Available online at www.sciencedirect.com ScienceDirect Journal of Approximation Theory 207 (2016) 207–231 www.elsevier.com/locate/jat Full length article Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in H γ Glenn Byrenheid a , Dinh D˜ung b,∗ , Winfried Sickel c , Tino Ullrich a a Hausdorff-Center for Mathematics, 53115 Bonn, Germany b Information Technology Institute, Vietnam National University, 144, Xuan Thuy, Hanoi, Viet Nam c Friedrich-Schiller-University Jena, Ernst-Abbe-Platz 2, 07737 Jena, Germany Received 13 November 2014; received in revised form January 2016; accepted 11 February 2016 Available online March 2016 Communicated by Hans G Feichtinger Abstract We investigate the rate of convergence of linear sampling numbers of the embedding H α,β (Td ) ↩→ H γ (Td ) Here α governs the mixed smoothness and β the isotropic smoothness in the space H α,β (Td ) of hybrid smoothness, whereas H γ (Td ) denotes the isotropic Sobolev space If γ > β we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on “energy-norm based sparse grids” for the classical trigonometric interpolation This complements earlier work by Griebel, Knapek and D˜ung, Ullrich, where general linear approximations have been considered In addition, we α (Td ) ↩→ H γ (Td ) and achieve optimality for Smolyak’s algorithm applied study the embedding Hmix mix to the classical trigonometric interpolation This can be applied to investigate the sampling numbers for α (Td ) ↩→ L (Td ) for < q ≤ ∞ where again Smolyak’s algorithm yields the the embedding Hmix q optimal order The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation β = γ (including the embedding into L (Td )) c 2016 Elsevier Inc All rights reserved ⃝ ∗ Corresponding author E-mail address: dinhzung@gmail.com (D D˜ung) http://dx.doi.org/10.1016/j.jat.2016.02.012 c 2016 Elsevier Inc All rights reserved 0021-9045/⃝ 208 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Introduction The efficient approximation of multivariate functions is a crucial task for the numerical treatment of several real-world problems Typically the computation time of approximating algorithms grows dramatically with the number of variables d Therefore, one is interested in reasonable model assumptions and corresponding efficient algorithms In fact, a large class of solutions of the electronic Schrăodinger equation in quantum chemistry does not only belong to Sobolev spaces with mixed regularity, one also knows additional information in terms of isotropic smoothness properties, see Yserentant’s recent lecture notes [37] and the references therein This type of regularity is precisely expressed by the spaces H α,β (Td ), defined in Section Here, the parameter α reflects the smoothness in the dominating mixed sense and the parameter β reflects the smoothness in the isotropic sense We aim at approximating such functions in an energytype norm, i.e., we measure the approximation error in an isotropic Sobolev space H γ (Td ) This is motivated by the use of Galerkin methods for the H (Td )-approximation of the solution of general elliptic variational problems see, e.g., [1,2,17,15,18,25] The present paper can be seen as a continuation of [12], where finite-rank approximations in the sense of approximation numbers were studied The latter are defined as am (T : X → Y ) := inf A:X →Y rank A≤m sup ∥T f − A f ∥Y , m ∈ N, ∥ f ∥ X ≤1 where X, Y are Banach spaces and T ∈ L(X, Y ), where L(X, Y ) denotes the space of all bounded linear operators T : X → Y In contrast to that, we restrict the class of admissible algorithms even further in this paper and deal with the problem of the optimal recovery of H α,β functions from only a finite number of function values, where the optimality in the worst-case setting is commonly measured in terms of linear sampling numbers m      gm (T : X → Y ) := inf inf sup T f − f (x )ψ (·)   , m ∈ N j j m (x j )mj=1 ⊂Td (ψ j ) j=1 ⊂Y ∥ f ∥ X ≤1 j=1 Y Here, X ⊂ C(Td ) denotes a Banach space of functions on Td and T ∈ L(X, Y ) The inclusion of X in C(Td ) is necessary to give a meaning to function evaluations at single points x j ∈ Td In what follows, we somewhere use the abbreviations am (T ) := am (T : X → Y ) and gm (T ) := gm (T : X → Y ) if X and Y are already defined We will mainly focus on the situation X = H α,β (Td ) and Y = H γ (Td ) The condition α > γ − β ensures a compact embedding I1 : H α,β (Td ) → H γ (Td ) (1.1) such that we can ask for the asymptotic decay of the sampling numbers gm (I1 : H α,β (Td ) → H γ (Td )) in m By investing more isotropic smoothness γ ≥ in the target space H γ (Td ) than β ∈ R in the source space H α,β we encounter two surprising effects for the sampling numbers gm (I1 ) if γ > β The main result of the present paper is the following asymptotic order gm (I1 ) ≍ am (I1 ) ≍ m −(α+β−γ ) , m ∈ N, which shows, on the one hand, the asymptotic equivalence to the approximation numbers and, on the other hand, the purely polynomial decay rate, i.e., no logarithmic perturbation (see G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 209 Theorem 6.7) The asymptotic behavior of the approximation numbers am (I1 : H α,β (Td ) → H γ (Td )) (including the dependence of all constants on d) has been determined in [12], see also [5,20] The present paper is intended as a continuation of [12] for the sampling recovery problem For the non-periodic situation and more general spaces we refer to the recent paper [10] See also [11] for a survey on results and bibliography on sampling recovery on sparse grids of functions having a mixed and more generally, an anisotropic mixed smoothness In the critical cases, i.e., γ = β ≥ 0, we are currently not able to give the precise decay rate of gm (I2 : H α,β (Td ) → H β (Td )) (1.2) although we are dealing with a Hilbert space setting and additional smoothness in the target space However, the following statement is true for α > 1/2 (see Theorem 6.11) m −α (log m)(d−1)α ≍ am (I2 ) ≤ gm (I2 ) m −α (log m)(d−1)(α+1/2) , ≤ m ∈ N Note, that if γ = β = this includes the classical problem of finding the correct asymptotic behavior of the sampling numbers for the embedding α (Td ) → L (Td ), I3 : Hmix α (Td ) denotes the Sobolev space of dominating mixed fractional order α > 1/2 where Hmix All our proofs are constructive We explicitly construct sequences of sampling operators that yield the optimal approximation order Let us briefly describe the framework The sampling operators will be appropriate sums of tensor products of the classical univariate trigonometric interpolation 2m  f (tℓm ) Dm (t − tℓm ), (1.3) 2m + ℓ=0  where Dm (t) := |k|≤m eikt = sin((m + 1/2)t)/ sin(t/2) denotes the Dirichlet kernel and tℓm are its zeros It is well-known that Jm f −−−−→ f in L (T) for every f ∈ H s (T) with s > 1/2 Jm f (t) := m→∞ Due to a telescoping series argument we may also write f = J1 f + ∞  (J2k − J2k−1 ) f k=1 We put for k ∈ N0  J k − J2k−1 ηk := J1 if k > 0, if k = We define qk := ηk1 ⊗ · · · ⊗ ηkd , k ∈ Nd0 , (1.4) via the usual tensorization procedure This construction allows for proving  the useful tool of a sampling representation of functions f ∈ H α,β (Td ) by the series f = k∈Nd qk ( f ) with the equivalent norm  1 ∥ f |H α,β (Td )∥+ = 22(α|k|1 +β|k|∞ ) ∥qk ( f )∥22 k∈Nd0 210 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 which is generated by function evaluations of f only (see Theorem 3.6 for details) Finally, for a given finite ∆ ⊂ Nd0 we define the general sampling operator Q ∆ as  Q ∆ := qk (1.5) k∈∆ Our degree of freedom will be the set ∆ We will choose ∆ according to the different situations we are dealing with That means in particular that ∆ may depend on the parameters of the function classes of interest The most interesting case is represented by the index set ∆(ξ ) = ∆(α, β, γ ; ξ ) := {k ∈ Nd0 : α|k|1 − (γ − β)|k|∞ ≤ ξ }, ξ > 0, (1.6) or by an ε-modification of it given by ∆ε (ξ ) = ∆(ε, α, β, γ ; ξ ) := {k ∈ Nd0 : (α − ε) |k|1 − (γ − β − ε)|k|∞ ≤ ξ }, ξ > 0, (1.7) and ε > chosen smaller than min{α, γ − β} These index sets will be used in connection with the embedding (1.1) The set of sampling points used by (1.5) will be called “energy-norm based sparse grid” (see Figs and 2) Fig d = 2, α = 2, β = 0, γ = 1, ξ = 20 Fig d = 2, α = 1, ξ = 20 This phrase stems from the works of Bungartz, Griebel and Knapek [1,2,15,17,18] and refers to the special case where the error is measured in the “energy space” H (Td ) These authors were the first observing the potential of this modification of the classical Smolyak sparse grid [32] The index set (1.6) has been considered for approximation numbers in [12], and the index sets (1.6) and (1.7) for sampling numbers in the recent paper [10] Note, that the approximation scheme in this paper is build upon the classical trigonometric interpolation, see (1.3), using the Dirichlet kernel itself Hence, the sets of equidistant interpolation nodes in (1.3) are in general not nested For more comments in this direction and how to modify the setting to guarantee a nestedness property we refer to Remark 4.4 α (Td ) The paper is organized as follows In Section we define and discuss the spaces Hmix α,β d and H (T ) Section is used to establish our main tool in all proofs involving sampling numbers, the so-called “sampling representation”, see Theorem 3.6 The next Section deals in a constructive way with estimates from above for the sampling numbers of the embedding (1.1) by evaluating the error norm ∥I − Q ∆ ∥ with the corresponding ∆ from (1.7) We deal with the limiting cases (1.2) leading to the classical Smolyak algorithm in Section In Section we G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 211 transfer our approximation results into the notion of sampling numbers and compare them to existing estimates for the approximation numbers Notation As usual, N denotes the natural numbers, N0 the non-negative integers, Z the integers and R the real numbers With T we denote the torus represented by the interval [0, 2π] The letter d is always reserved in Zd , Rd , Nd , and Td For < p ≤ ∞ and dfor the pdimension d 1/ p x ∈ R we denote |x| p = ( i=1 |xi | ) with the usual modification for p = ∞ We write  e j , j = 1, , d, for the respective canonical unit vector and 1¯ := dj=1 e j in Rd If X and Y are two Banach spaces, the norm of an operator A : X → Y will be denoted by ∥A : X → Y ∥ The symbol X ↩→ Y indicates that there is a continuous embedding from X into Y The relation an bn means that there is a constant c > independent of the context relevant parameters such that an ≤ cbn for all n belonging to a certain subset of N, often N itself We write an ≍ bn if an bn and bn an holds Sobolev-type spaces In this section we recall the definition of the function spaces under consideration here They α (Td ) are all of Sobolev-type In a first subsection we consider the periodic Sobolev spaces Hmix of dominating mixed fractional order α > In the second subsection the more general classes H α,β (Td ) are discussed 2.1 Periodic Sobolev spaces of mixed and isotropic smoothness All results in this paper are stated for function spaces on the d-torus Td , which is represented in the Euclidean space Rd by the cube Td = [0, 2π ]d , where opposite faces are identified The space L (Td ) consists of all (equivalence classes of) measurable functions f on Td such that the norm  1/2 ∥ f ∥2 := | f (x)|2 d x Td is finite All information on a function f ∈ L (Td ) is encoded in the sequence (ck ( f ))k of its Fourier coefficients, given by  f (x) e−ikx d x, k ∈ Zd ck ( f ) := (2π )d Td Indeed, we have Parseval’s identity  ∥ f ∥22 = (2π )d |ck ( f )|2 k∈Zd as well as f (x) =  ck ( f ) eikx k∈Zd with convergence in L (Td ) α (Td ) of dominating mixed Definition 2.1 Let α ≥ The periodic Sobolev space Hmix d smoothness α is the collection of all f ∈ L (T ) such that d    α 1/2 ∥ f ∥#H α (Td ) := |ck ( f )|2 + |k j |2 < ∞ mix k∈Zd j=1 212 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 We also need the (isotropic) Sobolev spaces H γ (Td ) Definition 2.2 Let γ ≥ The periodic Sobolev space H γ (Td ) of smoothness γ is the collection of all f ∈ L (Td ) such that   γ 1/2 |ck ( f )|2 + |k|22 < ∞ ∥ f ∥#H γ (Td ) := k∈Zd Remark 2.3 It is elementary to check α (Td ) ↩→ H α (Td ) H αd (Td ) ↩→ Hmix In addition it is known that H γ (Td ) ↩→ C(Td ) if and only if H γ (Td ) ↩→ L ∞ (Td ) if and only if γ > d/2, see [29] 2.2 Hybrid type Sobolev spaces α (Td ) obtained by adding isotropic To define the scale H α,β (Td ) we look for subspaces of Hmix smoothness This motivates the following definition Definition 2.4 Let α ≥ and β ∈ R such that α + β ≥ The generalized periodic Sobolev space H α,β (Td ) is the collection of all f ∈ L (Td ) such that ∥ f ∥#H α,β (Td ) :=  |ck ( f )|2 k∈Zd d   + |k j |2 α  (1 + |k|22 )β 1/2 < ∞ j=1 α,0 d α (Td ) and H d β d Remark 2.5 (i) Obviously we have Hmix (T ) = Hmix mix (T ) = H (T ), β ≥ More important for us will be the embedding 0,β H α,β (Td ) ↩→ H γ (Td ) if ≤ γ ≤ α + β (ii) Spaces of such a type have been first considered by Griebel and Knapek [17] Also in the non-periodic context they play a role in the description of the fine regularity properties of certain eigenfunctions of Hamilton operators in quantum chemistry, see [37] The periodic α,β spaces Hmix (Td ) also occur in the recent works [12,16] A first step towards the sampling representation in Theorem 3.6 will be the following equivalent characterization of Littlewood–Paley type We will work with the dyadic blocks on the Fourier side As usual, δℓ ( f ), ℓ ∈ Nd0 , represents that part of the Fourier series of f supported in a dyadic block Pℓ := Pℓ1 × · · · × Pℓd , where P j := {ℓ ∈ Z : ≤ |ℓ| <  δℓ ( f ) := ck ( f ) eikx j−1 (2.1) 2j} and P0 = {0} In other words, k∈Pℓ Hence, for all f ∈ L (Td ) we have the Littlewood–Paley decomposition  f = δℓ ( f ) ℓ∈Nd0 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 213 Lemma 2.6 Let α ≥ and β ∈ R such that α + β ≥ Then   1/2  H α,β (Td ) = f ∈ L (Td ) : ∥ f ∥ H α,β (Td ) := 22(α|k|1 +β|k|∞ ) ∥δk ( f )∥22 (independent of g and ℓ) such that |ℓ|1 ( 1p − q1 ) ∥g∥q ≤ C2 ∥g∥ p holds for every g ∈ T ℓ and every ℓ ∈ Nd0 Proof A proof can be found in [23, Theorem 3.3.2] To give a meaning to point evaluations of functions it is essential that the spaces under consideration contain only continuous functions To be more precise, they contain equivalence classes of functions having one continuous representative Theorem 2.8 Let α > 0, β ∈ R such that min{α + β, α + βd } > 21 Then H α,β (Td ) ↩→ C(Td ) Proof Applying Lemma 2.7 yields   ∥δk ( f )∥∞ = 2α|k|1 +β|k|∞ 2−(α|k|1 +β|k|∞ ) ∥δk ( f )∥∞ k∈Nd0 k∈Nd0  2α|k|1 +β|k|∞ 2−(α|k|1 +β|k|∞ ) |k|1 ∥δk ( f )2 kNd0 Employing Hăolders inequality we find 1   1  2 ∥δk ( f )∥∞ ≤ 2−2(α|k|1 +β|k|∞ ) 2|k|1 22(α|k|1 +β|k|∞ ) ∥δk ( f )∥2 k∈Nd0 k∈Nd0 ≤  k∈Nd0 k∈Nd0 2−2(α|k|1 +β|k|∞ ) 2|k|1 1 ∥ f ∥ H α,β (Td ) 214 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Using |k|∞ ≤ |k|1 ≤ d|k|∞ gives in case β ≥   β 2−2(α|k|1 +β|k|∞ ) 2|k|1 ≤ 2−2(α+ d − )|k|1 < ∞, k∈Nd0 k∈Nd0 whenever α + βd > 12 For the case β < observe that   2−2(α|k|1 +β|k|∞ ) 2|k|1 ≤ 2−2(α+β− )|k|1 < ∞ k∈Nd0 k∈Nd0 if α + β > 12 Since C(Td ) is a Banach space, the sum absolute convergence Further  f = δk ( f )  k∈Nd0 δk ( f ) belongs to C(Td ) due to its k∈Nd0 holds in L (Td ) Consequently, the equivalence class f representative ∈ H α,β (Td ) has a continuous Remark 2.9 (i) With essentially the same proof technique as above the assertion in Theorem 2.8 can be refined as follows Let α ≥ and β ∈ R such that α + β ≥ Then it holds the embedding  α+β/d Hmix (Td ) : β ≥ 0, H α,β (Td ) ↩→ α+β Hmix (Td ) : β < This embedding immediately implies Theorem 2.8 (ii) The restrictions in Theorem 2.8 are almost optimal Indeed, let g ∈ H α+β (T), then the function f (x1 , , xd ) := g(x1 ), x ∈ Rd , belongs to H α,β (Td ) Hence, from H α,β (Td ) ↩→ C(Td ) we derive H α+β (T) ↩→ C(T) which is known to be true if and only if α + β > 1/2 In case α = we know H α,β (Td ) = H β (Td ) Hence, H 0,β ↩→ C(Td ) if and only if β/d > 1/2 We will need the following Bernstein type inequality Lemma 2.10 Let min{α, α + β − γ } > and ℓ ∈ Nd0 Then ∥ f ∥ H α,β (Td ) ≤ 2α|ℓ|1 +(β−γ )|ℓ|∞ ∥ f ∥ H γ holds for all f ∈ T ℓ Proof Indeed, for f ∈ T ℓ , we have  ∥ f ∥2H α,β (Td ) = 22(α|k|1 +β|k|∞ ) ∥δk ( f )∥22 ki ≤ℓi i=1, ,d ≤ max 22(α|k|1 +(β−γ )|k|∞ ) ki ≤ℓi i=1, ,d  ki ≤ℓi i=1, ,d ≤ 22(α|ℓ|1 +(β−γ )|ℓ|∞ ) ∥ f ∥2H γ 22γ |k|∞ ∥δk ( f )∥22 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 215 Sampling representations Our main aim in this section consists in deriving a specific Nikol’skij-type representation for the spaces H α,β (Td ) Specific in the sense, that the building blocks in the decomposition originate from associated sampling operators of type (1.4) First we need some technical lemmas Lemma 3.1 Let α > 0, β ∈ R, min{α, α + β} > and ψ(k) := α|k|1 + β|k|∞ , k ∈ Nd0 Then for ε = min{α, α + β} ψ(k) ≤ ψ(k ′ ) − ε(|k ′ |1 − |k|1 ) holds for all k ′ , k ∈ Nd0 with k ′ ≥ k component-wise Proof Let k ′ ≥ k This implies ψ(k) = ψ(k ′ ) − α|k ′ − k|1 − β(|k ′ |∞ − |k|∞ ) (3.1) We need to distinguish two cases Case If β ≥ we have as an immediate consequence of (3.1) ψ(k) ≤ ψ(k ′ ) − α|k ′ − k|1 Case Let β < From (3.1) and |k ′ |∞ − |k|∞ ≤ |k ′ − k|∞ ≤ |k ′ − k|1 we obtain ψ(k) ≤ ψ(k ′ ) − (α + β)|k ′ − k|1 Recall the linear operator qk has been defined in (1.4) Let us settle the following cancellation property Lemma 3.2 Let ℓ, k ∈ Nd0 with kn < ℓn for some n ∈ {1, , d} Let further f ∈ T k and qℓ be the operator defined in (1.4) Then qℓ ( f ) = Proof Since f ∈ T k we have  f = am eimx k |m j |≤2 j j=1, ,d and qℓ ( f )(x) =  am qℓ (eim· )(x) = kj |m j |≤2 j=1, ,d  am kj |m j |≤2 j=1, ,d Due to 2ℓn −1 ≥ 2kn ≥ m n we have ηℓn (eim n · )(xn ) = (I2ℓn − I2ℓn −1 )(eim n · )(xn ) = which implies qℓ ( f ) = d  j=1 ηℓ j (eim j · )(x j ) 216 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Now we are in the position to prove Nikol’skij’s type representation theorems for the spaces H α,β (Td ) Proposition 3.3 Let min{α, α + β} > 1/2 Then every function f ∈ H α,β (Td ) can be represented by the series  f = qk ( f ) (3.2) k∈Nd0 converging unconditionally in H α,β (Td ), and satisfying the condition  22(α|k|1 +β|k|∞ ) ∥qk ( f )∥22 ≤ C∥ f ∥2H α,β (Td ) (3.3) k∈Nd0 with a constant C = C(α, β, d) > Proof Step We first prove (3.3) for f ∈ H α,β (Td ) We choose α˜ such that < α˜ < min{α, α + β} (3.4) For any ℓ ∈ Nd0 we have that  f = δℓ+k ( f ) (3.5) k∈Zd with the convention δm ( f ) := if m ∈ Zd \ Nd0 Linearity of qℓ and the triangle inequality implies      ∥qℓ ( f )∥2 ≤  qℓ (δℓ+k ( f )) ≤ ∥qℓ (δℓ+k ( f ))∥2 k∈Zd k∈Zd Using δm ( f ) ∈ T m for m ∈ Nd0 and Lemma 3.2 we find  ∥qℓ ( f )∥2 ≤ ∥qℓ (δℓ+k ( f ))∥2 k∈Nd0 Lemma in [30] together with univariate approximation results of the Jm , see [26], gives  ˜ ∥qℓ ( f )∥2 2−α|ℓ| ∥δℓ+k ( f )∥ H α˜ (Td ) mix k∈Nd0 Applying Lemma 2.10 yields  ˜ ∥qℓ ( f )∥2 2α|k| ∥δℓ+k ( f )∥2 (3.6) k∈Nd0 This together with Lemma 3.1 leads to  ˜ 2α|ℓ+k|1 +β|ℓ+k|∞ ∥δ 2α|ℓ|1 +β|ℓ|∞ ∥qℓ ( f )∥2 2(α−min{α,α+β})|k| ℓ+k ( f )∥2 k∈Nd0 (3.7) G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 217 We proceed taking the ℓ2 (Nd0 )-norm with respect to the index ℓ on the left-hand side of (3.7) The triangle inequality in ℓ2 (Nd0 ) yields   α|ℓ| +β|ℓ| ∞ ∥q ( f ) ∥  2 ℓ ℓ2 (Nd ,ℓ)   α|ℓ+k| +β|ℓ+k|  (α−min{α,α+β})|k| ˜ ∞ ∥δ 1 2  ℓ+k ( f ) ∥2 ℓ (Nd ,ℓ) k∈Nd0 Since k is always positive, we can estimate as follows  α|ℓ| +β|ℓ|   α|ℓ| +β|ℓ|  ∞ ∥q ( f ) ∥  ∞ ∥δ ( f ) ∥  2 2 ℓ ℓ ℓ (Nd ) ℓ 0  d (N0 ) ˜ 2(α−min{α,α+β})|k| k∈Nd0 Due to the choice of α˜ in (3.4) the sum is converging and we obtain   α|ℓ| +β|ℓ|   α|ℓ| +β|ℓ| ∞ ∥δ ( f ) ∥  ∞ ∥q ( f ) ∥  2 2 ℓ ℓ ℓ (Nd ) ℓ (Nd ) 2 This proves (3.3)  Step The representation f = k∈Nd qk ( f ) in H α,β (Td ) and the unconditional convergence of it can be achieved using standard arguments in connection with the density of trigonometric polynomials in H α,β (Td ) For more details we refer to [3] Proposition 3.4 Let β ∈ R, min{α, α+β} > and ( f k )k∈Nd a sequence with f k ∈ T k satisfying  2(α|k|1 +β|k|∞ ) 2 ∥ f k ∥2 < ∞ k∈Nd0  Assume that the series k∈Nd f k converges in L (Td ) to a function f Then f ∈ H α,β (Td ), and moreover, there is a constant C = C(α, β, d) > such that  ∥ f ∥2H α,β (Td ) ≤ C 22(α|k|1 +β|k|∞ ) ∥ f k ∥22 k∈Nd0 Proof Similar to (3.5) for ℓ ∈ Nd0 we write f as the series  f = f ℓ+k k∈Zd with f ℓ+k := for k + ℓ ∈ Zd \ Nd0 Clearly, δℓ : L (Td ) → L (Td ) is an orthogonal projection The projection properties of the operator δℓ together with f k ∈ T k yields  ∥δℓ ( f )∥2 ≤ ∥δℓ ( f ℓ+k )∥2 k∈Nd0 Thanks to ∥ δℓ ∥ L (Td )→L (Td ) = we conclude  ∥δℓ ( f )∥2 ≤ ∥ f ℓ+k ∥2 k∈Nd0 This together with Lemma 3.1 yields  2α|ℓ|1 +β|ℓ|∞ ∥δℓ ( f )∥2 2− min{α,α+β}|k|1 2α|ℓ+k|1 +β|ℓ+k|∞ ∥ f ℓ+k ∥2 k∈Nd0 (3.8) 218 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Finally, the same arguments as in the proof of Proposition 3.3 yield   α|ℓ| +β|ℓ|   α|ℓ| +β|ℓ| ∞ ∥ f ∥ ∞ ∥δ ( f )∥  2 2 ℓ ℓ (Nd ) ℓ ℓ (Nd ) 2 0 Since the left-hand side coincides with ∥ f ∥ H α,β (Td ) Proposition 3.4 is proved After one more definition we proceed to the main result of this section Definition 3.5 Let min{α, α + β} > 21 We define ∥ f ∥+ := H α,β (Td )  22(α|k|1 +β|k|∞ ) ∥qk ( f )∥22 1 k∈Nd0 for all f ∈ H α,β (Td ) Theorem 3.6 Let min{α, α + β} > 12 Then a function f on Td belongs to the space H α,β (Td ), if and only if f can be represented by the series (3.2) converging in H α,β (Td ) and satisfying the condition (3.3) Moreover, the norm ∥ f ∥ H α,β (Td ) is equivalent to the norm ∥ f ∥+ H α,β (Td ) Proof This result is an easy consequence of Propositions 3.3 and 3.4, applied with f k = qk ( f ) Remark 3.7 (i) The restriction min{α, α + β} > is essentially optimal, see Remark 2.9 (ii) The potential of sampling representations was first recognized for HăolderNikolskij type spaces of mixed smoothness by D˜ung [7,8] Sampling representations for non-periodic functions in connection with tensor product B-spline series have been treated in [9,10] (iii) For an extension of the present sampling characterizations and approximation results to Sobolev spaces build on L p (Td ) we refer to [4] Sampling on energy-norm based sparse grids In this section we consider the quality of approximation by sampling operators using energynorm based sparse grids In fact, a suitable sampling operator Q ∆ uses a slightly larger set ∆ε (ξ ) compared to ∆(ξ ) with the same combinatorial properties, where ∆ε (ξ ) is defined in (1.7) and ∆(ξ ) in (1.6), see Lemma 6.3 Theorem 4.1 Let α > 0, γ ≥ and β < γ such that min{α, α + β} > 21 Let further < ε < γ − β < α Then there exists a constant C = C(α, β, γ , ε, d) > such that ∥ f − Q ∆ε (ξ ) f ∥ H γ (Td ) ≤ C 2−ξ ∥ f ∥ H α,β (Td ) holds for all f ∈ H α,β (Td ) and all ξ > G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 219 Proof Step The triangle inequality in H γ (Td ), Lemma 2.10, and afterwards Hăolders inequality yield    ∥qk ( f )∥ H γ (Td ) qk ( f ) γ d ≤ ∥ f − Q ∆ε (ξ ) f ∥ H γ (Td ) =  H (T ) k̸∈∆ε (ξ )  ≤ γ |k|∞ k̸∈∆ε (ξ ) ∥qk ( f )∥2 k̸∈∆ε (ξ )  = 2α|k|1 +β|k|∞ 2−(α|k|1 +β|k|∞ ) 2γ |k|∞ ∥qk ( f )∥2 k̸∈∆ε (ξ ) ≤   2−2α|k|1 +2(γ −β)|k|∞ 1 k̸∈∆ε (ξ ) ×   k̸∈∆ε (ξ ) 22(α|k|1 +β|k|∞ ) ∥qk ( f )∥22 1 Applying Theorem 3.6 we have   1 22(α|k|1 +β|k|∞ ) ∥qk ( f )∥22 ≤ ∥ f ∥ H α,β (Td ) k̸∈∆ε (ξ ) Consequently, we obtain the following inequality ∥ f − Q ∆ε (ξ ) f ∥ H γ (Td ) ≤   2−2α|k|1 +2(γ −β)|k|∞ k̸∈∆ε (ξ ) 1 ∥ f ∥ H α,β (Td ) (4.1) Step Now we consider the sum  2−2α|k|1 +2(γ −β)|k|∞ ≤ k̸∈∆ε (ξ ) d   i=1 2−2α|k|1 +2(γ −β)|k|∞ , k̸∈∆ε (ξ ) k∈K i where K i := {k ∈ Nd0 : ki = |k|∞ } i = 1, , d (4.2) We want to find a proper upper bound for  2−2α|k|1 +2(γ −β)|k|∞ k̸∈∆ε (ξ ) k∈K i For simplicity we restrict ourselves to the case i = with k1 = |k|∞ and set k˜ := (k2 , , kd ) (4.3) ˜ holds for all k ∈ Indeed, |k|1 = k1 + |k| Nd0 So the following equivalence is true k ̸∈ ∆ε (ξ ) ⇐⇒ (α − ε)|k|1 − ((γ − β) − ε)k1 > ξ ˜ + (α − (γ − β))k1 > ξ ⇐⇒ (α − ε)|k| ⇐⇒ k1 > ˜1 ξ − (α − ε)|k| α − (γ − β) 220 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Using this equivalence we can proceed with   ˜ 2−2α|k|1 2−2α|k|1 +2(γ −β)|k|∞ = k̸∈∆ε (ξ ) k∈K  ˜ d−1 k∈N =  ˜ ∞ −1, k1 >max |k| ˜  2−2α|k|1 ˜ k∈I 2−2αk1 +2(γ −β)k1 ˜ ξ −(α−ε)|k| α−(γ −β)  2−2αk1 +2(γ −β)k1 (4.4)  (4.5) ˜∞ k1 ≥|k| ˜  +  2−2α|k|1 2−2αk1 +2(γ −β)k1 , ˜ ξ −(α−ε)|k| k1 > α−(γ −β) k̸˜ ∈ I1 where   ˜1 ξ − (α − ε)|k| d−1 ˜ ˜ I = k ∈ N0 : < |k|∞ α − (γ − β) First we compute an upper bound for the sum in (4.4) Because of ˜ ˜ 22((γ −β)−α)|k|∞ ≤ 2−2(ξ −[α−ε])|k|1 we conclude   2−2αk1 +2(γ −β)k1 ≤ C ˜ k1 ≥|k| ˜∞ k∈I if k˜ ∈ I1 ,  ˜ ˜ 2−2α|k|1 22((γ −β)−α)|k|∞ ˜ k∈I   2−2αk1 +2(γ −β)k1 ˜ k1 ≥|k| ˜∞ k∈I 2−2ξ  ˜ 2−2ε|k|1 ˜ k∈I −2ξ Here the constant behind does not depend on ξ Step Next, we estimate the sum in (4.5) Similarly as above we find    ˜ ˜ ˜ 2−2αk1 +2(γ −β)k1 2−2α|k|1 2−2(ξ −(α−ε)|k|1 ) 2−2α|k|1 k̸˜ ∈ I1 k1 > ˜ ξ −(α−ε)|k| α−(γ −β) k̸˜ ∈ I1 2−2ξ As a consequence we have  2−2α|k|1 +2(γ −β)|k|∞ 2−2ξ k̸∈∆ε (ξ ) This together with (4.1) proves the claim The previous result includes the case γ = Let us state this special case separately Corollary 4.2 Let α > 0, β < such that α + β > constant C = C(α, β, ε, d) > such that ∥ f − Q ∆ε (ξ ) f ∥2 ≤ C2−ξ ∥ f ∥ H α,β (Td ) holds for all f ∈ H α,β (Td ) and ξ > and < ε < −β < α Then there is a 221 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Remark 4.3 Estimates of sampling operators of Smolyak-type with respect to the non-periodic α ([0, 1]d ) → H γ ([0, 1]d ) may be found also in the papers [1,2,15,25] The embeddings I : Hmix authors have used energy-norm based sparse grids in case α = and γ = The smoothness restriction α ≤ in the source space is caused by the use of the hierarchical Faber system (hat function) Using trigonometric interpolation we not encounter any smoothness restrictions here and so the algorithm is able to exploit arbitrarily high smoothness However, the above mentioned authors cared for the dependence of all constants on the dimension d, an important issue in high-dimensional approximation, which we have ignored here Let us also mention [12] in this respect Remark 4.4 We have already mentioned in the introduction that the interpolation nodes in (1.3) are not nested in general For the theoretical purposes in this paper it does not play a role However, for practical issues nestedness properties might decrease the computational effort significantly This can be fixed by using a small modification of Jm , which we call J˜m , defined by  2m−1 f (π ℓ/m) D˜ m (t − π ℓ/m), J˜m f (t) := 2m ℓ=0 where D˜ m (x) := Dm (x) − e−imx = e−i(m−1)x (ei2mx − 1)(ei x − 1)−1 A similar operator has been studied recently in [16] Sampling on Smolyak grids In this section we intend to apply our new method to situations where the classical Smolyak algorithm is used On the one hand we give shorter proofs for existing results and extend some of them concerning the used approximating operators on the other hand 5.1 The mixed–mixed case γ α (Td ) measuring the error in H d We consider sampling operators for functions in Hmix mix (T ) The associated operator Q ∆ is this time given by the set ∆(ξ ) which is defined in (1.6) Theorem 5.1 Let γ > and α > max{γ , 1/2} Then there is a constant C = C(α, γ , d) > such that ∥ f − Q ∆(ξ ) f ∥ H γ mix (T d) ≤ C2−ξ ∥ f ∥ H α mix (T d) α (Td ) and ξ > holds for all f ∈ Hmix γ Proof We employ Proposition 3.4 to Hmix (Td ) with the sequence ( f k )k∈Nd given by  q (f) fk = k : k ̸∈ ∆(ξ ), : k ∈ ∆(ξ ) 222 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Note, that the only restriction for Proposition 3.4 is γ > Clearly, f − Q ∆(ξ ) f = and hence  ∥ f − Q ∆(ξ ) f ∥2H γ (Td ) 22γ |k|1 ∥ f k ∥22 mix  k∈Nd0 fk k∈Nd0 =  k̸∈∆(ξ ) 22(γ −α)|k|1 22α|k|1 ∥qk ( f )∥22  ≤ 2−2ξ 22α|k|1 ∥qk ( f )∥22 k∈Nd0 Applying Theorem 3.6 (here we need α > 1/2) completes the proof since  22α|k|1 ∥qk ( f )∥22 ∥ f ∥2H α (Td ) mix k∈Nd0 As a direct consequence of Theorem 5.1, we obtain the following result for the weaker error norm ∥ · ∥ H γ (Td ) Corollary 5.2 Let α > that and < γ < α Then there is a constant C = C(α, γ , d) > such ∥ f − Q ∆(ξ ) f ∥ H γ (Td ) ≤ C2−ξ ∥ f ∥ H α mix (T d) α (Td ) and ξ > holds for all f ∈ Hmix Remark 5.3 Sampling with Smolyak operators has some history Closest to us are Temlyakov [33,35,34] and D˜ung [7–10], see also [13,27,28,30] In almost all contributions preference was given to situations where the target space was L q (Td ) Let us also refer to a recent publication of Griebel and Hamaekers [16] 5.2 The case α > γ − β = Now we are interested in the embedding I : H α,β (Td ) → H β (Td ) The sampling operator Q ∆(ξ ) is determined by ∆(ξ ) from (1.6) with γ = β Let us simplify the structure by considering the index sets ∆(αm) for m ∈ N which consists of all k ∈ Nd0 satisfying |k|1 ≤ m Theorem 5.4 Let β = γ ≥ and α > 21 Then there is a constant C = C(α, β, d) > such that ∥ f − Q ∆(αm) f ∥ H β (Td ) ≤ C2−mα m d−1 ∥ f ∥ H α,β (Td ) holds for all f ∈ H α,β (Td ) and m ∈ N Proof We proceed as in proof of Theorem 4.1 The triangle inequality in H β (Td ) yields       ∥ f − Q ∆(αm) f ∥ H β (Td ) =  qk ( f ) β d ≤ ∥qk ( f )∥ H β (Td ) k̸∈∆(αm) H (T ) k̸∈∆(αm) 223 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Applying Lemma 2.10 gives ∥ f − Q ∆(αm) f ∥ H β (Td ) 2β|k|∞ ∥qk ( f )∥2  k̸∈∆(αm) Proceeding with Hăolders inequality leads to 2 ∥ f − Q ∆(αm) f ∥2 ≤ 2−2α|k|1 22(α|k|1 +β|k|∞ ) ∥qk ( f )∥22 |k|1 >m |k|1 >m Employing the upcoming lemma and Theorem 3.6 finishes the proof Lemma 5.5 Let α > Then  2−2α|k|1 m d−1 2−2αm |k|1 >m holds for all m > Proof This lemma is well known, but see, e.g., [3] for all details 5.3 The case γ = From Theorem 5.4 we immediately obtain the special case (γ = β = 0) ∥ f − Q ∆(αm) f ∥2 ≤ C2−mα m d−1 ∥ f ∥H α mix (T d) , m ∈ N, compare with [27,30] With our methods we can additionally show an error bound for L ∞ (Td ) instead of L (Td ) Theorem 5.6 Let α > 12 Then there is a constant C = C(α, d) > such that ∥ f − Q ∆(αm) f ∥∞ ≤ C2−m(α− ) m d−1 ∥ f ∥H α mix (T d) α (Td ) and m ∈ N holds for all f ∈ Hmix Proof As above with Lemma 2.7 we conclude      ∥ f − Q ∆(αm) f ∥∞ =  qk ( f ) ≤ ∞ k̸∈∆(αm) ≤   ∥qk ( f )∥∞ k̸∈∆(αm) 2|k|1 /2 2α|k|1 2−α|k|1 ∥qk ( f )∥2 |k|1 >m ≤   2−2|k|1 (α− ) |k|1 >m 1   |k|1 >m 22α|k|1 ∥qk ( f )∥22 1 Applying Lemma 5.5 and Theorem 3.6 proves the claim Now we turn to the case < q < ∞ The following result allows for comparing the present situation with the results in Section 5.1 Lemma 5.7 Let < q < ∞ Then  1/2 ∥ f ∥q ∥δk ( f )∥q2 k∈Nd0  22|k|1 (1/2−1/q) ∥δk ( f )∥22 1/2 = ∥f∥ k∈Nd0 holds true for any f ∈ L q (Td ), where the right-hand side may be infinite 1−1 q Hmix (Td ) 224 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Proof The proof of the first relation in Lemma 5.7 is elementary using the Littlewood–Paley decomposition in L q (Td ) together with q/2 ≥ 1, see for instance [34, Theorem 0.3.2, Page 20] The second relation follows by an application of Nikol’skij’s inequality in Lemma 2.7 Remark 5.8 Let us mention that Lemma 5.7 can be refined to   q 1/q ∥ f ∥q 2q|k|1 (1/2−1/q) ∥δk ( f )∥2 k∈Nd0 For this deep result we refer to [34, Lemma II.2.1] and to [9, Lemma 5.3] as well as [24, Lemma 1] for non-periodic versions In a more general context this embedding is a special case of a Jawerth–Franke type embedding, see [19] Sampling numbers In this section we will restate the approximation results from Sections and in terms of the number of degrees of freedom We additionally show the asymptotic optimality with regard to sampling numbers of the sampling operators considered in Sections and This requires estimates of the rank of the corresponding sampling operators A lower bound for the rank is deduced from the fact that the respective sampling operators reproduce trigonometric polynomials from modified hyperbolic crosses H∆ Recall that our approximation scheme is based on the classical trigonometric interpolation We have used several times the fact that the operator Jm defined in (1.3) reproduces univariate trigonometric polynomials of degree less than or equal to m What concerns the operator Q ∆ in (1.5) we can prove the following general reproduction result Lemma 6.1 Let ∆ ⊂ Nd0 be a solid finite set meaning that k ∈ ∆ and ℓ ≤ k implies ℓ ∈ ∆ Then Q ∆ reproduces trigonometric polynomials with frequencies in  H∆ := Pk , (6.1) k∈∆ where Pk is defined in (2.1) Proof One may follow the arguments in the proof of [30, Lemma 1], but see [3] for all details The previous result immediately implies the relation  rank Q ∆ ≥ 2|k|1 k∈∆ if ∆ ⊂ Nd0 is solid Lemma 6.2 Let α > 0, γ ≥ and β < γ such that < γ − β ≤ α (i) The index sets ∆(α, β, γ ; ξ ) defined in (1.6) and ∆(ε, α, β, γ ; ξ ) defined in (1.7) are solid sets in the sense of Lemma 6.1 for every ξ > (ii) The index set ∆(α; ξ ) is defined in (1.6) with γ = β is a solid set for every ξ > G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 225 Proof The second result is trivial We prove the first one Let ψ(k) := α|k|1 − (γ − β)|k|∞ The set ∆(ξ ) consists of all k ∈ Nd0 with ψ(k) ≤ ξ Applying Lemma 3.1 yields ψ(k ′ ) ≤ ψ(k) ≤ ξ for all k ′ ≤ k ∈ ∆(ξ ) That means all the k ′ also belong to ∆(ξ )  In the next lemma we give sharp estimates for k∈∆(ξ ) 2|k|1 with ∆(ξ ) from (1.6) Lemma 6.3 Let α > 0, γ ≥ 0, β ∈ R such that γ > β and α > γ − β Then  ξ 2|k|1 ≍ α−(γ −β) k∈∆(ξ ) holds for all ξ ≥ α − (γ − β), where the constants behind “≍” only depend on α, γ − β, and d Proof Step First we deal with the upper bound We are going to use the same notation as in (4.2) and (4.3) We obtain the following inequality  2|k|1 ≤ k∈∆(ξ ) d   2|k|1 i=1 k∈K i ∩∆(ξ ) By symmetry it will be enough to deal with i = Hence   2|k|1 ≤ d 2|k|1 k∈∆(ξ ) k∈K ∩∆(ξ ) Now we want to decompose the summation over k Since k1 ≥ |k|∞ we find k ∈ ∆(ξ ) ⇐⇒ α|k|1 − (γ − β)k1 ≤ ξ ˜ + k1 ) − (γ − β)k1 ≤ ξ ⇐⇒ α(|k| ˜1 ξ − α|k| ⇐⇒ k1 ≤ α − (γ − β) This implies ˜∞≤ |k| ˜1 ξ − α|k| ˜ ∞ ≤ ξ ˜ + (α − (γ − β))|k| ⇐⇒ α|k| α − (γ − β) We shall use these inequalities to produce an appropriate decomposition of K ∩ ∆(ξ ) which results in  2|k|1 ≤ d k∈∆(ξ ) ˜  2|k|1 ˜ d−1 k∈N ˜ +(α−(γ −β))|k| ˜ ∞ ≤ξ α|k| ξ ˜ ξ −α|k| α−(γ −β)  ˜∞ k1 =|k|  α−(γ −β) ˜ +(α−(γ −β))|k| ˜ ∞ ≤ξ α|k| ξ α−(γ −β) since α/(α − (γ − β)) > , 2k1 −α˜ α−(γ −β) ˜1 |k| 226 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Step We prove the lower bound First we claim that   ξ k ∗ := (1, 0, , 0) ∈ ∆(ξ ) α − (γ − β) Indeed, k ∗ ∈ ∆(ξ ) ⇐⇒ α|k ∗ |1 − (γ − β)|k ∗ |∞ ≤ ξ     ξ ξ − ((γ − β) − ε) ≤ξ ⇐⇒ (α − ε) α − (γ − β) α − (γ − β)   ξ ≤ ξ ⇐⇒ (α − (γ − β)) α − (γ − β) Obviously, the last inequality is true Consequently   ξ  ξ ∗ −1 2|k|1 ≥ 2|k |1 = α−(γ −β) ≥ α−(γ −β) k∈∆(ξ ) The proof is complete Corollary 6.4 Let α > 0, γ ≥ 0, β ∈ R such that γ > β and α > γ − β Let further ∆(ξ ) as in (1.6) ξ (i) The sampling operator Q ∆(ξ ) uses at most C2 α−(γ −β) function values, where the constant C > only depends on α, γ − β and d (ii) The rank of the linear operator Q ∆(ξ ) satisfies ξ rank Q ∆(ξ ) ≍ α−(γ −β) , ξ ≥ α − (γ − β), where the constants behind “≍” only depend on α, γ − β, and d Proof Clearly, Jm f uses 2m + values of function f , hence ηm f is using ≤ 2m+2 function values This implies that qk f applies ≤ 22d 2|k|1 function values As a consequence of Lemma 6.3 we find that Q ∆(ξ ) f is using  ξ 2|k|1 ≍ α−(γ −β) k∈∆(ξ ) function values of f Part (ii) follows from Lemma 6.1 and the lower bound in Lemma 6.3 Let us now count the degrees of freedom for a classical Smolyak grid Lemma 6.5 For any d ∈ N and m ∈ Nd0 , we have the inequality  m + d − d−1  e(m + d − 1) d−1  2m ≤ 2|k|1 ≤ 2m+1 d −1 d − |k| ≤m Proof This assertion is a direct consequence of [12, Lemma 3.10] together with the well-known relation  N n  N   eN n ≤ ≤ n n n G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 227 Corollary 6.6 Let m ∈ N and ∆ = {k ∈ Nd0 : |k|1 ≤ m} (i) The sampling operator Q ∆ is using at most Cm d−1 2m function values, where C decays super-exponentially in d (ii) The rank of the linear operator Q ∆ satisfies rank Q ∆ ≍ m d−1 2m , m ∈ N Proof Part (i) follows from the fact that qk ( f ) uses 22d 2|k|1 function values for any k together with the upper bound in Lemma 6.5 The second assertion can be derived by using the reproduction properties of Q ∆ , see Lemma 6.1, and the lower bound in Lemma 6.5 Now we are in position to formulate our results in terms of sampling numbers Theorem 6.7 Let α, β, γ ∈ R such that min{α, α + β} > 1/2, γ ≥ and < γ − β < α Then it holds gm (I1 : H α,β (Td ) → H γ (Td )) ≍ am (I1 : H α,β (Td ) → H γ (Td )) ≍ m −(α−γ +β) , m ≥ Proof Step Let α > γ − β > We claim an (I : H α,β (Td ) → H γ (Td )) ≍ n −α+γ −β , n ∈ N This has been proved in [12, Theorem 4.7], however, with the additional restriction that 2(γ −β) > α > γ −β For the convenience of the reader we give a proof without this restriction The lower bound is a consequence of a well-known abstract result (see [36, Theorem 1] or [21, Theorem 1.4, p 405]) on lower bounds for linear n-widths, namely Lemma 6.8 Let L n+1 be an n + 1-dimensional subspace in a Banach space X , and Bn+1 (r ) := { f ∈ L n+1 : ∥ f ∥ X ≤ r } Then λn (Bn+1 (r ), X ) ≥ r Here λn (Bn+1 (r ), X ) denotes the linear n-width of the set Bn+1 (r ) in X We apply this lemma with X = H γ and L n+1 to be the subspace of all trigonometric polynomials with frequencies in H∆(ξ ) from (6.1) with ∆(ξ ) = ∆(α, β, γ ; ξ ) and ξ chosen accordingly From Lemma 6.3 we get n ≍ 2ξ/(α−(γ −β)) We immediately see the Bernstein type inequality ∥ f ∥ H α,β 2ξ ∥ f ∥ H γ , f ∈ L n+1 (6.2) Hence, by choosing r := 2−ξ we get from (6.2) that Bn+1 (r ) is contained in the unit ball of H α,β Finally, by Lemma 6.8 we conclude an (I ) ≥ λn (Bn+1 (2−ξ ), H γ ) = 2−ξ ≍ n −(α−(γ −β)) This proves the claim Step By Step we conclude m −(α−(γ −β)) am (I1 : H α,β (Td ) → H γ (Td )) ≤ gm (I1 : H α,β (Td ) → H γ (Td )), m ∈ N 228 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Suppose < ε < γ − β Let Dε (ξ ) be the number of function values the operator Q ∆ε (ξ ) is using Then Theorem 4.1 yields  D (ξ ) α−(γ −β) ε Dε (ξ )−(α−(γ −β)) ∥ f ∥ H α,β (Td ) ∥ f − Q ∆ε (ξ ) f ∥ H γ (Td ) 2ξ/(α−(γ −β)) Applying Corollary 6.4(i) with α − ε and γ − β − ε shows that Dε (ξ ) ξ/(α−(γ −β)) ≤ C1 (ε, α, γ − β, d) This proves the estimate from above in case m = Dε The corresponding estimate for all m follows by a simple monotonicity argument Remark 6.9 In case β = Griebel and Hamaekers recently proved a similar upper bound for gm (I1 ) (see [16, Lemma 9]) Under the conditions of Theorem 6.7 the family of sampling operators Q ∆ε (ξ ) for < ε < γ − β is optimal in order A non-periodic version of Theorem 6.7 has been proved recently by D˜ung [10] The next two theorems collect results for sampling numbers which are based on Smolyak’s algorithm Theorem 6.10 Let α > 1/2 and suppose < γ < α (i) We have for m ≥ γ γ α α (Td ) → Hmix (Td )) (Td ) → Hmix (Td )) ≍ am (I5 : Hmix gm (I5 : Hmix ≍ m −(α−γ ) (log m)(d−1)(α−γ ) (ii) Let < q < ∞ Then we have for m ≥ α α gm (I4 : Hmix (Td ) → L q (Td )) ≍ am (I4 : Hmix (Td ) → L q (Td )) ≍m −α+ 21 − q1 (d−1) (α− 12 + q1 ) (log m) (iii) In case q = we obtain for all m ≥ α m −α (log m)(d−1) α ≍ am (I3 : Hmix (Td ) → L (Td )) α (Td ) → L (Td )) ≤ gm (I3 : Hmix m −α (log m)(d−1) (α+ ) (iv) In case q = ∞ it holds for all m ≥ α α gm (I4 : Hmix (Td ) → L ∞ (Td )) ≍ am (I4 : Hmix (Td ) → L ∞ (Td )) ≍ m −α+ (log m)(d−1) α Proof Step The relation α am (I : Hmix (Td ) → L (Td )) ≍ m −α (log m)α(d−1) , ≤ m ∈ N, (6.3) holds true for any α > and is due to Galeev [14] In case α > γ ≥ we use a simple lifting argument to obtain γ α−γ α am (I : Hmix (Td ) → Hmix (Td )) ≍ am (I : Hmix (Td ) → L (Td )) ≍ m −(α−γ ) (log m)(d−1)(α−γ ) , ≤ m ∈ N Note, that a similar lifting argument does in general not apply for sampling numbers gm G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 229 Step Proof of (i) By Step we obtain for m ≥ γ m −(α−γ ) (log m)(d−1)(α−γ ) α am (I5 : Hmix (Td ) → Hmix (Td )) α ≤ gm (I5 : Hmix (Td ) → H γ (Td )) Concerning the estimate from above we apply Theorem 5.1 with ∆(ξ ) := ∆(α − γ , 0, 0; ξ ) and ξ = (α − γ )n for n ∈ N This gives ∥ f − Q ∆((α−γ )n) f ∥ H γ mix (T d) 2−(α−γ )n , (6.4) n ∈ N Let D(n) be the number of function values used by Q ∆((α−γ )n) f By Corollary 6.6(i), (ii) we know that D(n) ≍ n d−1 2n and log D(n) ≍ n Rewriting (6.4) gives ∥ f − Q ∆((α−γ )n) f ∥ H γ mix (T d) D(n)−(α−γ ) (log D(n))(d−1)(α−γ ) Obvious monotonicity arguments complete the proof Step Proof of (ii) The estimate from below for the approximation numbers is due to Romanyuk [22] The corresponding estimate from above for the sampling numbers is an immediate consequence of Lemma 5.7 together with (i), where γ = 1/2 − 1/q Step Proof of (iii) For the lower bound by approximation numbers we follow Step The upper bound is obtained similar to (i) applying Theorem 5.4 and Corollary 6.6(i), (ii) Step Proof of (iv) The estimate from below for the approximation numbers is due to Temlyakov [35] Let us mention that this lower bound is also implied by a recent general result by Cobos, Kăuhn and Sickel [6] The estimate from above for sampling numbers follows from Theorem 5.6 combined with Corollary 6.6(i), (ii) in the same way as in (i) Theorem 6.11 Let α > 1/2 and β ≥ Then it holds m −α (log m)α(d−1) ≍ am (I2 : H α,β (Td ) → H β (Td )) ≤ gm (I2 : H α,β (Td ) → H β (Td )) m −α (log m)(d−1)(α+ ) , m ≥ Proof Step From (6.3) we obtain by a simple lifting argument for β > 0, α am (I : H α,β (Td ) → H β (Td )) ≍ am (I : Hmix (Td ) → L (Td )) ≍ m −α (log m)α(d−1) , ≤ m ∈ N Step Let D(n) be the number of function values used by the operator Q ∆(ξ ) with respect to the index set ∆(ξ ) := ∆(α, 0, 0; ξ ) with ξ = αn and n ∈ N Due to Corollary 6.6 we have D(n) ≍ n d−1 2n and log D(n) ≍ n Then Theorem 5.4 yields ∥ f − Q ∆(αn) f ∥ H β (Td ) ≤ C2−nα n d−1 ∥ f ∥ H α,β (Td ) [D(n)]−α [log(D(n))](d−1)(α+ ) ∥ f ∥ H α,β (Td ) A standard monotonicity argument concludes the proof 230 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 Remark 6.12 As we have mentioned before, not all the results in Theorem 6.10 are new A non-periodic version of (ii) has been proved recently by D˜ung [9] What concerns (iii) let us refer to [9,28,30,31] where even more general situations are treated Part (iv) reproduces a result due to Temlyakov [35] Note, that our methods allow for proving this result in the framework of classical trigonometric interpolation, see Theorem 5.6, whereas Temlyakov had to use de la Vall´ee-Poussin sampling operators In any case, it is remarkable that Smolyak’s algorithm yields optimal bounds here Acknowledgments The authors would like to thank the organizers of the HCM-workshop “Discrepancy, Numerical Integration, and Hyperbolic Cross Approximation”, where this work has been initiated, for providing a pleasant and fruitful working atmosphere In addition, the authors would like to thank the Hausdorff-Center for Mathematics (HCM) and the Bonn International Graduate School (BIGS) for providing additional financial support to finish this work Dinh D˜ung’s research work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 102.01-2014.02, and a part of it was done when he was working as a research professor at the Vietnam Institute for Advanced Study in Mathematics (VIASM) He would like to thank the VIASM for providing a fruitful research environment and working condition References [1] H.-J Bungartz, M Griebel, A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives, J Complexity 15 (1999) 167–199 [2] H.-J Bungartz, M Griebel, Sparse grids, Acta Numer 13 (2004) 147–269 [3] G Byrenheid, Dinh D˜ung, W Sickel, T Ullrich, Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in H γ , arXiv:1408.3498 [4] G Byrenheid, T Ullrich, Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations, Preprint, Bonn [5] A Chernov, Dinh D˜ung, New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness, J Complexity 32 (2016) 92121 [6] F Cobos, T Kăuhn, W Sickel, Optimal approximation of Sobolev functions in the sup-norm, Preprint, Leipzig, 2014 [7] Dinh D˜ung, On optimal recovery of multivariate periodic functions, in: S Igary (Ed.), Harmonic Analysis, in: Satellite Conf Proceedings, Springer, Tokyo, 1991, pp 96–105 [8] Dinh D˜ung, Non-linear approximations using sets of finite cardinality or finite pseudo-dimension, J Complexity 17 (2001) 467–492 [9] Dinh D˜ung, B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness, J Complexity 27 (2011) 541–567 [10] Dinh D˜ung, Sampling and cubature on sparse grids based on a B-spline quasi-interpolation, Found Comput Math (2015) http://dx.doi.org/10.1007/s10208-015-9274-8 [11] Dinh D˜ung, V.N Temlyakov, T Ullrich, Hyperbolic cross approximation, ArXiv e-prints, 2015 arXiv:1601 03978 [math.NA] [12] Dinh D˜ung, T Ullrich, N -widths and ε-dimensions for high-dimensional approximations, Found Comput Math 13 (2013) 965–1003 [13] Dinh D˜ung, T Ullrich, Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square, Math Nachr 288 (2015) 743–762 [14] E.M Galeev, On linear widths of classes of periodic functions of several variables, Vestnik MGU, Ser Mat.-Mekh (4) (1987) 13–16 [15] M Griebel, Sparse grids and related approximation schemes for higher dimensional problems, in: Proc Foundations of Comp Math., Santander 2005, in: London Math Soc Lect Notes Series, vol 331, Cambridge Univ Press, Cambridge, 2006, pp 106–161 G Byrenheid et al / Journal of Approximation Theory 207 (2016) 207–231 231 [16] M Griebel, J Hamaekers, Fast Discrete Fourier Transform on Generalized Sparse Grids, Sparse grids and Applications, in: Lecture Notes in Computational Science and Engineering, Springer, 2014, pp 75–108 [17] M Griebel, S Knapek, Optimized tensor-product approximation spaces, Constr Approx 16 (4) (2000) 525–540 [18] M Griebel, S Knapek, Optimized general sparse grid approximation spaces for operator equations, Math Comp 78 (268) (2009) 2223–2257 [19] M Hansen, J Vybiral, The Jawerth–Franke embedding of spaces with dominating mixed smoothness, Georgian Math J 16 (4) (2009) 667682 [20] T Kăuhn, W Sickel, T Ullrich, Approximation of mixed order Sobolev functions on the d-torus: asymptotics, preasymptotics, and d-dependence, Constr Approx 42 (2015) 353–398 [21] G.G Lorentz, M von Golitschek, Y Makovoz, Constructive Approximation Advanced Problems, in: Grundlehren der Mathematischen Wissenschaften, vol 304, Springer-Verlag, Berlin, 1996 [22] A.S Romanyuk, Linear widths of the Besov classes of periodic functions of many variables II, Ukrain Math Zh 53 (2001) 965–977 [23] H.J Schmeißer, H Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, Chichester, 1987 [24] H.-J Schmeisser, W Sickel, Spaces of functions of mixed smoothness and their relations to approximation from hyperbolic crosses, J Approx Theory 128 (2004) 115150 [25] Ch Schwab, E Săuli, R.A Todor, Sparse finite element approximation of high-dimensional transport-dominated diffusion problems, ESAIM Math Model Numer Anal 42 (05) (2008) 777–819 [26] W Sickel, Some remarks on trigonometric interpolation on the n-torus, Z Anal Anwend 10 (1991) 551–562 [27] W Sickel, Approximate recovery of functions and Besov spaces of dominating mixed smoothness, in: Constructive theory of functions, Varna 2002, DARBA, Sofia, 2003, pp 404–411 [28] W Sickel, Approximation from sparse grids and function spaces of dominating mixed smoothness, Banach Center Publ 72 (2006) 271–283 s type, Z Anal [29] W Sickel, H Triebel, Hăolder inequalities and sharp embeddings in function spaces of B sp,q and F p,q Anwend 14 (1995) 105–140 [30] W Sickel, T Ullrich, Smolyak’s algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness, East J Approx 13 (2007) 387–425 [31] W Sickel, T Ullrich, Spline interpolation on sparse grids, Appl Anal 90 (2011) 337–383 [32] S.A Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl Akad Nauk 148 (1963) 1042–1045 [33] V.N Temlyakov, Approximation recovery of periodic functions of several variables, Mat Sb 128 (1985) 256–268 [34] V.N Temlyakov, Approximation of Periodic Functions, Nova Science, New York, 1993 [35] V.N Temlyakov, On approximate recovery of functions with bounded mixed derivative, J Complexity (1993) 41–59 [36] V.M Tikhomirov, Widths of sets in function spaces and the theory of best approximations, Uspekhi Mat Nauk 15 (3) (1960) 81–120 English translation in Russian Math Survey 15 (3) (1960) 75–111 [37] H Yserentant, Regularity and Approximability of Electronic Wave Functions, in: Lecture Notes in Mathematics, vol 2000, Springer-Verlag, Berlin, 2010  ... approximation results into the notion of sampling numbers and compare them to existing estimates for the approximation numbers Notation As usual, N denotes the natural numbers, N0 the non-negative integers,... section we recall the definition of the function spaces under consideration here They α (Td ) are all of Sobolev- type In a first subsection we consider the periodic Sobolev spaces Hmix of dominating... Nd0 Proof A proof can be found in [23, Theorem 3.3.2] To give a meaning to point evaluations of functions it is essential that the spaces under consideration contain only continuous functions To