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

DSpace at VNU: Multivariate approximation by translates of the Korobov function on Smolyak grids tài liệu, giáo án, bài...

Journal of Complexity 29 (2013) 424–437 Contents lists available at ScienceDirect 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 ), < 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 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 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 ), ≤ 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: dinhzung@gmail.com (D Dung) 0885-064X/$ – see front matter © 2013 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.jco.2013.06.002 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 = 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 > 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) :=  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 ≤ 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 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) ̸= 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 = 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 ) , where (·, ·)K r (Td ) denotes the inner product on the Hilbert space K2r (Td ) For a definitive treatment of 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 ), < 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 < p < ∞, r > 1, p ̸= 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 < p < ∞, p ̸= 2, r > 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 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 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 < 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 +   κr · − 2π l 2m +  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 428 D D˜ ung, C.A Micchelli / Journal of Complexity 29 (2013) 424–437 Theorem 2.3 If < 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) =   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 ∗ f = 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 =  2m + l∈Z [m+1] Sm (g ) 2π l   2m +  θm · − 2π l  2m + , (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 ) =  2m + l∈Z [2m+1] Sm ( g ) 2π l  2m +   (Sm κr ) · − 2π l 2m +  (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 + 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 ≤ p ≤ ∞ we define the lp (A)-norm of a by ∥a∥p,A :=       1/p p |al | , ≤ p < ∞, l∈A sup{|al | : l ∈ A}, p = ∞ D D˜ ung, C.A Micchelli / Journal of Complexity 29 (2013) 424–437 429 Also, we introduce the mapping Wm : Tm → R2m defined at f ∈ Tm as Wm (f ) = 2π l   f  2m +  : l ∈ Z [2m + 1] Lemma 2.5 If < 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 = 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 < 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 :=  2m + l∈Z [2m+1] Sm (g )  2π l 2m +  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 +   =   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 + 2π ilj  = gˆ (j)e 2m+1 |j|∈Z [m+1] Hence,  bj = gˆ (j), gˆ (j − 2m − 1), ≤ j ≤ m, m + ≤ j ≤ 2m (2.10) We decompose the set Z¯ [m] as a disjoint union of finite sets each containing 2m + 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˜ ung, C.A Micchelli / Journal of Complexity 29 (2013) 424–437 431 We combine this inequality with inequalities (2.11) and r > 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 < p < ∞ and r > 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 < p ≤ ∞ and r > However, this would go beyond the scope of this paper We end this section by providing an improvement of Theorem 2.3 when p = 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 : 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 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]) 432 D D˜ 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 < p < ∞, p ̸= 2, r > 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 = it follows from Theorems 2.3 and 2.10 Assume that the lemma is true for d − 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 < p < ∞, p ̸= 2, r > 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 − 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 < p < ∞, p ̸= 2, r > 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 ) 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 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 < p < ∞, p ̸= 2, r > 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 | = 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 , 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 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  |λk |2 , k∈H(q) where λk was defined earlier before Definition 1.1 Since ζ2  |λk |2 ≤ ζ 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 = 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) L2 (Td ) (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 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Kpr (T) by linear combinations of translates of the Korobov function κr and prove an upper bound for the approximation error... 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(·