Approximation by translates of a single function of functions in space induced by the convolution with a given function

Applied Mathematics and Computation 361 (2019) 777–787

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Approximation by translates of a single function of functions in space induced by the convolution with a given function ˜ a, Charles A. Micchelli b, Vu Nhat Huy c,d,∗ Dinh Dung a

Vietnam National University, Information Technology Institute, 144 Xuan Thuy, Hanoi, Vietnam Department of Mathematics and Statistics, SUNY Albany, Albany 12222, USA c College of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam d Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam b

a r t i c l e

i n f o

MSC: 41A46 41A63 42A99 Keywords: Function spaces induced by the convolution with a given function Reproducing kernel Hilbert space Approximation by arbitrary linear combinations of n translates of a single function

a b s t r a c t We study approximation by arbitrary linear combinations of n translates of a single function of periodic functions. We construct some linear methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of the Lp -approximation convergence rate by these methods, when n → ∞, for 1 < p < ∞. We also prove a lower bound of the quantity of best approximation of this class by arbitrary linear combinations of n translates of arbitrary function, for the particular case p = 2. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The purpose of this paper is to improve and extend the ideas in the recent papers [10,11] on approximation by translates of the multivariate Korobov function. The motivation for the results given in [10,11], 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 Machine Learning is described in the papers [10,18] and is not reviewed in detail here. Nonetheless, in this regard, we recall that the observation presented in [18] provide necessary and sufficient conditions for sections of a reproducing kernel to be dense in continuous functions in the corresponding Hilbert space. This result begs the question of the convergence rate of approximation by sections of a reproducing kernel. We refer the reader to [10,11] for detailed survey and bibliography on the problems considered in the present paper. Here, in this paper, we introduce a weighted Hilbert space of multivariate periodic functions and provide insights into this question. The results presented here also extend other norms on multivariate periodic functions and these results are presented separately in this paper. The problem of periodic approximation of translates of a single function is related to the problem of sampling recovery by translates of trigonometric kernels or B-splines (see [12] for a survey and bibliography), the problem of approximation from a principal shift-invariant space generated by the scaled functions ϕ h := ϕ (·/h) (see [5,6,13,15] and the references there), and its periodic counterpart (see [4,14,17,20] and the references there), the problem of sampling recovery and integration based on B-spline quasi-intepolation representations (see [7–9] and the references there), the problem of scattered data B-spline ∗

Corresponding author at: College of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam. E-mail addresses: [email protected], huyvn0 0 [email protected] (V.N. Huy).

https://doi.org/10.1016/j.amc.2019.06.034 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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intepolation and integration (see [1,2,19,21] and references there). One of the main questions considered in approximation from a principal shift-invariant space is a condition on ϕ such that the principal shift-invariant space (or linear operator into this space) provides approximation order r for Sobolev spaces Wpr . We shall begin to formulate the problems which are investigated in the present paper with a description of the notation used throughout the paper. In this regard, we merely follow closely the presentation in [10,11]. 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 simply 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 L p (Td ), 1 ≤ p < ∞, the space of integrable functions on Td equipped with the norm

 f  p := (2π )−d/p



Td

| f (x )| p dx



1/p

and we shall only consider only real valued functions on Td . However, all the results in this paper are true in the complex setting. Also, we will use the Fourier series of a real valued function in complex form.  For vectors x := (xl : l = 1, 2, . . . , d ) and y := (yl : l = 1, 2, . . . , d ) in Td we use (x, y ) := dl=1 xl yl for the inner product of x with y. Given any integrable function f on Td and any lattice vector j = ( j : l = 1, 2, . . . , d ) ∈ Zd , we let fˆ(j ) denote the l

jth Fourier coefficient of f defined by the equation

fˆ(j ) := (2π )−d



Td

f (x ) e−i(j,x ) dx.

Frequently, we use the superscript notation Bd to denote the cross product of d copies of a given set B in Rd . Let 1 ≤ p ≤ ∞ and p be defined be the equation 1/p + 1/p = 1. Assume that ϕ λ,d belongs to L p (Td ) and can be represented as the Fourier series

ϕλ,d =



λ−1 e i ( j,· ) j

(1.1)

j∈Z d

in distributional sense for some sequence λ := (λj : j ∈ Zd ) with nonzero components. Notice that if (λ−1 : j ∈ Zd ) is abj solutely summable, the function ϕ λ,d is continuous on Td . In the case that d = 1 we merely write ϕ λ for the univariate function ϕ λ,1 . The special case when d = 1, r ∈ (0, ∞) and λ is given by the equation



λj =

| j |r 1

if j = 0, if j = 0.

In this case, the function ϕ λ corresponds to the Korobov function which was the focus of study in [10]. In general, we introduce a subspace of L p (Td ) defined as

λ,p (Td ) := { f : f = ϕλ,d ∗ g, g ∈ L p (Td )} with the norm  f  usual, define it at x

:= g p , where d λ,p (T ) d ∈ T by equation

( f1 ∗ f2 )(x ) := (2π )−d



Td

we denote the convolution of any two functions f1 and f2 on Td , as f1 ∗ f2 , and as

f1 (y ) f2 (x − y ) dy,

whenever the integrand is in L1 (Td ). The space λ,2 (Td ) is particularly interesting as it has an interpretation in Machine Learning which is described in detail in the papers [10,18]. We investigate the approximation of functions in λ,p (Td ) by linear combinations of n equidistant translates of a function ϕ β of the form given in Eq. (1.1) where the sequence β may be, in general different from λ. With this purpose we introduce the trigonometric polynomial Hm defined at x ∈ Td as

Hm (x ) :=

 |k|∞

βk ikx e = λk ≤m



αk eikx ,

(1.2)

|k|∞ ≤m

β

where |k|∞ := max 1 ≤ j ≤ d |kj | and αk := λk for |k|∞ ≤ m. For a function f ∈ λ,p (Td ) represented as f = ϕλ,d ∗ g, g ∈ L p (Td ), k we define the linear operator

Qm ( f ) :=

1

( 2m + 1 )d



Vm (g)(δm l )ϕβ ,d (· − δm l ),

(1.3)

l∈Zd+ : |l|∞ ≤2m

where δm := 2π /(2m + 1 ) and Vm (g) := Hm ∗ g. In the case if the function ϕ β is smoother than ϕ λ , we shall give estimates of the convergence rate of the error  f − Qm ( f ) p of the approximation of f ∈ λ,p (Td ) by the linear operators Qm when  −2 m → ∞. Thus, we show that the error  f − Qm ( f ) p is typically estimated via the partial series remaining |k|∞ >0 λmk in  −1 the case p = 2 (the reproducing kernel case), and k∈N λmk in the case 1 < p < ∞, d = 1.

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As in the paper [10], in the case p = 2 we shall also obtain a lower bound for the convergence rate as n → ∞ of the quantity Mn (Uλ,2 (Td ))2 which gives information about the best choice of ψ for the class Uλ,2 (Td ), where

Uλ,p (Td ) = { f ∈ λ,p (Td ) :

 f λ,p (Td ) ≤ 1}

and

Mn (Uλ,p (Td )) p :=

inf

ψ ∈L p ( Td )

Mn (Uλ,p (Td ), ψ ) p :=

inf

sup

inf

ψ ∈L p ( Td ) f ∈W cl ∈R , yl ∈T

n  f− cl ψ ( · − y l ) p . d l=1

This paper is organized in the following manner. In Section two we provide estimates of the error  f − Qm ( f ) p of the univariate Lp -approximation of f ∈ λ,p (Td ) by the linear operators Qm separately for p = 2 and 1 < p < ∞. In Section three we extend the results in Section two multivariate case and continue this line of investigation by relying upon observations of Maiorov [16] as a means to establish lower bounds of the quantity Mn (Uλ,2 (Td ))2 . 2. A linear method of univariate approximation In this section, we introduce a method of approximation induced by translates of the function defined in Eq. (1.1) in the univariate case (d = 1). We do this in some greater generality. To the end, we start with the functions ϕ λ , ϕ β of the form given in Eq. (1.1). Our goal is to obtain an estimate for the error of approximating a function f ∈ λ,p (T ) by Qm (f) a linear combination of 2m + 1 translates of the function ϕβ ∈ L p (T ). 2.1. Error estimates for functions in the space λ,2 (T ) For every m, k ∈ Z, we denote by k¯ = k¯ (m, k ) the unique integer in [−m, m] such that (k − k¯ )/(2m + 1 ) is an integer,

γk = αk /βk , and

Im,k = { j ∈ Z : (2m + 1 )k − m ≤ j ≤ (2m + 1 )k + m},

m,k = max{|γ j | : j ∈ Im,k }.

Clearly, k¯ = k − (2m + 1 ) j for all k ∈ Im,j . Theorem 2.1. We put

εm = max{ sup |λ−1 |, k



|k|>m

| k |∈ N

2 }. m,k

Then there exists a positive constant c such that for all m ∈ N and f ∈ λ,2 (T ) we have that

 f − Qm ( f )2 ≤ c εm  f λ,2 (T) . Proof. We define the kernel Pm (x, t) for x, t ∈ T as

 1 ϕβ (x − δm l )Hm (δm l − t ) 2m + 1 2m

Pm (x, t ) :=

l=0

and easily obtain from our definition (1.3) the equation

Qm ( f )(x ) =

1 2π



T

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

We now use Eq. (1.1), the definition of the trigonometric polynomial Hm given in Eq. (1.2) and the easily verified fact, for k, s ∈ Z, s ∈ [−m, m], that

1 2m + 1

2m 

e

ik(t−δm l ) is(δm l−t )

e

l=0

to conclude that

Pm (x, t ) =



=

⎧ ⎨0,

k−s Z ∈ 2m + 1 k−s if ∈Z 2m + 1

if

⎩ei(k−k¯ )t ,

γk eikx e−ik¯ t .

k∈Z

We again use the formula for the function ϕ λ given in Eq. (1.1) to get that

Pm (x, t ) − ϕλ (x − t ) =



¯

eikx (γk e−ikt − λ−1 e−ikt ). k

k∈Z

For |k| ≤ m we have that k = k¯ ≤ 2m and the above expression becomes

Pm (x, t ) − ϕλ (x − t ) =



|k|>m

γk eikx e−ik¯ t −



|k|>m

λ−1 eikx e−ikt . k

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From this equation and the definition of f we deduce the formula

Qm ( f )(x ) − f (x ) =





γk gˆ(k¯ )eikx −

|k|>m

|k|>m

λ−1 gˆ(k )eikx =: Am (x ) − Bm (x ). k

By the triangle inequality we have

Qm ( f ) − f 2 ≤ Am 2 + Bm 2 .

(2.4)

Parseval’s identity gives us the equation



Am 22 =

|γk |2 |gˆ(k¯ )|2

|k|>m

 

=

|γk |2 |gˆ(k − (2m + 1 ) j )|2

| j|∈N k∈Im, j

 

≤

|m, j |2 |gˆ(k − (2m + 1 ) j )|2 .

| j|∈N k∈Im, j

Hence, by our assumption on ε m and the inequality



|gˆ(k − (2m + 1 ) j )|2 ≤ g22 ,

k∈Im, j

we obtain that



Am 22 ≤

2 2 2 2 m, j g2 ≤ εm g2 .

(2.5)

| j |∈ N

Next, by using Parseval’s identity again, we have that

Bm 22 =



|k|>m

|λ−2 ||gˆ(k )|2 ≤ sup |λ−1 | k k |k|>m



|gˆ(k )|2 ≤ εm2 g22 .

|k|>m

This, together with (2.4) and (2.5), proves the theorem.



Definition 2.2. The sequence {θk : k ∈ Z} will be called a nondecreasing-type sequence if there exists a positive constant c such that θ k ≥ cθ l for all k, l ∈ Z satisfying the inequality |k| ≥ |l|.



Theorem 2.3. Let

| βk | | λk | : k ∈ Z



and {|λk | : k ∈ Z} be nondecreasing-type sequences. Then there exists a positive constant c such

that for all m ∈ N and f ∈ λ,2 (T ) we have that

 f − Qm ( f )2 ≤ c  f λ,2 (T)



λ−2 . mk

k∈N

Proof. From our hypothesis we have that |γk | = |λ−1 ||βk¯ /λk¯ ||λk /βk | ≤ c1 |λ−1 |, k k

sup

|k|>m

|λ−1 | ≤ c2 |λ−1 m | ≤ c2 k



| k |∈ N

λ−2 ≤ c3 mk



λ−2 mk

k∈N

c4 |λ−1 | mk

and m,k ≤ for all k ∈ N with some positive constants c1 , c2 , c3 and c4 . From these inequalities and Theorem 2.1, the proof of the result is complete.  From this theorem we have the following result. Corollary 2.4. Let |λk | = |βk | for all k ∈ Z, and



| λk | |k|r : k ∈ Z



be a nondecreasing-type sequence for some r >

exists a positive constant c such that for all m ∈ N and f ∈ λ,2 (T ) we have that

 f − Qm ( f )2 ≤ c |λ−1 m | f λ,2 (T ) . Proof. We see from the hypothesis that

|λmk | |λm | ≥ c |mk|r |m|r for some positive constant c and then from which it follows that

|λmk | ≥ c  kr |λm |

1 2.

Then there

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for all m, k ∈ N. Hence, we conclude that

c



λ−2 ≤ |λ−1 m | mk



k∈N

|k|−2r .

k∈N

1 2

Note that, since r >

we have



k∈N k

−2r

< ∞ and then by applying Theorem 2.3 we complete the proof.



|2

Corollary 2.5. Let |βk | = |λk for all k ∈ Z, and {|λk | : k ∈ Z} be a nondecreasing-type sequence. Then there exists a positive constant c such that for all m ∈ N and f ∈ λ,2 (T ) we have that

 f − Qm ( f )2 ≤ c  f λ,2 (T)



λ−2 . mk

k∈N

Proof. The inclusions ϕλ , ϕβ ∈ L2 (T ) and the hypothesis yield that {β k /λk }l is the nondecreasing-type sequence. Hence, by applying Theorem 2.3 we prove the corollary.  Similarly to the proof of Corollary 2.4 we can prove the following fact. Corollary 2.6. Let |βk | = |λk |2 for all k ∈ Z and



| λk | |k|r : k ∈ Z



be a nondecreasing-type sequence r >

1 2.

Then there exists a

positive constant c such that for all m ∈ N and f ∈ λ,2 (T ) we have that

 f − Qm ( f )2 ≤ c |λ−1 m | f λ,2 (T ) . Corollary 2.7. Let r > 12 and λk = βk = |k|r for all k ∈ Z, k = 0, and λ0 = 1. Then there exists a positive constant c such that for all m ∈ N and f ∈ λ,2 (T ) we have that

 f − Qm ( f )2 ≤ c m−r  f λ,2 (T) Remark 2.8. Note that under the assumptions of Corollary 2.5, Kλ (x, y ) := ϕβ (x − y ) is the reproducing kernel for the Hilbert space λ,2 (T ). This means, for every function f ∈ λ,2 (T ) and x ∈ T, we have that

f (x ) = ( f, Kλ (·, x ))λ,2 (T ) , where (·, · )λ,2 (T ) denotes the inner product on the Hilbert space λ,2 (T ). It is known that the linear span of the set of functions Kλ := {Kλ (· − y ) : y ∈ T} is dense in the Hilbert space λ,2 (T ). Under a certain restriction on the sequence λ, Corollaries 2.5, 2.6 and 2.7 give an explicit rate of the error of the linear approximation of f ∈ λ,2 (T ) by the function Qm (f) belonging Kλ . For a definitive treatment of reproducing kernels, see, for example, [3]. Corollary 2.7 has been proven as Theorem 2.10 in [10] where λ,2 (T ) is the Korobov space K2r (T ). 2.2. Error estimates for functions in the space λ,p (T ) For this purpose, we define, for m ∈ N, the quantity



εm := max



|k|>m

|(λ−1 )|, k



|γk | +

|k|>m





|γk(2m+1)+m | ,

(2.6)

k∈Z

where γ k is defined as in Theorem 2.1, and θk := θk − θk+1 for the sequence {θk : k ∈ Z}. Now, we are ready to state the following result. Theorem 2.9. If 1 < p < ∞ then there exists a positive constant c such that for all f ∈ λ,p (T ) and m ∈ N, we have that

 f − Qm ( f ) p ≤ c εm  f λ,p (T) .

(2.7)

Proof. In order to prove (2.7) we need an auxiliary result which is a direct corollary of the well-known Marcinkiewicz multiplier theorem, see, e.g., [10, Lemma 2.7]. For r, s ∈ Z and an integrable function g on T we introduce the trigonometric polynomial

Fr,s (g, x ) :=



gˆ(k )eikx .

r≤k≤s

Then there exists an absolute positive constant c such that for all g ∈ L p (T ) and r, s ∈ Z we have that

Fr,s (g) p ≤ c g p .

(2.8)

For m, k ∈ Z, let k¯ = k¯ (m, k ) be the number defined at the beginning of Section 2.1. In a completely similar way as in the proof of Theorem 2.1, we can establish the formula

Qm ( f )(x ) − f (x ) =



|k|>m

γk gˆ(k¯ )eikx −



|k|>m

λ−1 gˆ(k )eikx k

(2.9)

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for a function f ∈ λ,p (T ) represented as f = ϕλ ∗ g, g ∈ L p (T ). The next step is to decompose each sum above into two parts. Specifically, we write the first sum above as





γk gˆ(k¯ )eikx +

γk gˆ(k¯ )eikx .

(2.10)

k<−m

k>m

− + We call the the first sum in Eq. (2.10) A+ m (x ) and the other Am (x ). Now, we readily rewrite Am (x ) in the form

A+ m (x ) =

 

γk eikx gˆ(k¯ ).

(2.11)

j∈N k∈Im, j

We shall express the right hand side of Eq. (2.11) in an alternate form by using summation by parts. For this purpose, we introduce the modified difference operator defined on vectors as

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

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

A+ m (x ) =

 

γk eikx gˆ(k − j (2m + 1 ))

j∈N k∈Im, j

=

 

γk ei j(2m+1)x F−m,k− j(2m+1) (g, x ).

j∈N k∈Im, j

Consequently, according to (2.8) and the Hölder inequality, there exists a positive constant c1 such that

A+m  p ≤ c1 g p

 

|γk |.

j∈N k∈Im, j

From this inequality and the definition of ε m , given in Eq. (2.6), we conclude that

A+m  p ≤ c1 εm g p . A bound on A− m  p follows by a similar argument and yields the inequality

A−m  p ≤ c1 εm g p . There still remains the task of bounding the second sum in Eq. (2.9). As before, we split it into two parts





λ−1 eikx gˆ(k ) + k

λ−1 eikx gˆ(k ) k

k<−m

k>m

− and call the first sum above B+ m (x ) and the second sum Bm (x ). As before, summation by parts yields the alternate form

B+ m (x ) =



λ−1 F (g, x ) k m+1,k

k≥m+1

from which we deduce that

B+m (x ) p ≤



|λ−1 |g p . k

k≥m+1

Therefore, by (2.6) we obtain that

B+m (x ) p ≤ c1 εm g p and, in a similar way, we prove that

B−m  p ≤ c1 εm g p . Combining our remarks above proves the result.



Now, we are ready to state the following result. Theorem 2.10. Let 1 < p < ∞ and βk = β−k , λk = λ−k . Assume that {(log(βk /λk ))/2k : k ∈ N} and {λk : k ∈ N} are nondecreasing, positive sequences. Then there exists a positive constant c such that for all f ∈ λ,p (T ) and m ∈ N, we have that

 f − Qm ( f ) p ≤ c  f λ,p (T)



λ−1 . mk

(2.12)

k∈N

Proof. Since {λk : k ∈ N} be a nondecreasing sequence, we have that |λ−1 | = λ−1 − λ−1 for all k ∈ N and then it follows k k k+1 that



|k|>m

|λ−1 | ≤ 2λ−1 m . k

(2.13)

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From the inequalities log(βk+1 /λk+1 )/2k+1 ≥ log(βk /λk )/2k ≥ log(βk−1 /λk−1 )/2k−1 > 0 we have that

log(βk+1 /λk+1 ) ≥ 2 log(βk /λk ) ≥ log(βk /λk ) + log(βk−1 /λk−1 ) and also that

log(βk+1 /λk+1 ) − log(βk /λk ) ≥ log(βk−1 /λk−1 ) ≥ log(βk+1 /λk+1 ) − log(βk /λk ). Then, it follows from the hypothesis that log γk = log λ−1 + log(βk /λk ) − log(βk /λk ) and {λk : k ∈ N} are both nondecreasing, k positive sequences, from which we deduce for all k ∈ N that

γk+1 ≤ γk ≤ λ−1 . k Consequently, we conclude that



|γk | +

|k|>m



|γk(2m+1)+m | = 2(γm+1 +

k∈Z



|γk(2m+1)+m | ) ≤ 4

k∈N



λ−1 . mk

(2.14)

k∈N

From inequalities (2.13), (2.14) and Theorem 2.9 we confirm (2.12) which completes the proof of the theorem.



Remark 2.11. Note that for the sequence λ defined as

λj =

 | j|r if j = 0, if j = 0,

1

λ,p becomes the Korobov space K pr (T ) and from Theorem 2.10 we derive the following estimate

 f − Qm ( f ) p ≤ cm−r  f Kpr (T) , which have been proven in [10]. From Theorem 2.10 we immediately derive the following corollary. Corollary 2.12. Let 1 < p < ∞, λk = βk = e−s|k| for all k ∈ Z where s > 0. Then there exists a positive constant c such that for all f ∈ λ,p (T ) and m ∈ N, we have that

 f − Qm ( f ) p ≤ c e−sm  f λ,p (T) , and consequently,

M2m+1 (Uλ,p (T ), ϕβ ) p ≤ c e−sm . Definition 2.13. Let β > 0. A function b : R → R will be called a mask of type β if b is an even function, twice continuously differentiable such that for t > 0, b(t ) = (1 + |t | )−β Fb (log |t | ) for some function Fb : R → R, where for some constant c(b)|Fb(k ) (t )| ≤ c (b) for all t > 1 and k = 0, 1. A sequence {bk : k ∈ Z} will be called a sequence mask of type β if there exists a mask b of type β such that bk = b(k ) for all k ∈ Z.  := β −1 . In the next two theorems and their proofs we use the abbreviated notation:  λk := λ−1 and β k k k Theorem 2.14. Let 1 < p < ∞ and the sequence { λk : k ∈ Z} = {βk : k ∈ Z} be a sequence mask of type r > 1. Then there exists a positive constant c such that for all f ∈ λ,p (T ) and m ∈ N,

 f − Qm ( f ) p ≤ c m−r  f λ,p (T) . Proof. According the hypothesis we have, by Theorem 2.9, that

 f − Qm ( f ) p ≤ cεm  f λ,p (T) , where

εm =



| λk | +

k>m



| λk(2m+1)+m |.

k∈Z +

Note that for k ∈ N

| λk | = | λ (k ) −  λ(k + 1 )| = | λ ( c )|, 

where c ∈ (k, k + 1 ). Therefore, we have for k ∈ N, that

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

| λ (c )| = |(1 + c )−(r+1) ( 

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Consequently, we conclude that



| λk | ≤ 2c( λ)

k>m

r+1  r + 1 −r (1 + k )−(r+1) ≤ c( λ) m . r r

We also have that



(2.15)

k>m

| λk(2m+1)+m | ≤ c( λ)

k∈Z +

where c1 =



(1 + k(2m + 1 ) + m)−r ≤ c( λ )c1 m−r ,

(2.16)

k∈Z +



1 k∈N kr

. We complete the proof by using Eqs. (2.15) and (2.16).



Definition 2.15. A function b : R → R will be called a function of exponent-type if b is two 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} will be called a sequence mask of exponent-type if there exists a function b of exponent-type such that bk = b(k ) for all k ∈ Z. Theorem 2.16. Let 1 < p < ∞ and the sequence { λk : k ∈ Z} be a sequence mask of type r > 1, the sequence {βk : k ∈ Z} be sequence mask of exponent-type. Then there exists a positive constant c such that for all f ∈ λ,p (T ) and m ∈ N,

 f − Qm ( f ) p ≤ c m−r  f λ,p (T) . Proof. We have proven in Theorem 2.14 that



| λk | ≤ cm−r .

|k|>m

Also, we have that

F (|k| ) β |γk | = | λk k | = |(1 + k )−r Fλ (log |k| )e−s(k−k¯ ) β | Fβ (k¯ ) βk¯ ≤ |(1 + k )−r c λe−s(k−k) | ≤ ck−(r+1) , ¯

and so we obtain that



| λk | +

|k|>m



| λk(2m+1)+m | ≤ cm−r .

k∈Z

The proof is complete.



3. Multivariate approximation 3.1. Error estimates for functions in the space λ,2 (Td ) We introduce Qm (f) a linear combination of 2m + 1 translates of the function ϕβ ∈ L2 (Td ), which is defined as in (1.3), and we put

εm := max{ sup |λ−1 |; k |k|∞ ≥m



|j|∞ >0

2 }, m, j

where γk = βk−1 αk¯ and k¯ = k¯ (m, k ) is the unique vector in Zd such that |k¯ |∞ ≤ m and max |γk+(2m+1 )j |.

k j −k¯ j 2m+1

∈ Z for all j ∈ Nd , and m,j =

|k|∞ ≤m

Definition 3.1. The sequence {θk : k ∈ Zd } will be called a non decreasing-type sequence if θ k ≥ cθ l for all k, l ∈ Zd satisfying the inequalities |kj | ≥ |lj |, j ∈ Nd . Theorem 3.2. There exists a positive constant c such that for all f ∈ λ,2 (Td ) and m ∈ N, we have that

 f − Qm ( f )2 ≤ cεm  f λ,2 (Td ) . Proof. Let us give a sketch of a proof which generalizes the proof of Theorems 2.1. 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∈[0,2m]d

and easily obtain from our definition (1.3) the equation

Qm ( f )(x ) =

1

( 2π )d



Td

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

(3.1)

D. D˜ ung, C.A. Micchelli and V.N. Huy / Applied Mathematics and Computation 361 (2019) 777–787

785

We now use Eq. (1.1), the definition of the trigonometric polynomial Hm given in Eq. (1.2) and the easily verified fact, for k, s ∈ Zd , s ∈ [−m, m]d , that



1 ( 2m + 1 )d

eik(t−δm l ) eis(δm l−t ) =

l∈[0,2m]d

to conclude that



Pm (x, t ) =

⎧ ⎪ ⎨0,

kj − sj Z for some j ∈ [1, d] ∈ 2m + 1 kj − sj if ∈ Z for all j ∈ [1, d] 2m + 1

if

⎪ ⎩ei(k−k¯ )t ,

γk eikx e−ikt .

(3.2)

k∈Z d

where γk := βk−1 αk¯ , k¯ ∈ [−m, m]d . We again use this formula and the formula for the function ϕ λ,d given in Eq. (1.1) and the equation to get that

Qm ( f )(x ) − f (x ) = Am (x ) − Bm (x ), where



Am (x ) := k∈Z d

and

\

γk gˆ(k¯ )eikx

[−m,m]d



Bm (x ) :=

(3.3)

k∈Zd \[−m,m]d

λ−1 gˆ(k )eikx . k

By the triangle inequality we have

Qm ( f ) − f 2 ≤ Am 2 + Bm 2 .

(3.4)

The norm Am 2 can be estimated as





Am 22 ≤

|m,j |2 |gˆ(k )|2 ,

|j|∞ >0 k∈[−m,m]d

Hence, by the hypothesis of ε m and the inequality



|gˆ(k )|2 ≤ g22 ,

k∈[−m,m]d

we obtain



Am 22 ≤

2 2 2 2 m, j g2 ≤ εm g2 .

(3.5)

|j|∞ >0

Next, for the norm Bm 2 we have that



Bm 22 =

k∈Zd \[−m,m]d

|λ−2 ||gˆ(k )|2 ≤ k

sup

k∈Zd \[−m,m]d

|λ−2 | k

This together with (3.4) and (3.5) proves the theorem.



|gˆ(k )|2 ≤ εm2 g22 .

k∈Zd \[−m,m]d



In a similar way to the proof of Theorem 2.1 we can prove the following theorem. Theorem 3.3. Let {|βk /λk | : k ∈ Zd } and {|λk | : k ∈ Zd } be non decreasing-type sequence. Then there exists a positive constant c such that for all m ∈ N and f ∈ λ,2 (Td ) we have that

 f − Qm ( f )2 ≤ c  f λ,2 (Td )  Corollary 3.4. Let

| βk | d |k|r∞ |λk | : k ∈ Z

such that for all m ∈ N and f ∈ λ,2





|k|∞ >0

λ−2 . mk

be a non decreasing-type sequence for some r >

( Td )

we have that

 f − Qm ( f )2 ≤ c sup |λ | f λ,2 (Td ) . |k|∞ >m

−1 k

Proof. We see from the hypothesis that for all k with |k|∞ ≤ m, we have that

|βk+(2m+1)j | |βk | ≤ c1 |k|r∞ |λk | |k + (2m + 1 )j|r∞ |λk+(2m+1)j |

d 2.

Then there exists a positive constant c

D. D˜ ung, C.A. Micchelli and V.N. Huy / Applied Mathematics and Computation 361 (2019) 777–787

786

and then it follows that

|λk+(2m+1)j | |βk | ≤ c1 sup |λ−1 ||j|−r ∞. k |βk+(2m+1)j | |λk | |k|∞ >m

|γk+(2m+1)j | = |λ−1 | k+(2m+1 )j Hence, we conclude that

m,j ≤ c1 sup |λ−1 ||j|−r ∞ k |k|∞ >m

and so



−1 2 ≤c m, 1 sup |λk | j

|k|∞ >m

|j|∞ >0

Note that for 2r > d the series

 |j|∞ >0

∞

j=1

r −1 |j|−2 ∞ ≤ c1 sup |λk |

jd−1 j−2r

|k|∞ >m

∞ 

jd−1 j−2r .

j=1

is convergent which completes the proof of the corollary.

Corollary 3.5. Let βk = λ2k for all k ∈ Zd , and the sequence



| λk | d |k|r : k ∈ Z

exists a positive constant c such that for all m ∈ N and f ∈ λ,2

( Td )





be non decreasing-type for some r >

d 2.

Then there

we have that

 f − Qm ( f )2 ≤ c sup |λ | f λ,2 (Td ) . −1 k

|k|∞ >m

Corollary 3.6. Let βk = αk for all k ∈ Zd ; the sequence



a positive constant c such that for all m ∈ N and f ∈ λ,2

| λk | d |k|r : k ∈ Z

( Td )



is nondecreasing-type for some r >

d 2.

Then there exists

we have that

 f − Qm ( f )2 ≤ c sup |λ−1 | f λ,2 (Td ) . k |k|∞ >m

3.2. Convergence rate Theorem 3.7.  Let  : R+ → R+ be a nondecreasing function such that  (2t) ≤ c (t) for all t > 1. If λk = (|k|2 ) for all k ∈ Zd , k21 + · · · + k2d , then there exist positive constants c and c such that for all n ∈ N,

where |k|2 :=

c /((n log n )1/d ) ≤ Mn (Uλ,2 )2 ≤ c /(n1/d ). Proof. Let s be any natural number and set

s∗ = |{k : k ∈ Zd ,

|k|2 ≤ s}|

where |A| denoted by the number of elements of a finite set A. Then, for some positive constants c1 and c2 , we have that

c1 sd ≤ s∗ ≤ c2 sd .

(3.6)

Let n be any natural number satisfying n ≥ 10. We define the positive integers

m = c3 n log n + 1 and s = s(n ) as the largest natural number satisfying the inequality

m ≥ s∗ , where c3 =



4ec1 . Then it follows from (3.6) that there exists a positive number c4 independent of n such that

s ≤ c4 (n log n )1/d .

(3.7)

We consider the set of trigonometric polynomials

Fn,s =

  ω k eikx :

 |k | = 1, |k|2 ≤ s ,

|k|2 ≤s

where

  ω = m−1/2 max |λk | −1 . |k|2 =s

Let

f (x ) =

 |k|2 ≤s

fˆk eikx

D. D˜ ung, C.A. Micchelli and V.N. Huy / Applied Mathematics and Computation 361 (2019) 777–787

be any polynomial from Fn,s . Since

 f  λ = ω

 

λ2k



1/2

≤ ω max |λk | |k|2 =s

|k|2 ≤s

   1/2 1

|k|2 ≤s

 

= ω max |λk | s∗

1/2

|k|2 =s

787

≤ 1,

f belongs to Uλ,2 and consequently, Fn,s ⊂ Uλ,2 . Take an arbitrary function ψ from L2 (Td ). Then it follows from a result in [16] that there exists a function f∗ ∈ Fn,s and a positive number c5 such that for any linear combination of translates of ψ

h (x ) =

n 

bl ψ ( x − al ),

l=1

we have that

   f ∗ − h2 ≥ c5 ωm1/2 ≥ c5 max |λk | −1 . |k|2 ≤s

Therefore, by using (3.7) and  (2t) ≤ c (t) for all t > 1, there exists a positive constant c independent of n such that

Mn (Uλ,2 )2 ≥ Mn (Fn,s )2 ≥  f ∗ − h2



≥ c5 max |λk | |k|2 ≤s



−1

≥ c5 ((s ))−1 ≥ c /((n log n )1/d )

which proves the lower bound of the theorem. By using Corollary 3.4 for βk = |k|d∞ |λk | and m = (2u + 1 )d < n, there exists a positive constant c6 independent of n such that

Mn (Uλ,2 )2 ≤ Mm (Uλ,2 )2 ≤ Mm (Uλ,2 , ψ )2 ≤ c6 sup where ψ (x ) =

 k∈Z d

Mn (Uλ,2 )2 ≤ c6

√

|k|∞ >u

|λ−1 |, k

λk eikx and u =  d n/2 − 1. That gives us the inequality sup

k∈Zd \[−u,u]d

|λ−1 | k

and then we get that

Mn (Uλ,2 )2 ≤ c6 ((u ))−1 . Hence, by using  (2t) ≤ c (t) for all t > 1, we obtain the remaining desired result

Mn (Uλ,2 )2 ≤ c /(n1/d ). The proof is complete.



Acknowledgments ˜ Dinh Dung’s research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 102.01–2017.05. Charles Micchelli wishes to acknowledge partial support from NSF, US under Grant DMS 1522339. In addition, the authors thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for providing a fruitful research environment and working condition. References [1] G. Allasia, R. Cavoretto, A. De Rossi, Numerical integration on multivariate scattered data by lobachevsky spline, Int. J. Comput. Math. 90 (2013) 2003–2018. [2] G. Allasia, R. Cavoretto, A. De Rossi, Lobachevsky spline functions and interpolation to scattered data, Comput. Appl. Math. 32 (2013) 71–87. [3] N. Aronszajn, Theory of reproducing kernels, Trans. Am. Math. Soc. 68 (1950) 337–404. [4] R. Bergmann, J. Prestin, Multivariate anisotropic interpolation on the torus, in: Approximation Theory XIV: San Antonio 2013, in: Springer Proceedings in Mathematics & Statistics, 83, Springer, 2014, pp. 27–44. [5] C. de Boor, R. DeVore, A. Ron, Approximation from shift-invariant subspaces of L2 (Rd ), Trans. Am. Math. Soc. 341 (1994) 787–806. [6] C. de Boor, A. Ron, Fourier analysis of the approximation power of principal shift-invariant spaces, Constr. Approx. 8 (1992) 427–462. ˜ , B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness, J. Complex. 27 (2011) 541–567. [7] D. Dung ˜ , Optimal adaptive sampling recovery, Adv. Comput. Math. 34 (2011) 1–41. [8] D. Dung ˜ , Sampling and cubature on sparse grids based on a b-spline quasi-interpolation, Found. Comput. Math. 16 (2016) 1193–1240. [9] D. Dung ˜ , C.A. Micchelli, Multivariate approximation by translates of the Korobov function on Smolyak grids, J. Complex. 29 (2013) 424–437. [10] D. Dung ˜ , C.A. Micchelli, Corrigendum to “multivariate approximation by translates of the Korobov function on smolyak grids”, J. Complex. 35 (2016) [11] D. Dung 124–125. [J. Complexity 29 (2013) 424–437]. ˜ [12] D. Dung, V.N. Temlyakov, T. Ullrich, Hyperbolic Cross Approximation, in: Advanced Courses in Mathematics–CRM Barcelona, Springer, Birkhäuser Basel, 2018. [13] K. Jetter, D.X. Zhou, Order of linear approximation from shift-invariant spaces, Constr. Approx. 11 (1995) 423–438. [14] Y. Jiang, Y.S. Xu, B-spline quasi-interpolation on sparse grids, J. Complex. 27 (2011) 466–488. [15] G.C. Kyriazis, Approximation from shift-invariant spaces, Constr. Approx. 11 (1995) 141–164. [16] V. Maiorov, Almost optimal estimates for best approximation by translates on a torus, Constr. Approx. 21 (2005) 1–20. [17] H.N. Mhaskar, J. Prestin, Polynomial frames: a fast tour, Approximation Theory XI: Gatlinburg 2004, Nashboro Press, Brentwood, TN, 2005, pp. 287–318. [18] C.A. Micchelli, Y. Xu, H. Zhang, Universal kernels, J. Mach. Learn. Res. 7 (2006) 2651–2667. [19] L.L. Schumaker, Spline Functions: Computational Methods, SIAM, 2015. [20] F. Sprengel, A class of periodic function spaces and interpolation on sparse grids, Numer. Func. Anal. Optim. 21 (20 0 0) 273–293. [21] H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005.