be a exponent - type. Then there exists a positive constant c such that for all
f ∈ Φλ,p (T) and m ∈ N,
f − Qm (f ) p ≤ cm−r f Φλ,p (T)
(2.57)
and
Qm (f )

p

≤c f
16

Φλ,p (T)

(2.58)

Proof. We have known in Theorem 2.22 that
|∆λk | ≤ cm−r .
|k|>m/2

We also have
|γk | = |λk

Fβ (|k|)
βk
| ≤ |(1 + k)−r cλe−s(k−k ) | ≤ ck −(r+1) .
| = |(1 + k)−r Fλ (log |k|)e−s(k−k )
Fβ (k )
βk

So
|λk(2m+1)+m | ≤ ck −r .

|∆λk | +
|k|>m/2

k∈Z

The proof is complete.

3

Multivariate approximation

For m ∈ Zd+ , let the multivariate operator Qm in Φλ,p (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 ∈ Nd } and kj denotes
the jth coordinate of k.
Set Zd−1 := {k ∈ Zd : kj ≥ −1, j ∈ Nd }. For k ∈ Z−1 , we define the univariate operator Tk in
Φλ,p (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 Φλ,p (Td ) in the
manner of the definition of (3.1) as
d

Tk :=

Tki .
i=1

Set |k| :=

j∈Nd

|kj | for k ∈

Zd−1 .

Lemma 3.1 Let 1 < p < ∞, p = 2, r > 1 or p = 2, r > 1/2 and the sequence {λk }k∈Z be a mask of
type r. Then we have for any f ∈ Φλ,p (Td ) and k ∈ Zd−1 ,
Tk (f )

p

≤ C2−r|k| f

Φλ,p (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 ?? and ??. Assume
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

17


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 Theorem 2.22 and Corollary 2.12 and the induction assumption
Tk (f )

p

=

Tk Tkd (f )

= 2

−r|k |

= 2

−r|k|

Tkd (f )
f

2−r|k | Tkd (f )

p,xd

p,x

2

p,xd Kpr (Td−1 ),x

Kpr (Td−1 ),x

−r|k |

−rkd

2

f

p,xd
Kpr (T),xd Kpr (Td−1 ),x

Φλ,p (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
d

qk :=

qk j .
j=1

For k ∈ Zd+ , we write k → ∞ if kj → ∞ for each j ∈ Nd .
Theorem 3.2 Let 1 < p < ∞, p = 2, r > 1 or p = 2, r > 1/2 and the function λ be a mask of type
r. Then every f ∈ Φλ,p (Td ) can be represented as the series
f =

qk (f )

(3.2)

k∈Zd+

converging in Lp -norm, and we have for k ∈ Zd+ ,
qk (f )

≤ C2−r|k| f

p

Φλ,p (Td )

with some constant C independent of f and k.
Proof. Let f ∈ Φλ,p (Td ). In a way similar to the proof of Lemma 3.1, we can show that
f − Q2k (f )

max 2−rkj f

p

j∈Nd

Φλ,p (Td ) ,

and therefore,
f − Q2k (f )

p

→ 0, k → ∞,

where 2k = (2kj : j ∈ Nd ). On the other hand,
Q2k =

qs (f ).
sj ≤kj , j∈Nd

This proves (3.2). To prove (3.3) we notice that from the definition it follows that
(−1)|e| Tke ,

qk =
e⊂Nd

18

(3.3)


where ke is defined by kie = ki if i ∈ e, and kie = ki − 1 if i ∈
/ e. Hence, by Lemma 3.1
qk (f )

p

≤

Tke (f )

2−r|k

p

e|

f

2−r|k| f

Φλ,p (Td )

Φλ,p (Td ) .

e⊂Nd

e⊂Nd

For approximation of f ∈ Φλ,p (Td ), we introduce the linear operator Pm , m ∈ Z+ , by
qk (f ).

Pm (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 function ϕλ,d on a Smolyak grid. The following theorem gives an upper
bound for the error of the approximation of functions f ∈ Φλ,p (Td ) by the operator Pm .
Theorem 3.3 Let 1 < p < ∞, p = 2, r > 1 or p = 2, r > 1/2 and the sequence {λk }k∈Z be a mask
of type r. Then, we have for every m ∈ Z+ and f ∈ Φλ,p (Td ),
f − Pm (f )

p

≤ C 2−rm md−1 f

Φλ,p (Td )

with some constant C independent of f and m.
Proof. From Theorem 3.2 we deduce that
f − Pm (f )

p

=

qk (f )

p

≤

|k|>m

qk (f )

p

|k|>m

2−r|k| f

Φλ,p (Td )

f

|k|>m

2

4
4.1

−rm

2−r|k|

Φλ,p (Td )
|k|>m

m

d−1

f

Φλ,p (Td ) .

Convergence rate and optimality
Generalization of Dinh D˜
ung’s and Charles Micchelli’s result

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

Td

2πsj
2kj +1 +1

consists of all such vectors and is given as
Gd (m) := {yk,s : |k| ≤ m, s ∈ ⊗j∈Nd Z[2kj +1 + 1]}.

A simple computation confirms, for m → ∞ that
|Gd (m)| =

(2kj +1 + 1)
|k|≤m j∈Nd

19

2d md−1 ,

: j ∈ Nd . The Smolyak


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 equation (3.4) we see that the range of Pm is contained in the subspace
span {ϕλ,d (· − y) : y ∈ Gd (m)}.
Other words, Pm defines a multivariate method of approximation by translates of the function ϕλ,d
on the sparse Smolyak grid Gd (m). An upper bound for the error of this approximation of functions
from Φλ,p (Td ) is given in Theorem 3.3.
Theorem 4.1 If 1 < p < ∞, p = 2, r > 1 or p = 2, r > 1/2 and the sequence {λk }k∈Z be a mask of
type r, then
Mn (Uλ,p , ϕλ,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
Theorem 3.3 we have that
Mn (Uλ,p (Td ), ϕλ,d ) ≤ sup{ f − Pm (f )

Lp (Td )

: f ∈ Uλ,p (Td )}

2−rm md−1

2m md−1 and by

n−r (log n)r(d−1) .

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
[18].
Lemma 4.2 (V. Maiorov) If m, l, q ∈ N satisfy the inequality l log( 4emq
l )≤
t ∈ Em and a positive constant c such that
inf{ t − x

2

m

4,

then there is a vector

: 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 .
Theorem 4.4 If r > 1/2 and the sequence {λk }k∈Z be a mask of type r, then we have that
n−r (log n)r(d−2)

Mn (Uλ,2 )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 [18, Theorem 1.1]. For every positive number a
we define a subset H(a) of lattice vectors given by
H(a) :=

k : k = (kj : j ∈ Nd ) ∈ Zd ,

|kj | ≤ a .
j∈Nd

20

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)
and
m(log m)−d+1

q

(4.3)

as n → ∞. Also, we readily confirm that
l
log
n→∞ m

4emq
l

lim

=

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
Y := ζ
tk χk : t = (tk : k ∈ H(q)) ∈ E|H(q)| .
k∈H(q)

If f ∈ Y is written in the form
f =ζ

tk χk ,
k∈H(q)

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

2
L2 (Td )

≤ ζ2

|λk |2 ,
k∈H(q)

where λk was defined earlier before Definition ??. Since
|λk |2 ≤ ζ 2 q 2r |H(q)| = m−1 |H(q)|,

ζ2
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
cj ϕ(· − yj ).

v=
j∈N [n]

By the Bessel inequality we readily conclude for
tk χk ∈ Y

f =ζ
k∈H(q)

that

2

f −v

2
L2 (Td )

≥ ζ2
k∈H(q)

ϕ(k)

ˆ
tk −
ζ
21

cj ei(yj ,k) .
j∈N [n]

(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 ∈ Nd ) 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
kl
zj,l

zkj =
l∈Nd

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

ϕ(k)
ˆ
ζ

cj zk .

j∈H(q)

We only need to consider the real part of qk , namely, pk = Re qk since we have that




2




ϕ(k)
ˆ
inf
tk −
cj ei(yj ,k) : cj ∈ R, yj ∈ Td ≥ inf
|tk − pk (u)|2 : u ∈ Rl .




ζ
k∈H(q)

j∈N [n]

k∈H(q)

Therefore, by Lemma 4.2 and (4.3) we conclude there is a vector t0 = (t0k : k ∈ H(q)) ∈ Ehq and the

corresponding function
f0 = ζ
t0k χk ∈ Y
k∈H(q)

for which there is a positive constant c such that for every v of the form
cj ϕ(· − yj ),

v=
j∈N [n]

we have that
f0 − v

1

L2 (Td )

≥ cζm 2 = q −r

n−r (log n)r(d−2)

which proves the lower bound of (4.1).

4.2

Generalization of Maiorov’s result

Definition 4.5 A function Ψ : R+ → R+ satisfies ∆2 condition if there exists a constan C > 0
such that Ψ(2t) ≤ cΨ(t) for all t > 1.

Theorem 4.6 Assume that λk = Ψ(|k|) and Ψ is nondecreasing and satisfies ∆2 condition. Then we
have that
1/Ψ((n log n)1/d )
Mn (Uλ,2 )2
1/Ψ(n1/d ).
Proof. Let n be any natural numbers. Set
m = c2 n log n
and
m = |{k ∈ Zd :
22

|k| ≤ s}|.


Then
c sd ≤ m ≤ c sd .
We consider the set consisting of trigonometric polynomials
Fn,s = {ω

ke

ikx

| k | = 1}

:

|k|≤s

where

−1

ω = m−1/2 max |λk |
|k|=s

Let f (x) = |k|≤s fˆk eikx be any polynomial from Fn,s , and let h(x) =
function from the class Hn (ϕβ ). Then it follows from ([17]) that
f −h

λ2k

=ω

Φλ

− al ) be any

≥ cωm1/2 .

2

Note that
f

n
l=1 bl ϕβ (x

1/2

.

|k|≤s

Hence
f

Φλ

λ2k

=ω

1/2

1/2

≤ ω max |λk |

1

|k|=s

|k|≤s

So
f −h

2

≥ cωm1/2 = c max |λk |

−1

|k|=s

Hence
Mn (Uλ,2 )2 ≥ c max |λk |

= 1.

|k|≤s

.

−1

.

|k|=s

So
Mn (Uλ,2 )2 ≥ c(Ψ(s))−1
where s = (c2 n log n)1/d .
We have known for n = ud that
Mn (Uλ,2 )2 ≤ c1

sup
k∈Zd \[−u,u]d

|λ−1
k |

and then
Mn (Uλ,2 )2 ≤ c1 (Ψ(u))−1 .
Hence
c1 /Ψ(n1/d ) ≥ Mn (Uλ,2 )2 ≥ c/Ψ((n log n)1/d ).
The proof is complete.
Corollary 4.7 Assume that 1 ≤ p ≤ ∞, λk = Ψ(|k|p ) and Ψ is nondecreasing and satisfies ∆2
condition. Then we have that
1/Ψ((n log n)1/d )

Mn (Uλ,2 )2
23

1/Ψ(n1/d ).


Acknowledgments
Dinh Dung’s research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under Grant No. 102.01-2014.02. A part of this paper was done when
Dinh D˜
ung and Vu Nhat Huy were working at and Charles Micchelli was visiting Vietnam Institute
for Advanced Study in Mathematics (VIASM). The authors thank VIASM for providing a fruitful
research environment and working condition.

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