Multivariate approximation by translates of tensor product kernel on Smolyak grids

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 ddimensional torus denoted by T d 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 dtorus by T. Functions on T d are identified with functions on R d which are 2π periodic in each variable. We shall denote by Lp(T d ), 1 ≤ p < ∞, the space of integrable functions on T d equipped with the norm

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: dinhzung@gmail.com. 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|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 km 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. 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Zygmund, Trigonometric Series, Third edition, Volume I & II combined, Cambridge University Press, 2002. 25 [...]... 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... Journal of Complexity, 27(6)(2011), 541–567 [13] Dinh D˜ ung and T Ullrich, n-Widths and ε-dimensions for high dimensional sparse approximations, Foundations Comp Math (2013), http://dx.doi.org/10.1007/s10208-013-9149-9 [14] Dinh Dung and Charles A Michelli, Multivariate approximation by translates of the Korobov function on Smolyak grids 24 [15] W.J Gordon, Blending function methods of bivariate and multivariate. .. 53-64, 1977 [7] Dinh D˜ ung, Number of integral points in a certain set and the approximation of functions of several variables, Math Notes, 36(1984), 736-744 [8] Dinh D˜ ung, Approximation of functions of several variables on tori by trigonometric polynomials, Mat Sb 131(1986), 251-271 [9] Dinh D˜ ung, On interpolation recovery for periodic functions, In: Functional Analysis and Related Topics (Ed... 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... interpolation and approximation, SIAM Numerical Analysis, Vol 8, 158-177, 1977 [16] M Griebel and S Knapek, Optimized general sparse grid approximation spaces for operator equations, Math Comp., 78(2009)(268),2223–2257 [17] V Maiorov, On best approximation by radial functions, J Approx Theory 120(2003), 36–70 [18] V Maiorov, Almost optimal estimates for best approximation by translates on a torus, Constructive... thank VIASM for providing a fruitful research environment and working condition References [1] N Aronszajn, Theory of reproducing kernels, Trans Amer Math Soc 68(1950), 337-404 [2] N K Bari, A Treatise on Trigonometric Series, Volume II, The Macmillian Company, 1964 [3] K.I Babenko, On the approximation of periodic functions of several variables by trigonometric polynomials, Dokl Akad Nauk USSR 132(1960),... ung, On optimal recovery of multivariate periodic functions, In: Harmonic Analysis (Conference Proceedings, Ed S Igary), Springer-Verlag 1991, Tokyo-Berlin, pp 96-105 [11] Dinh D˜ ung, Non-linear approximation using sets of finite cardinality or finite pseudo-dimension, J of Complexity 17(2001), 467- 492 [12] Dinh D˜ ung, B-spline quasi-interpolant representations and sampling recovery of functions... Ullrich, Spline interpolation on sparse grids, Applicable Analysis 90(2011), 337-383 [26] S.A Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl Akad Nauk 148(1963), 1042–1045 [27] V Temlyakov, Approximation of periodic functions, Nova Science Publishers, Inc., New York, 1993 [28] G Wahba, Interpolating surfaces: High order convergence rates and their... Tractability of Multivariate Problems, Volume I: Linear Information, EMS Tracts in Mathematics, Vol 6, Eur Math Soc Publ House, Z¨ urich 2008 [23] W Sickel and T Ullrich, The Smolyak algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness, East J Approx 13(4)(2007), 387–425 [24] W Sickel and T Ullrich, Tensor products of Sobolev-Besov spaces and applications to approximation from... Meir, Lower bounds for multivariate approximation by affine-invariant dictionaries, IEEE Transactions on Information Theory 47(2001), 1569–1575 [20] C.A Micchelli and G Wahba, Design problems for optimal surface interpolation, in Approximation Theory and Applications, Z.Ziegler (ed.), 329-348, Academic Press, 1981 [21] C A Micchelli, Y Xu, H Zhang, Universal kernels, Journal of Machine Learning Research ... Michelli, Multivariate approximation by translates of the Korobov function on Smolyak grids 24 [15] W.J Gordon, Blending function methods of bivariate and multivariate interpolation and approximation, ... 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... 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