Found Comput Math DOI 10.1007/s10208-015-9274-8 Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation Dinh Dung ˜ Received: 30 March 2014 / Revised: February 2015 / Accepted: 30 June 2015 © SFoCM 2015 Abstract Let X n = {x j }nj=1 be a set of n points in the d-cube Id := [0, 1]d , and n d n = {ϕ j } j=1 a family of n functions on I We consider the approximate recovery d of functions f on I from the sampled values f (x ), , f (x n ), by the linear sampling algorithm L n (X n , n , f ) := nj=1 f (x j )ϕ j The error of sampling recovery is measured in the norm of the space L q (Id )-norm or the energy quasi-norm of the γ isotropic Sobolev space Wq (Id ) for < q < ∞ and γ > Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, α,β in particular, spaces B p,θ of a “hybrid” of mixed smoothness α > and isotropic smoothness β ∈ R, and spaces B ap,θ of a nonuniform mixed smoothness a ∈ Rd+ We constructed asymptotically optimal linear sampling algorithms L n (X n∗ , ∗n , ·) on special sparse grids X n∗ and a family ∗n of linear combinations of integer or half integer translated dilations of tensor products of B-splines We computed the asymptotic order of the error of the optimal recovery This construction is based on B-spline α,β quasi-interpolation representations of functions in B p,θ and B ap,θ As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces Keywords Linear sampling algorithms · Optimal sampling recovery · Cubature formulas · Optimal cubature · Sparse grids · Besov-type spaces of anisotropic smoothness · B-spline quasi-interpolation representations Communicated by Albert Cohen B Dinh D˜ung dinhzung@gmail.com Information Technology Institute, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam 123 Found Comput Math 41A15 · 41A05 · 41A25 · 41A58 · 41A63 Mathematics Subject Classification Introduction The aim of the present paper is to construct linear sampling algorithms and cubature formulas on sparse grids based on a B-spline quasi-interpolation, and study their optimality in the sense of asymptotic order for functions on the unit d-cube Id := [0, 1]d , having an anisotropic smoothness The error of sampling recovery is measured in the norm of the space L q (Id )-norm or the energy norm of the isotropic Sobolev γ space Wq (Id ) for < q < ∞ and γ > For convenience, we use somewhere the convention Wq0 (Id ) := L q (Id ) Let X n = {x j }nj=1 be a set of n points in Id , n = {ϕ j }nj=1 a family of n functions on Id If f is a function on Id , for approximately recovering f from the sampled values f (x ), , f (x n ), we define the linear sampling algorithm L n (X n , n , ·) by n L n (X n , n, f ) := f (x j )ϕ j (1.1) j=1 Let B be a quasi-normed space of functions on Id , equipped with the quasi-norm · B For f ∈ B, we measure the recovery error by f − L n (X n , n , f ) B Let W ⊂ B To study optimality of linear sampling algorithms of the form (1.1) for recovering f ∈ W from n of their values, we will use the quantity of optimal sampling recovery rn (W, B) := inf Xn , f − L n (X n , sup f ∈W n n, f) B Further, let n = {λ j }nj=1 be a sequence of n numbers For a f ∈ C(Id ), we want to approximately compute the integral I ( f ) := Id f (x) dx by the cubature formula n In (X n , n, f ) := λ j f (x j ) j=1 To study the optimality of cubature formulas for f ∈ W , we use the quantity of optimal cubature i n (W ) := inf sup |I ( f ) − In (X n , n , f )| Xn , n f ∈W Recently, there has been increasing interest in solving approximation and numerical problems that involve functions depending on a large number d of variables Without further assumptions, the computation time typically grows exponentially in 123 Found Comput Math d, and the problems become intractable already for mild dimensions d This is the so-called curse of dimensionality [2] In sampling recovery and numerical integration, a classical model in attempt to overcome it which has been widely studied, is to impose certain mixed smoothness or more general anisotropic smoothness conditions on the function to be approximated, and to employ sparse grids for construction of approximation algorithms for sampling recovery or integration We refer the reader to [6,24,34,35] for surveys and the references therein on various aspects of this direction Sparse grids for sampling recovery and numerical integration were first considered by Smolyak [38] He constructed the following grid of dyadic points (m) := {2−k s : k ∈ D(m), s ∈ I d (k)}, where D(m) := {k ∈ Zd+ : |k|1 ≤ m} and I d (k) := {s ∈ Zd+ : ≤ si ≤ 2ki , i ∈ [d]} Here and in what follows, we use the notations: x y := (x1 y1 , , xd yd ); 2x := d |xi | for x, y ∈ Rd ; [d] denotes the set of all natural (2x1 , , 2xd ); |x|1 := i=1 numbers from to d; xi denotes the ith coordinate of x ∈ Rd , i.e., x := (x1 , , xd ) Observe that (m) is a sparse grid of the size 2m m d−1 in comparing with the standard full grid of the size 2dm In approximation theory, Temlyakov [40–42] and the author of the present paper [13–15] developed Smolyak’s construction for studying the asymptotic order of rn (W, L q (Td )) for periodic Sobolev classes W pa and Nikol’skii classes H pa having nonuniform mixed smoothness a = (a1 , , ad ) ∈ Rd with different a j > 0, where Td denotes the d-dimensional torus For the uniform mixed smoothness α1, Temlyakov [43] investigated sampling recovery for periodic Sobolev classes W pα1 and Nikol’skii classes H pα1 , and recently, Sickel and Ullrich [36] for periodic Besov classes α1 , where := (1, 1, , 1) ∈ Rd For nonperiodic functions of mixed smoothU p,θ ness, linear sampling algorithms have been recently studied by Triebel [45] (d = 2), D˜ung [18], Sickel and Ullrich [37], using the mixed tensor product of B-splines and Smolyak grids (m) Smolyak grids are a counterpart of hyperbolic crosses which are frequency domains of trigonometric polynomials widely used for approximations of functions with a bounded mixed smoothness These hyperbolic cross trigonometric approximations are initiated by Babenko [1] For further surveys and references on the topic see [12,21,41,43], and the more recent contributions [36,46] In computational mathematics, the sparse grid approach was first considered by Zenger [51] Numerical integration using sparse grids was investigated in [23] For nonperiodic functions of mixed smoothness of integer order, linear sampling algorithms on sparse grids have been investigated by Bungartz and Griebel [6] employing hierarchical Lagrangian polynomials multilevel basis and measuring the approximation error in the L -norm and energy H -norm There is a very large number of papers on sparse grids in various problems of approximations, sampling recovery and integration with applications in data mining, mathematical finance, learning theory, numerical solving of PDE and stochastic PDE, etc to mention all of them The reader can see the surveys in [6,24,30] and the references therein For recent further developments and results, see in [4,22,27–29] 123 Found Comput Math Quasi-interpolation based on scaled B-splines with integer knots possesses good local and approximation properties for smooth functions, see [9, pp 63–65], [8, pp 100–107] It can be an efficient tool in some high-dimensional approximation problems, especially in applications ones Thus, one of the important bases for sparse grid high-dimensional approximations having various applications is the Faber functions (hat functions) which are piecewise linear B-splines of second order [4,6,22,24,27–29] The representation by Faber basis can be obtained by the B-spline quasi-interpolation (see, e g., [18]) In the recent paper [18], by using a quasiinterpolation representation of functions by mixed high-order B-spline series, we have constructed linear sampling algorithms L n (X n , n , f ) on Smolyak grids (m), for α1 , which is defined as the unit ball functions on Id from the nonperiodic Besov class U p,θ d of the Besov space B α1 p,θ of functions on I having uniform mixed smoothness α For various < p, θ, q ≤ ∞ and α > 1/ p, we proved upper bounds for the worst-case error sup f ∈U α1 f − L n (X n , n , f ) q which in some cases, coincide with the asymptotic p,θ order α1 rn (U p,θ , L q (Id )) (d−1)b n −α+(1/ p−1/q)+ log2 n, (1.2) where b = b(α, p, θ, q) > and x+ := max(0, x) for x ∈ R In the paper [21], we have obtained the asymptotic order of optimal sampling α1 for recovery on Smolyak grids in the L q (Id )-quasi-norm of functions from U p,θ < p, θ, q ≤ ∞ and α > 1/ p It is necessary to emphasize that any sampling algorithm on Smolyak grids always gives a lower bound of recovery error of the form (d−1)b n, b > Unfortunately, as in the right side of (1.2) with the logarithm term log2 in the case when the dimension d is very large and the number n of samples is rather (d−1)b n which grows fast exponentially in d To mild, the main term becomes log2 avoid this exponential growth, we impose on functions other anisotropic smoothness and construct appropriate sparse grids for functions having them Namely, we extend α,β a for the above study to functions on Id from the classes U p,θ for α > 0, β ∈ R, and U p,θ a ∈ Rd with a1 < a2 ≤ · · · ≤ ad , which are defined as the unit ball of the Besov-type α,β α,β spaces B p,θ and B ap,θ , respectively The space B p,θ and B ap,θ are certain sets of functions with bounded mixed modulus of smoothness Both of them are generalizations in α,β different ways of the space B α1 p,θ of mixed smoothness α The space B p,θ is a “hybrid” β of the space B α1 p,θ and the classical isotropic Besov space B p,θ of smoothness β α,β The space B p,θ is a Besov-type generalization of the Sobolev type space H α,β = α,β B2,2 The latter space has been introduced in [30] for solutions of the following elliptic variational problems a(u, v) = ( f, v) for all v ∈ H γ , where f ∈ H −γ and a : H γ × H γ → R is a bilinear symmetric form satisfying the conditions a(u, v) ≤ λ u H γ v H γ and a(u, u) ≥ μ u 2H γ By use of tensor-product biorthogonal wavelet bases, the authors of these papers constructed so-called optimized sparse grid subspaces for finite element approximations of the solution having H α,β -regularity, whereas the approximation error is measured in the energy norm of isotropic Sobolev space H γ They generalized the construction of [5] for a hyperbolic cross approxi- 123 Found Comput Math mation of the solution of Poisson’s equation to elliptic variational problems A generalization H α,β (R3 ) N of the space H α,β of functions on (R3 ) N , based on isotropic Sobolev smoothness of the space H (R3 ), has been considered by Yserentant [48– 50] for solutions u : (R3 ) N → R : (x1 , , x N ) → u(x1 , , x N ) of the electronic Schrödinger equation H u = λu for eigenvalue problem where H is the Hamilton operator He proved that the eigenfunctions are contained in the intersection of spaces H 1,0 (R3 ) N ∩ ∩ϑ 1/ p and α > (γ − β)/d if β > γ , and α > γ − β if β < γ (with the additional restriction < q < ∞ in the case γ > 0) Then we explicitly constructed a set n such that |G( n )| ≤ n and sup α,β f ∈U p,θ f − L n n (X n , n, f) γ Wq (Id ) α,β γ rn U p,θ , Wq (Id ) n −α−(β−γ )/d+(1/ p−1/q)+ , β > γ , β < γ n −α−β+γ +(1/ p−1/q)+ , (1.4) From (1.4) for the case γ = 0, p = 1, we derived that α,β i n U p,θ n −α−β/d+(1/ p−1)+ , β > 0, β < n −α−β+(1/ p−1)+ , 123 Found Comput Math α,β The set n is specially constructed for the class of U p,θ , depending on the relationship between < p, θ, q, τ ≤ ∞ and α, β, respectively The grids G( n ) are sparse and have much smaller number of sample points than the corresponding standard full grids and Smolyak grids, but give the same error of the sampling recovery on the both latter ones The construction of asymptotically optimal linear sampling algorithms L n n (X n , n , ·) is essentially based on quasi-interpolation representations by α,β B-spline series of functions f ∈ B p,θ with a discrete equivalent quasi-norm in terms of the coefficient function-valued functionals of this series Moreover, for the sampling recovery in the L -norm, L n n (X n , n , ·) generates an asymptotically optimal cubature formula (see Sect for details) a , we preliminarily notice the following For To discuss results on the class U p,θ the nonuniform mixed smoothness a with < a1 = · · · = aν < aν+1 ≤ · · · ≤ ad , it is known that in many approximation problems asymptotic characteristics of corresponding function classes with smoothness a the extra log n appears in the form (log n)(ν−1)b (see, for example, [12,41] and references there) In the case ν = 1, the extra log n disappears independently of b This makes the problem of finding the optimal rate in the case ν = much easier than in the case ν > Thus, it was proven in [40] that for ≤ p ≤ ∞, r > 1/ p, a (Td ), L q (Td )) rn (U p,∞ n −a1 (log n)(ν−1)(a1 +1) (1.5) a−(1/ p−1/q) + and wellCombining this with the well-known embedding B ap,θ → Bq,∞ known lower bounds in the univariate case, we obtain for the case ν = 1, a (Td ), L q (Td )) rn (U p,θ n −a1 +(1/ p−1/q)+ (0 < θ ≤ ∞) It is important to emphasize that linear sampling algorithms constructed in [40] which give the upper bounds (1.5) and which are asymptotically optimal for the case ν = 1, are developed from a construction in [38], but essentially based on extended nonuniform Smolyak grids These grids are a counterpart of extended hyperbolic crosses suggested by Teljakovskii [39] (for further development of Teljakovskii’s construction in hyperbolic cross approximation of functions having one or several nonuniform mixed smoothness, see [12,41] for surveys and references there) Extended nonuniform Smolyak grids and their modifications then were used in sampling recovery problems in [13–15,27,28,40–42] In the present paper, we are interested in constructing asymptotically optimal linear a with nonuniform mixed sampling algorithms for the nonperiodic Besov class U p,θ smoothness a More precisely, if < p, θ, q ≤ ∞ and a ∈ Rd with 1/ p < a1 < a2 ≤ · · · ≤ ad , we explicitly constructed a set n such that |G( n )| ≤ n and the sampling algorithm L n n (X n , sup a f ∈U p,θ f − L n n (X n , n, n, f) a , i e., f ) is asymptotically optimal for the class U p,θ q a rn (U p,θ , L q (Id )) n −a1 +(1/ p−1/q)+ (1.6) The construction of the sampling algorithms L n n (X n , n , f ) on the grids G( n ) is similar to that in [40] for the case ν = The main contribution of the present paper 123 Found Comput Math is a theorem on quasi-interpolation representation by B-spline series of functions f ∈ B ap,θ with a discrete equivalent quasi-norm in terms of the coefficient functionvalued functionals of this series This theorem plays a key role in constructing the a , asymptotically optimal linear sampling algorithms L n n (X n , n , f ) for the class U p,θ as well in proving the relation (1.6) In the present paper, we consider only two kinds of anisotropic smoothness spaces α,β B p,θ and B ap,θ However, our constructions and methods of proofs of results can be extended to other kinds of anisotropic smoothness, see examples in Remark at the end of Sect We are restricted to compute the asymptotic order of rn with respect only to n when n → ∞, not analyzing the dependence on the number of variables d Recently, in [20], Kolmogorov n-widths dn (U, H γ ) and ε-dimensions n ε (U, H γ ) in space H γ of periodic multivariate function classes U have been investigated in highdimensional settings, where U is the unit ball in H α,β or its subsets We computed the accurate dependence of dn (U, H γ ) and n ε (U, H γ ) as a function of two variables n, d or ε, d Although n is the main parameter in the study of convergence rate with respect to n when n → ∞, the parameter d may affect this rate when d is large It is interesting and important to investigate optimal sampling recovery and cubature in such high-dimensional settings In the recent paper [19], we have constructed linear algorithms of sampling recovery and cubature formulas on Smolyak grids of periodic d-variate functions having Lipschitz-Hölder mixed smoothness based on B-spline quasi-interpolation, and established upper and lower estimates of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit in d and n when the number d of variables and the number n of sampled function values may be very large The present paper is organized as follows In Sect 2, we give definitions of Besov-type spaces B p,θ of functions with bounded mixed modulus of smoothness, α,β in particular, spaces B p,θ and B ap,θ , and prove theorems on quasi-interpolation representation by B-spline series, with relevant discrete equivalent quasi-norms In Sect 3, we construct linear sampling algorithms on sparse grids of the form (1.3) for function α,β a , and prove upper bounds for the error of recovery by these classes U p,θ and U p,θ algorithms In Sect 4, we prove the sparsity and asymptotic optimality of the linear α,β sampling algorithms constructed in Sect 3, for the quantities rn (U p,θ , L q (Id )) and a , L (Id )), and establish their asymptotic orders In Sect 5, we extend the invesrn (U p,θ q α,β γ tigations of Sects and to the quantities rn (U p,θ , Wq (Id )) for γ > In Sect 6, we discuss the problem of optimal cubature formulas for numerical integration in terms α,β a ) of i n (U p,θ ) and i n (U p,θ Function Spaces and Quasi-Interpolation Representations 2.1 Function Spaces Let us first introduce spaces B p,θ of functions with bounded mixed modulus of smoothα,β ness and Besov-type spaces B p,θ and B ap,θ of functions with anisotropic smoothness, 123 Found Comput Math as well fractional isotropic Sobolev and Besov spaces W pα and B αp,θ , and give necessary knowledge of them Let G be a domain in R For univariate functions f on G, the r th difference operator r is defined by h r r r f (x + j h) ( f, x) := (−1)r − j h j j=0 If e is any subset of [d], for multivariate functions on Gd , the mixed (r, e)th difference operator r,e h is defined by r,e h := r hi , r,∅ h := I, i∈e where the univariate operator rh i is applied to the univariate function f by considering f as a function of variable xi with the other variables held fixed, and I ( f ) := f for functions f on Gd Denote by L p (Gd ) the quasi-normed space of functions on Gd with the pth integral quasi-norm · p,Gd for < p < ∞, and the sup-norm · ∞,Gd for p = ∞ Let r,e d ωre ( f, t) p,Gd := sup h ( f ) p,Gd (r,h,e) , t ∈ R+ , |h i |≤ti ,i∈e be the mixed (r, e)th modulus of smoothness of f , where Gd (r, h, e) := {x ∈ Gd : xi , xi + r h i ∈ G, i ∈ e} (in particular, ωr∅( f, t) p,Gd = f p,Gd ) For x, x ∈ Rd , the inequality x ≤ x (x < x) means xi ≤ xi (xi < xi ), i ∈ [d] : Rd+ → R+ be a function satisfying Denote: R+ := {x ∈ R : x ≥ 0} Let conditions (t) > 0, t > 0, t ∈ Rd+ , (t) ≤ C (t ), t ≤ t , t, t ∈ (2.1) Rd+ , (2.2) and for a fixed γ ∈ Rd+ , γ ≥ 1, there is a constant C = C (γ ) such that for every λ ∈ Rd+ with λ ≤ γ , (λ t) ≤ C (t), t ∈ Rd+ (2.3) te For e ⊂ [d], we define the function e : Rd+ → R+ by e (t) := (t e ), where ∈ Rd+ is given by t ej = t j if j ∈ e, and t ej = otherwise If < p, θ ≤ ∞, we introduce the quasi-semi-norm | f | B (e) for functions p,θ f ∈ L p (Gd ) by 123 Found Comput Math | f |B p,θ (e) := (in particular, | f | B B p,θ (Gd ) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p,θ (∅) Id {ωre ( f, t) p,Gd / e (t)} sup ωre ( f, t) p,Gd / t∈Id = f p,Gd ) 1/θ ti−1 dt θ i∈e e (t), , θ < ∞, θ = ∞, For < p, θ ≤ ∞, the Besov-type space is defined as the set of functions f ∈ L p (Gd ) for which the quasi-norm f B p,θ (Gd ) := | f |B p,θ (e) e⊂[d] is finite In what follows, we assume that the function satisfies the conditions (2.1)– (2.3) Bn ( f ) if An ( f ) ≤ C Bn ( f ) with C an absolute We use the notations: An ( f ) Bn ( f ) constant not depending on n and/or f ∈ W, and An ( f ) Bn ( f ) if An ( f ) An ( f ) Put Z+ := {s ∈ Z : s ≥ 0} and Zd+ (e) := {s ∈ Zd+ : si = and Bn ( f ) 0, i ∈ / e} for a set e ⊂ [d] Lemma 2.1 Let < p, θ ≤ ∞ Then we have the following quasi-norm equivalence f B p,θ ( Gd ) ωre ( f, 2−k ) p,Gd / (2−k ) B1 ( f ) := e⊂[d] θ 1/θ k∈Zd+ (e) with the corresponding change to sup when θ = ∞ Proof This lemma follows from properties of mixed modulus of smoothness ωre ( f, t) p,Gd and the properties (2.1)–(2.3) of the function We prove it for completeness The lemma will be proven if we show that for every e ⊂ [d], | f |B p,θ (e) ωre ( f, 2−k ) p / (2−k ) θ 1/θ , (2.4) k∈Zd+ (e) with the corresponding change to sup when θ = ∞ Let us prove this semi-norms equivalence for instance, for e = [d], ≤ p < ∞ and < θ < ∞ The general case can be proven in a similar way with a slight modification Put D(k) := {x ∈ Rd+ : k ≤ x < k + 1} and use the abbreviation ωr ( f, ·) p := ωr[d] ( f, ·) p By (2.1)–(2.3), we have (2−x ) (2−k ), x ∈ D(k), k ∈ Zd+ (2.5) From the monotonicity of ωr ( f, ·) in each variable and the inequality d ωr ( f, c t) p ≤ (1 + c j )r ωr ( f, t) p , c ∈ Rd+ , c > 0, j=1 123 Found Comput Math we obtain ωr ( f, 2−x ) p ωr ( f, 2−k ) p , x ∈ D(k), k ∈ Zd+ (2.6) Setting I (k) := {t ∈ Id : 2−k−1 ≤ t ≤ 2−k }, by (2.5) and (2.6), we have | f |θB p,θ ([d]) = I (k) k∈Zd+ ti−1 dt {ωr ( f, t) p / (t)}θ i∈[d] {ωr ( f, 2−x ) p / (2−x )}θ dx = D(k) k∈Zd+ {ωr ( f, 2−k ) p / (2−k )}θ k∈Zd+ α,β Let us define the Besov-type spaces B ap,θ (Gd ) and B p,θ (Gd ) of functions with anisotropic smoothness as particular cases of B p,θ (Gd ) For a ∈ Rd+ , we define the space B ap,θ (Gd ) of mixed smoothness a by d B ap,θ (Gd ) := B p,θ (Gd ), where tiai , t ∈ Rd+ (t) = (2.7) i=1 α,β Let α ∈ R+ and β ∈ R with α + β > We define the space B p,θ (Gd ) as follows α,β B p,θ (Gd ) := B p,θ (Gd ), where (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ d i=1 d i=1 β tiα inf t j , β ≥ 0, j∈[d] (2.8) tiα sup j∈[d] β tj , β < The definition (2.8) seems different for β > and β < However, it can be well interpreted in terms of the equivalent discrete quasi-norm B1 ( f ) in Lemma 2.1 Indeed, the function in (2.8) for both β ≥ and β < satisfies the assumptions (2.1)–(2.3) and moreover, 1/ (2−x ) = 2α|x|1 +β|x|∞ , x ∈ Rd+ , where |x|∞ := max j∈[d] |x j | for x ∈ Rd Hence, by Lemma 2.1, we have the following quasi-norm equivalence f α,β B p,θ (Gd ) 2α|k|1 +β|k|∞ ωre ( f, 2−k ) p,Gd e⊂[d] θ 1/θ (2.9) k∈Zd+ (e) α,β with the corresponding change to sup when θ = ∞ The notation B p,θ (Gd ) becomes explicitly reasonable if we take the right side of (2.9) as a definition of the quasi-norm α,β α1 d d of the space B p,θ (Gd ) Notice that B α,0 p,θ (G ) = B p,θ (G ) However, in general, the 123 Found Comput Math for τ = θ ∗ , q ∗ , θ For simplicity, we prove the case ( p, θ, q) ∈ B for τ = and (1/ p − 1/q)+ ) = 0, the general case can be proven similarly In this particular case, we get f − R (ξ ) ( f ) q 2−(a,k) =: (ξ ) (3.13) sup a f ∈U p,θ k∈Zd+ \ (ξ ) It is easy to verify that for every ξ > 0, (ξ ) W (ξ ) 2−(a,x) dx, (3.14) where W (ξ ) := {x ∈ Rd+ : (a , x) > ξ } We put V (ξ, s) := {x ∈ W (ξ ) : ξ + s − ≤ (a, x) < ξ + s}, s ∈ N then from (3.14) we have (ξ ) 2−ξ ∞ 2−s |V (ξ, s)| (3.15) s=1 Let us estimate |V (ξ, s)| Put V ∗ (ξ, s) := V (ξ, s) − x ∗ , where x ∗ := (a1 )−1 ξ e1 For every y = x − x ∗ ∈ V ∗ (ξ, s), from the equation (a, x ∗ ) = ξ and the inequality (a, x) < ξ + s we get (a, y) < s and therefore, y j < s/a j , j ∈ [d] (3.16) On the other hand, for every x ∈ V (ξ, s), from the inequality (a, x) < ξ + s and (a, x) − ε(1 , x) = (a , x) > ξ we get (1 , x) < ε−1 s, where := (0, 1, 1, , 1) ∈ Rd This inequality together with the inequality a1 x1 + ad (1 , x) ≥ (a, x) ≥ ξ + s − gives x1 ≥ ξ/a1 + ((1 − ε−1 ad )s + 1)/a1 for every x ∈ V (ξ, s) Hence, for every y = x − x ∗ ∈ V ∗ (ξ, s), y1 ≥ ((1 − ε−1 ad )s + 1)/a1 , y j ≥ 0, j = 2, , d (3.17) This means that V ∗ (ξ, s) ⊂ V (s) for every ξ > 0, where V (s) ⊂ Rd is the box of all y ∈ Rd given by the conditions (3.16) and (3.17) Since |V (ξ, s)| = |V ∗ (ξ, s)| ≤ |V (s)| sd , by (3.13) and (3.15), we obtain sup a f ∈U p,θ 123 f −R (ξ ) q 2−ξ ∞ s=1 2−s s d 2−ξ Found Comput Math Remark The grids G( (ξ )) and G( (ξ )) defined for ( p, θ, q) ∈ A or ( p, θ, q) ∈ B with β > 0, were employed in [13–15] for sampling recovery of periodic functions from an intersection of spaces of different mixed smoothness Independently of our β paper, similar upper bounds of the error in W2 (Td )-norm of linear sampling algorithms by interpolation for functions from periodic space W2α1 (Td ) have recently obtained in [26] Sparsity and Optimality Denote by x the integer part of x ∈ R+ Lemma 4.1 Let < p, θ, q ≤ ∞ and α ∈ R+ , β ∈ R, β = 0, such that 1/ p < min(α, α + β) ≤ max(α, α + β) < r Then we have 2|k|1 |G( (ξ ))| 2ξ/ν , (4.1) k∈ (ξ ) where ν := α + β/d − (1/ p − 1/q)+ , β > 0, β < α + β − (1/ p − 1/q)+ , (4.2) Proof The first asymptotic equivalence in (4.1) follows from the definitions Let us prove the second one For simplicity we prove it for the case where p ≥ q, the general case can be proven similarly Let us first consider the case ( p, θ, q) ∈ B, β > It is easy to verify that for every ξ > 0, 2|k|1 k∈ (ξ ) W (ξ ) 2(1,x) dx, (4.3) where W (ξ ) := {x ∈ Rd+ : (α + ε/d)(1, x) + (β − ε)M(x) ≤ ξ } and M(x) := max j∈[d] x j for x ∈ Rd We put V (ξ, s) := {x ∈ W (ξ ) : ξ/ν + s − ≤ (1, x) < ξ/ν + s}, s ∈ Z+ From the inequalities β > ε and M(x) − (1, x)/d ≥ 0, x ∈ Rd+ , one can verify that for every x ∈ W (ξ ), (1, x) ≤ ξ/ν Hence, we have W (ξ ) 2(1,x) dx 2ξ/ν ξ/ν 2−s |V (ξ, s)| (4.4) s=0 123 Found Comput Math Let us estimate |V (ξ, s)| Put V ∗ (ξ, s) := V (ξ, s) − x ∗ , where x ∗ := (νd)−1 ξ From the equation (1, x ∗ ) = ξ/ν, we get for every y = x − x ∗ ∈ V ∗ (ξ, s), s − ≤ (1, y) < s (4.5) (α + ε/d)(1, y) + (β − ε)M(y) ≤ (4.6) and This means that V ∗ (ξ, s) ⊂ V (s) for every ξ > 0, where V (s) ⊂ Rd is the set of all y ∈ Rd given by the conditions (4.5) and (4.6) Notice that V (s) is a bounded s d−1 Hence, by the inequality polyhedron and |V (s)| |V (ξ, s)| = |V ∗ (ξ, s)| ≤ |V (s)|, (4.3) and (4.4), we prove the upper bound in (4.1): 2|k|1 2ξ/ν k∈ (ξ ) ∞ 2−s s d−1 2ξ/ν s=0 To prove the lower bound for this case, we take k ∗ := ξ/dν ∈ Zd+ It is easy to check k ∗ ∈ (ξ ) and consequently, 2|k|1 ≥ 2|k ∗| 2ξ/ν k∈ (ξ ) The case ( p, θ, q) ∈ B, β < can be proven similarly with a slight modification To prove the case ( p, θ, q) ∈ A, it is enough to put ε = in the proof of the case ( p, θ, q) ∈ B Lemma 4.2 Let < p, θ, q ≤ ∞ and a ∈ Rd+ satisfying the condition (3.11) and 1/ p < a1 < ad < r Then we have |G( 2|k|1 (ξ ))| k∈ 2ξ/(a1 −(1/ p−1/q)+ ) (4.7) (ξ ) Proof The first asymptotic equivalence in (4.7) follows from the definitions Let us prove the second one For simplicity, we prove it for the case where p ≥ q, the general case can be proven similarly Let us first consider the case ( p, θ, q) ∈ B It is easy to verify that for every ξ > 0, 2|k|1 k∈ 123 (ξ ) W (ξ ) 2(1,x) dx, (4.8) Found Comput Math where W (ξ ) := {x ∈ Rd+ : (a , x) ≤ ξ } We put V (ξ, s) := {x ∈ W (ξ ) : ξ/a1 + s − ≤ (1, x) < ξ/a1 + s}, s ∈ Z+ One can verify that for every x ∈ W (ξ ), (1, x) ≤ ξ/a1 Hence, we have 2(1,x) dx W (ξ ) ξ/a1 2ξ/a1 2−s |V (ξ, s)| (4.9) s=0 Let us estimate |V (ξ, s)| Put V ∗ (ξ, s) := V (ξ, s) − x ∗ , where x ∗ := (a1 )−1 ξ e1 From the equation (1, x ∗ ) = ξ/a1 , we get for every y = x − x ∗ ∈ V ∗ (ξ, s), s − ≤ (1, y) < s (4.10) and (a , y) ≤ (4.11) V ∗ (ξ, s) Rd This means that ⊂ V (s) for every ξ > 0, where V (s) ⊂ is the set of all y ∈ Rd given by the conditions (4.10) and (4.11) Notice that V (s) is a bounded s d−1 Hence, by the inequality polyhedron and |V (s)| |V (ξ, s)| = |V ∗ (ξ, s)| ≤ |V (s)|, (4.8) and (4.9), we obtain the upper bound in (4.7): k∈ |k|1 ξ/a1 (ξ ) ∞ 2−s s d−1 2ξ/a1 s=0 To prove the lower bound, we take k ∗ := ξ/a1 e1 ∈ Zd+ It is easy to check k ∗ ∈ and consequently, ∗ 2|k|1 ≥ 2|k |1 2ξ/a1 k∈ (ξ ) (ξ ) Remark The grids of sample points G( (ξ )) and G( (ξ )) are sparse and have much less elements than the standard dyadic full grids G( (ξ )) and Smolyak grids G( (ξ )) which give the same recovery error, where (ξ ) := {k ∈ Zd+ : λ|k|∞ ≤ ξ } and (ξ ) := {k ∈ Zd+ : λ|k|1 ≤ ξ } and the number λ := ν is as in (4.2) for G( (ξ )) and λ := a1 −(1/ p−1/q)+ for G( (ξ )) For instance, the linear sampling algorithms R i (ξ ) , i = 1, 2, on the grids G( i (ξ )) give the worst-case error sup α,β f ∈U p,θ f −R i (ξ ) (f) q 2−ξ 123 Found Comput Math The number of sample points in G( (ξ )) is |G( (ξ ))| 2dξ/ν and in G( (ξ )) is |G( (ξ ))| 2ξ/ν ξ d−1 , whereas due to Theorem 3.1 and Lemma 4.1 we can get the same error by the linear sampling algorithm R (ξ ) on the grids G( (ξ )) with the number of sample points |G( (ξ ))| 2ξ/ν The following two theorems show that the linear sampling algorithms R (ξ ) on sparse grids G( (ξ )), and R (ξ ) on sparse grids G( (ξ )) are asymptotically optimal in the sense of the quantity rn Theorem 4.1 Let < p, θ, q ≤ ∞ and α ∈ R+ , β ∈ R, β = 0, such that 1/ p < min(α, α + β) ≤ max(α, α + β) < r Assume that for a given n ∈ Z+ , ξn is the largest nonnegative number such that |G( (ξn ))| ≤ n (4.12) Then R (ξn ) defines an asymptotically optimal linear sampling algorithm for rn := α,β rn (U p,θ , L q ) by R (ξn ) ( f ) = L n (X n∗ , ∗ n, f (2−k s)ψk,s , f) = (4.13) (k,s)∈K ( (ξn )) where X n∗ := G( (ξn )) = {2−k s}(k,s)∈K ( have the following asymptotic orders sup α,β f −R (ξn ) ( f ) q (ξn )) , ∗ n := {ψk,s }(k,s)∈K ( (ξn )) , n −α−β/d+(1/ p−1/q)+ , β > 0, β < n −α−β+(1/ p−1/q)+ , rn f ∈U p,θ and we (4.14) Proof Upper bounds Due to Lemma 4.1, we have n 2ξn /ν |G( (ξn ))| ≤ n, where ν is as in (4.2) Hence, we find n −α+β/d−(1/ p−1/q)+ , β > 0, β < n −α+β−(1/ p−1/q)+ , 2−ξn (4.15) By Lemma 3.1 and (4.12), R (ξn ) is a linear sampling algorithm of the form (1.1) as in (4.13) and consequently, from Theorem 3.5, we get rn ≤ sup α,β f ∈U p,θ f −R (ξn ) ( f ) q 2−ξn These relations together with (4.15) prove the upper bounds of (4.14) 123 Found Comput Math Lower bounds We need the following auxiliary result If W ⊂ L q , then we have rn (W, L q (Id )) inf X n ={x j }mj=1 ⊂Id sup f f ∈W : f (x j )=0, j=1, ,n q (4.16) For the proof of this inequality see [33, Proposition 19] Since f q ≥ f p for p ≥ q, it is sufficient to prove the lower bound for the case p ≤ q Fix a number r = 2m with integer m so that max(α, α + β) < min(r , r − + 1/ p) We first treat the case β > Put k ∗ = k ∗ (η) := η1 for an integer η > m Consider the boxes J (s) ⊂ Id J (s) := {x ∈ Id : 2−η+m s j ≤ x j < 2−η+m (s j + 1), j ∈ [d]}, where Z (η) := {s ∈ Zd+ : ≤ s j ≤ 2η−m − 1, s ∈ Z (η), j ∈ [d]} For a given n, we find η satisfying the relations 2|k n ∗| 2d(η−m) = |Z (η)| ≥ 2n (4.17) Let X n = {x j }nj=1 be an arbitrary subset of n points in Id Since J (s) ∩ J (s ) = ∅ for s = s , and |Z (η)| ≥ 2n, there is Z ∗ (η) ⊂ Z (η) such that |Z ∗ (η)| ≥ n and X n ∩ {∪s∈Z ∗ (η) J (s)} = ∅ Consider the function g ∗ ∈ d ∗ r (k ) g ∗ := λ2−α|k ∗| (4.18) defined by −β|k ∗| ∞ +|k ∗| 1/ p Mk ∗ ,s+r /2 , (4.19) s∈Z ∗ (η) where Mk ∗ ,s+r /2 are B-splines of order r Since |Z ∗ (η)| g∗ and λ2−α|k q g∗ ∗| −β|k ∗| λ2−α|k p ∞ +(1/ p−1/q)|k ∗| −β|k ∗| ∞ 2|k ∗| ∗| , by (2.24) we have , (4.20) α,β Hence, by Corollary 2.1, there is λ > independent of η and n such that g ∗ ∈ U p,θ Notice that Mk ∗ ,s+m−1 (x), x ∈ / J (s), for every s ∈ Z ∗ (η), and consequently, by ∗ j (4.18) g (x ) = 0, j = 1, , n From (4.16), (4.17) and (4.20), we obtain rn g∗ q n −α−β/d+1/ p−1/q This proves the lower bound of (4.14) for the case β > 123 Found Comput Math We now consider the case β < We will use some notations which coincide with those in the proof of the case β > Put k ∗ = k ∗ (η) := (η, m, , m) for integer η > m Consider the boxes J (s) ⊂ Id J (s) := {x ∈ Id : 2−η+m s1 ≤ x1 < 2−η+m (s1 + 1)}, s ∈ Z (η), where Z (η) := {s ∈ Zd+ : ≤ s1 ≤ 2η−m − 1, s j = 0, j = 2, , d} For a given n, we find η satisfying the relations ∗ 2η−m = |Z (η)| ≥ 2n 2k n (4.21) Let X n = {x j }nj=1 be an arbitrary subset of n points in Id Since J (s) ∩ J (s ) = ∅ for s = s , and |Z (η)| ≥ 2n, there is Z ∗ (η) ⊂ Z (η) such that |Z ∗ (η)| ≥ n and X n ∩ {∪s∈Z ∗ (η) J (s)} = ∅ Consider the function g ∗ ∈ d ∗ r (k ) (4.22) defined by ∗ g ∗ := λ2−(α+β−1/ p)k1 Mk ∗ ,s+r /2 , (4.23) s∈Z ∗ (η) where Mk ∗ ,s+r /2 are B-splines of order r Since |Z ∗ (η)| g∗ and ∗ λ2−(α+β−1/ p+1/q)k1 , q g∗ ∗ 2k1 , by (2.24) we have (4.24) ∗ p λ2−(α+β)k1 α,β Hence, by Corollary 2.1, there is λ > independent of η and n such that g ∗ ∈ U p,θ Notice that Mk ∗ ,s+m−1 (x), x ∈ / J (s), for every s ∈ Z ∗ (η), and consequently, by (4.22) g ∗ (x j ) = 0, j = 1, , n From the inequality (4.16), (4.24) and (4.21), we obtain α,β g∗ q n −α−β+1/ p−1/q rn (U p,θ , L q ) This proves the lower bound of (4.14) for the case β < Theorem 4.2 Let < p, θ, q ≤ ∞ and a ∈ Rd+ satisfying the condition (3.11) and 1/ p < a1 < a2 ≤ · · · ≤ ad < r Assume that for a given n ∈ Z+ , ξn is the largest nonnegative number such that |G( 123 (ξn ))| ≤ n (4.25) Found Comput Math Then R (ξn ) defines an asymptotically optimal linear sampling algorithm for rn := α,β rn (U p,θ , L q (Id )) by R (ξn ) ( f ) = L n (X n∗ , ∗ n, f (2−k s)ψk,s , f) = (k,s)∈K ( where X n∗ := G( (ξn )) = {2−k s}(k,s)∈K ( have the following asymptotic order sup a f ∈U p,θ f −R (ξn ) ( f ) q (ξn )) , ∗ n rn (ξn )) := {ψk,s }(k,s)∈K ( (ξn )) , n −a1 +(1/ p−1/q)+ and we (4.26) Proof Upper bounds For a given n ∈ Z+ (large enough), due to Lemma 4.2 we can define ξ = ξn as the largest nonnegative number such that n 2ξn /(a1 −(1/ p−1/q)+ ) Hence, we find 2−ξn |G( (ξn ))| ≤ n n −a1 +(1/ p−1/q)+ (4.27) By Lemma 3.1 and (4.25), R (ξn ) is a linear sampling algorithm of the form (1.1) and consequently, from Theorem 3.2, we get rn ≤ sup a f ∈U p,θ f −R (ξn ) ( f ) q 2−ξn These relations together with (4.27) prove the upper bounds for (4.26) Lower bounds As in the proof of Theorem 4.1, it is sufficient to prove the lower bound for the case p ≤ q Fix a number r = 2m with integer m so that rd < min(r, r − + 1/ p) In the next steps, the proof is similar to the proof of the lower bound for the case β < in Theorem 4.1 Indeed, we can repeat almost all the details in it with replacing α + β by a1 Remark Concerning the asymptotically optimal sparse grids of sampling points α,β a , L ), it is worth to notice G( (ξn )) and G( (ξn )) for rn (U p,θ , L q ) and rn (U p,θ q the following Let set A and B be the sets of triples ( p, θ, q) introduced in Sect For every triple ( p, θ, q) ∈ A, we can define the best choice of family of asymptotically optimal sparse grids G( (ξn )) and G( (ξn )), whereas for a triple ( p, θ, q) ∈ B, there are many families of asymptotically optimal sparse grids G( (ξn )) and G( (ξn )) α,β a , L ), respectively depending on parameter ε > 0, for rn (U p,θ , L q ) and rn (U p,θ q Moreover, the parameter ε > plays a crucial role in the construction of asymptotically optimal sparse grids for ( p, θ, q) ∈ B Indeed, to understand the substance let us consider, for instance, the problem of asymptotically optimal sparse grids for even α,β the simplest case rn (U2,2 , L )) with β < Suppose that for this case instead the set (ξ ) := {k ∈ Zd+ : (α − ε)|k|1 + (β + ε)|k|∞ ≤ ξ }, 123 Found Comput Math we take the set ˜ (ξ ) := {k ∈ Zd+ : α|k|1 + β|k|∞ ≤ ξ } Then ˜ (ξ ) is a proper subset of (ξ ), i.e., the grid G( (ξ )) is essentially extended from G( ˜ (ξ )) by parameter ε However, |G( (ξ ))| |G( ˜ (ξ ))| On the other hand, α,β the grid G( ˜ (ξ )) cannot be asymptotically optimal for rn (U2,2 , L ) and because for this gird (3.5) is replaced by f − R ˜ (ξ ) ( f ) sup α,β f ∈U2,2 2−ξ ξ d−1 α,β γ (ξ )) holds for rn (U2,2 , W2 (Id )) with A similar optimal property of the grid G( γ > β (see the next section) Sampling Recovery in Energy Norm In this section, we extend the results on sampling recovery in the quasi-norm of L q (Id ) α,β of functions from B p,θ in Sects and 4, to sampling recovery in the energy norm of γ the isotropic Sobolev space Wq (Id ) with γ > and < q < ∞ We preliminarily γ study the sampling recovery in the norm of Bq,τ and then receive results on sampling γ d γ recovery in the norm of Wq (I ) as consequences of those in the norm of Bq,min(q,2) and the inequality (2.13) Put θ := (1 − 1/θ )−1 for ≤ θ ≤ ∞ Lemma 5.1 Let < p, θ ≤ ∞, ≤ q, τ ≤ ∞, < γ < min(r, r − + 1/ p) and {ψ} ψ : Zd+ → R+ Then for every f ∈ B p,θ , we have f − R (f) f γ Bq,τ ⎧ 2−ψ(k)+γ |k|∞ +(1/ p−1/q)+ |k|1 , ⎪ ⎨ sup d {ψ} B p,θ θ ≤ 1, k∈Z+ \ ⎪ ⎩ k∈Zd+ \ {2−ψ(k)+γ |k|∞ +(1/ p−1/q)+ |k|1 }θ 1/θ , θ > (5.1) Proof Let us prove the following Bernstein-type inequality g γ Bq,τ 2γ |k|∞ g q, ∀g ∈ d r (k), ∀k ∈ Zd+ , where we recall that rd (k) is the set of B-splines of the form (2.23) Let g ∈ and k ∈ Zd+ Due to (2.12) it is sufficient to show that 2γ s ωr,i (g, 2−s )q τ 1/τ 2γ k i g q (5.2) d r (k) (5.3) s∈Z+ with the usual change to sup when τ = ∞ Observe that g can be represented by d (k ) for s = k and g = for s = k Hence, the series (2.31) with gs = g ∈ r,i i q i s i 123 Found Comput Math applying Lemma 2.8, we prove (5.3) and therefore, (5.2) The inequality (5.2) together with (2.24) gives g 2γ |k|∞ +(1/ p−1/q)+ |k|1 g γ Bq,τ Since ≤ q, τ ≤ ∞, · d r (k) γ Bq,τ p, ∀g ∈ d r (k), ∀k ∈ Zd+ (5.4) is a norm Consequently, from the inclusions qk ( f ) ∈ {ψ} and (5.4), we obtain for every f ∈ B p,θ , f −R ( f ) γ Bq,τ ≤ qk ( f ) 2γ |k|∞ +(1/ p−1/q)+ |k|1 qk ( f ) γ Bq,τ k∈Zd+ \ p k∈Zd+ \ By use of these inequalities, in a way similar to the proof of Lemma 3.2, we prove the lemma Let < p, θ ≤ ∞, ≤ q, τ ≤ ∞ and α, γ ∈ R+ , β ∈ R be given We fix a number ε so that < ε < min(α − (1/ p − 1/q)+ , |γ − β|), and define the set (ξ ) for ξ > by ⎧ d ⎪ θ ≤ 1, ⎨{k ∈ Z+ : (α − (1/ p − 1/q)+ )|k|1 − (γ − β)|k|∞ ≤ ξ }, (ξ ) := {k ∈ Zd+ : (α − (1/ p − 1/q)+ + ε/d)|k|1 − (γ − β − ε)|k|∞ ≤ ξ }, θ > 1, β > γ , ⎪ ⎩ {k ∈ Zd+ : (α − (1/ p − 1/q)+ − ε)|k|1 − (γ − β + ε)|k|∞ ≤ ξ }, θ > 1, β < γ γ The following theorem gives an upper bound of the error in the quasi-norm of Bq,τ of the sampling recovery by the linear sampling operator R (ξ ) on the grids G( (ξ )) α,β for f ∈ U p,θ It can be proven in a similar way to the proof of Theorem 3.1 with a slight modification on the basis of Lemma 5.1 Theorem 5.1 Let < p, θ ≤ ∞, ≤ q, τ ≤ ∞, α, γ ∈ R+ and β ∈ R, β = γ , satisfy the conditions (γ − β)/d, β > γ , α> γ − β, β < γ, and 1/ p < min(α, α + β) ≤ max(α, α + β) < r, < γ < min(r, r − + 1/ p) Then we have the following upper bound sup α,β f ∈U p,θ f −R (ξ ) ( f ) Bq,τ γ 2−ξ (5.5) 123 Found Comput Math As the next step we need the asymptotic order of the cardinality of the grids G( (ξ )) It can be established in a similar way to Lemma 4.1 with a slight modification More precisely, we have the following Lemma 5.2 Under the assumptions of Theorem 5.1, we have |G( 2|k|1 (ξ ))| 2ξ/ν , (5.6) (ξ ) k∈ where α − (γ − β)/d − (1/ p − 1/q)+ , β > γ , β < γ α − (γ − β) − (1/ p − 1/q)+ , ν := From Theorem 5.1 and Lemma 5.2, by analogous technique and argument as in the proof of Theorem 4.1, we prove the following Theorem 5.2 Under the assumptions of Theorem 5.1, let for a given n ∈ Z+ , ξn be the largest nonnegative number such that |G( Then R α,β rn (U p,θ , (ξn ) defines γ Bq,τ ) by R (ξn ))| ≤ n an asymptotically optimal linear sampling algorithm for rn := (ξn ) ( f ) = L n (X n∗ , ∗ n, f (2−k s)ψk,s , f) = (k,s)∈K ( where X n∗ := G( (ξn )) = {2−k s}(k,s)∈K ( we have the following asymptotic orders sup α,β f ∈U p,θ f −R (ξn ) ( f ) Bq,τ γ rn (ξn )) , (ξn )) ∗ n := {ψk,s }(k,s)∈K ( (ξn )) , and n −α−(β−γ )/d+(1/ p−1/q)+ , β > γ , (5.7) β < γ n −α−β+γ +(1/ p−1/q)+ , Notice that although in general, for a set the upper bound (5.1) depends on θ and τ , for the special set (ξ ) the upper bound (5.5) does not depend on θ and τ , more precisely, these parameters go to a constant in Hence, the upper bound in (5.7) which is derived from (5.5) and (5.6), does not depend on θ and τ too From Theorem 5.2 and (2.13), we derive following theorem on optimal sampling γ α,β recovery in the energy norm of Wq (Id ) of the class U p,θ Theorem 5.3 Under the assumptions of Theorem 5.1, we have the following asymptotic orders for < q < ∞, α,β γ rn (U p,θ , Wq (Id )) 123 n −α−(β−γ )/d+(1/ p−1/q)+ , β > γ , β < γ n −α−β+γ +(1/ p−1/q)+ , Found Comput Math α,β γ Remark Asymptotically optimal linear sampling algorithms for rn (U p,θ , Wq (Id )) are γ α,β the same as for rn (U p,θ , Bq,min(q,2) ) For periodic functions, the asymptotic order of γ α,β rn (U2,2 , W2 (Td )) with β < γ recently has been obtained in [7] Optimal Cubature Every linear sampling algorithm L n (X n , formula In (X n , n , f ) where n n , ·) of the form (1.1) generates the cubature = {λ j }nj=1 , λ j = Id ϕ j (x) dx Hence, it is easy to see that |I ( f ) − In (X n , n, f )| ≤ f − L n (X n , n, f ) 1, and consequently, from the definitions, we have the following inequality i n (W ) ≤ rn (W )1 (6.1) Theorem 6.1 Let < p, θ ≤ ∞ and α ∈ R+ , β ∈ R such that 1/ p < min(α, α + β) ≤ max(α, α + β) < r Assume that for a given n ∈ Z+ , ξn is the largest nonnegative number such that |G( (ξn ))| ≤ n Then R (ξn ) α,β defines an asymptotically optimal cubature formula for i n (U p,θ ) by In (X n∗ , ∗ n, λk,s f (2−k s), f) = (k,s)∈K ( (ξn )) where X n∗ := G( (ξn )) = {2−k s}(k,s)∈K ( λ(k,s := Id (ξn )) , ∗ n := {λk,s }(k,s)∈K ( (ξn )) ψk,s (x)dx, and we have the following asymptotic orders sup α,β f ∈U p,θ |I ( f ) − In (X n∗ , ∗ n, f )| α,β i n (U p,θ ) n −α−β/d+(1/ p−1)+ , β > 0, β < n −α−β+(1/ p−1)+ , (6.2) 123 Found Comput Math Proof The upper bound of (6.2) follows from (6.1) and Theorem 4.1 To prove the lower bound of (6.2), we observe that i n (W ) ≥ inf X n ={x j }nj=1 ⊂Id sup f ∈W : f (x j )=0, j=1, ,n |I ( f )|, and for the functions g ∗ given in (4.19) and (4.23), we have I (g ∗ ) = g ∗ Hence, we can see that the lower bound is derived from the proof of the lower bound of Theorem 4.1 In a similar way, we can prove the following Theorem 6.2 Let < p, θ ≤ ∞ and a ∈ Rd+ satisfying the condition (3.11) and a1 > 1/ p Assume that for a given n ∈ Z+ , ξn is the largest nonnegative number such that |G( (ξn ))| ≤ n Then R (ξn ) a ) by defines an asymptotically optimal cubature formula for i n (U p,θ In (X n∗ , ∗ n, λk,s f (2−k s), f) = (k,s)∈K ( (ξn )) where X n∗ := G( (ξn )) = {2−k s}(k,s)∈K ( λk,s := Id (ξn )) , ∗ n := {λk,s }(k,s)∈K ( (ξn )) ψk,s (x)dx, and we have the following asymptotic order sup |I ( f ) − In ( a f ∈U p,θ ∗ n, X n∗ , f )| a i n (U p,θ ) n −a1 +(1/ p−1)+ Acknowledgments This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 102.01-2014.02 A part of this work was done when the author 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 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Math 4 1A1 5 · 4 1A0 5 · 4 1A2 5 · 4 1A5 8 · 4 1A6 3 Mathematics Subject Classification Introduction The aim of the present paper is to construct linear sampling algorithms and cubature formulas on sparse