We investigate linear algorithms of sampling recovery and cubature formulas on Smolyak grids of periodic dvariate functions having LipschitzH¨older mixed smoothness based on Bspline quasiinterpolation, and their optimality when the number d of variables and the number n of sampled function values may be very large. We establish upper and lower estimates of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit in d and n.
High-dimensional periodic sampling and cubature on Smolyak grids based on B-spline quasi-interpolation Dinh D˜ ung Vietnam National University, Hanoi, Information Technology Institute 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam dinhzung@gmail.com August 03, 2014 -- Version 1.3 Abstract We investigate linear algorithms of sampling recovery and cubature formulas on Smolyak grids of periodic d-variate functions having Lipschitz-H¨older mixed smoothness based on Bspline quasi-interpolation, and their optimality when the number d of variables and the number n of sampled function values may be very large. We establish upper and lower estimates of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit in d and n. Keywords and Phrases Linear sampling algorithm · Cubature formula · Smolyak grid · Lipschitz-H¨ older mixed smoothnesse space of mixed smoothness · Quasi-interpolation representations by B-spline series. Mathematics Subject Classifications (2000) 41A15 · 41A05 · 41A25 · 41A58 · 41A63. 1 Introduction The aim of this paper is to investigate linear algorithms of sampling recovery and cubature formulas on Smolyak grids of periodic d-variate functions having Lipschitz-H¨older mixed smoothness based on B-spline quasi-interpolation, and their optimality when the number d of variables and the number n of sampled function values may be very large. We stress on explicitly estimating from above and below the error of the optimal sampling recovery and the optimal integration on Smolyak grids as a function of two variables d and n. We consider only functions on Rd 1-periodic at each variable. It is convenient to consider them as functions defined in the d-torus Td = [0, 1]d which is defined as the cross product of d copies of the interval [0, 1] with the identification of the end points. To avoid confusion, we use the notation Id to denote the standard unit d-cube [0, 1]d . 1 For m ∈ N, the well known periodic Smolyak grid of sampled points Gd (m) ⊂ Td is defined as Gd (m) := {ξ = 2−k s : k ∈ Nd , |k|1 = m, s ∈ I d (k)}, (1.1) where I d (k) := {s ∈ Zd+ : sj = 0, 1, ..., 2kj − 1, j ∈ [d]}. Here and in what follows, we use the notations: xy := (x1 y1 , ..., xd yd ); 2x := (2x1 , ..., 2xd ); |x|1 := di=1 |xi | for x, y ∈ Rd ; [d] denotes the set of all natural numbers from 1 to d; xi denotes the ith coordinate of x ∈ Rd , i.e., x := (x1 , ..., xd ). The number |Gd (m)| of points in the grid Gd (m) is m d m d. Notice the grid Gd (m) is full if and only if m ≥ d. Sparse grids Gd (m) for sampling recovery and numerical integration were first considered by Smolyak [36]. Temlyakov [37] – [39] and the author of the present paper [9] – [11] developed Smolyak’s construction for studying the sampling recovery for periodic Sobolev classes and Nikol’skii classes having mixed smoothness. Recently, Sickel and Ullrich [34] have investigated the sampling recovery for periodic Besov classes having mixed smoothness. For non-periodic functions of mixed smoothness linear sampling algorithms on Smolyak grids have been recently studied by Triebel [40] (d = 2), Dinh D˜ ung [14], Sickel and Ullrich [35], using the mixed tensor product hat functions and more general, B-splines. In [16], we have constructed methods of approximation by arbitrary linear combinations of translates of the Korobov kernel κr,d on Smolyak grids of functions from the Korobov space K2r (Td ) which is a reproducing kernel Hilbert space with the associated kernel κr,d . In numerical applications for high-dimensional approximation problems, Smolyak grids was first considered by Zenger [42] in parallel algorithms for numerical solving PDEs. Numerical integration on Smolyak grids was investigated in [21]. For non-periodic functions of mixed smoothness of integer order, linear sampling algorithms on Smolyak grids have been investigated by Bungartz and Griebel [3] employing hierarchical Lagrangian polynomials multilevel basis. There is a very large number of papers on Smolyak grids and their modifications 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 [3, 29, 22] and the references therein. For recent further developments and results see in [25, 24, 26, 19, 2]. For univariate functions f on R, the rth difference operator ∆rh is defined by r ∆rh (f, x) := (−1)r−j j=0 r f (x + jh). j If u is any subset of [d], for multivariate functions on Rd the mixed (r, e)th difference operator ∆r,u h 2 is defined by ∆r,u h := ∆rhi , ∆r,∅ h = I, i∈u where the univariate operator ∆rhi is applied to the univariate function f by considering f as a function of variable xi with the other variables held fixed. Denote by C(Td ) the normed space of all bounded continuous functions on Td with the max norm · C(Td ) . For a function f on Td , let ωru (f, t) := ∆r,u h (f ) sup |hi | 1/p, ˚α )p ≤ sm (U α )p ≤ C(d, α, p) 2−αm md−1 . C (d, α, p) 2−αm md−1 ≤ ˚ sm (U ∞ ∞ (1.9) In high-dimensional approximation problems using function values information, the number m of sampled values is the main parameter in the study of convergence rates of the approximation error with respect to m going to infinity. However, the parameter d may hardly affect this rate when d is large. In the present paper, inspired by the relation (1.9), we establish upper and α ) and ˚ α ) explicitly in n and d as a function of two ˚∞ lower bounds for the quantities sm (U∞ sm (U p p variables m, d, in particular, upper bounds for the constant C(d, α, p) and C (d, α, p) in (1.9). To obtain these upper and lower bounds we construct linear sampling algorithms on Smolyak grids of the form (1.5) based on a B-spline quasi-interpolation series specially on Faber-Schauder series related to the well-known hat functions. As consequences we obtain upper and lower bounds for α ) and Int ˚ m (U ˚α )p of optimal cubature formula on Smolyak grids explicit the quantities Intm (U∞ p ∞ in n and d. Related to the problems investigated in the present paper, is problems of hyperbolic cross approximation of functions having mixed smoothness in high-dimensional setting in terms of n-widths and ε-dimensions which have been invesigated in [4, 17]. The paper is organized as follows. In Section 2, we establish upper and lower bounds we α) ˚∞ construct linear sampling algorithms on Smolyak grids based on Faber-Schauder series for ˚ sm (U p 4 α ) for 0 < α ≤ 2. As consequence, we derive upper and lower bounds for the error and sm (U∞ p α) ˚ sm (U ˚∞ of cubature formulas on Smolyak grids and of optimal integration on Smolyak grids Int p s s α ) . In Section 3, we extend the results for s (U α ) and Int (U α ) in Section 2 to and Intm (U∞ p m ∞ p m ∞ p the case of large mixed smoothness α > 2, based on B-spine quasi-interpolation representations α . In Section 4, we give some example of polynomials inducing generating for function from H∞ quasi-interpolation operators. 2 Sampling recovery based on Faber-Schauder series 2.1 Faber-Schauder series Let M2 (x) = (1 − |x − 1|)+ , x ∈ I, be the piece-wise linear B-spine with knot at 0, 1, 2. Since the support of functions M2 (2k+1 ·) for k ∈ Z+ is the interval [0, 2−k ] we can extend these functions to an 1-periodic function on the whole R. Denote this periodic extension by ϕk . The univariate periodic Faber-Schauder system of the hat functions is defined by F := {ϕk,s : s ∈ Z(k), k ∈ Z+ }, where Z(0) := {0} and Z(k) := {0, 1, ..., 2k−1 − 1} for k > 0, ϕ0,0 (x) := 1, x ∈ T, and for k > 0 and s ∈ Z(k) ϕk,s (x) := ϕk (x − 2s), x ∈ T. Let 1 λk,s (f ) := − ∆22−k (f, 2−k+1 s), k > 0, and λ0,0 (f ) := f (0). 2 Put Z d (k) := d i=1 Z(ki ). For k ∈ Zd+ , s ∈ Z d (k), define the tensor product hat functions d ϕk,s (x) := ϕki ,si (xi ), i=1 and the d-variate periodic Faber-Schauder system Fd by Fd := {ϕk,s : s ∈ Z d (k), k ∈ Zd+ }. Let λk,s (f ) be defined in the manner of the definition (3.10) as λk,s (f ) := λk1 ,s1 (λk2 ,s2 (...λkd ,sd (f ))). 5 Lemma 2.1 The d-variate periodic Faber-Schauder system Fd is a basis in C(Td ). Moreover, function f ∈ C(Td ) can be represented by the series λk,s (f )ϕk,s , f = k∈Zd+ (2.1) s∈Z d (k) converging in the norm of C(Td ). Proof. For the univariate case (d = 1), this lemma can be deduced from its well-known counterpart for non-periodic functions on I (see, e.g., [30, Theorem 1, Chapter VI]). For the multivariate case (d > 1), it can be proven by the tensor product argument. Put Zd+ (u) := {k ∈ Zd+ : supp k = u}, where supp k denotes the support of k, i.e., the subset of all j ∈ [d] such that kj = 0. α can be represented by the Theorem 2.1 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then a function f ∈ H∞ series (2.1) converging in the norm of C(Td ). Moreover, we have for every k ∈ Zd+ (u), qk (f ) ∞ ≤ sup s∈Z d (k) α (u) , |λk,s (f )| ≤ 2−|u| 2−α|k|1 |f |H∞ (2.2) and qk (f ) p α (u) , 0 < p < ∞. ≤ 2−|u| (p + 1)−|u|/p 2−α|k|1 |f |H∞ (2.3) Proof. The first part of the lemma on representation and convergence is in Lemma 2.1. Let us α we have for prove the inequalities (3.42) and (2.3). By the definitions of ϕk,s , λk,s (f ) and f H∞ d d every k ∈ Z+ (u) and x ∈ T , |qk (f )(x)| ≤ sup |λk,s (f )| s∈Z d (k) ϕk,s (x) ≤ and consequently, qk (f ) ∞ ≤ sup |λk,s (f )| s∈Z d (k) ≤ sup s∈Z d (k) j∈u sup s∈Z d (k) s∈Z d (k) 1 − ∆22−kj f (xk,s (u)) 2 ≤ 2−|u| ωru (f, 2−k ) α (u) . ≤ 2−|u| 2−α|k|1 |f |H∞ 6 |λk,s (f )|, The inequality (3.42) is proven. Let us prove (2.3). We first consider the case 1 ≤ p < ∞. We have for every k ∈ Zd+ (u), qk (f ) p p p λk,s (f )ϕk,s (x) dx = Td ≤ s∈Z d (k) ϕk,s (x) dx Td s∈Z d (k) = p |λk,s (f )|p sup s∈Z d (k) |λk,s (f )|p |Z d (k)| sup |ϕk,0 (x)|p dx Td s∈Z d (k) 2−kj = p d |λk,s (f )| |Z (k)| sup s∈Z d (k) 0 j∈u α (u) 2−|u| 2−α|k|1 |f |H∞ ≤ |2kj xj |p dxj 2 p 2−|u| 2|k|1 2|u| (p + 1)−|u| 2−|k|1 = 2−|u| (p + 1)−|u| 2−pα|k|1 |f |pH α (u) . ∞ This proves (2.3) for 1 ≤ p < ∞. The case 0 < p < 1 can be proven in a similar way starting from the inequality qk (f ) p p Td s∈Z d (k) 2.2 |ϕk,s (x)|p dx. |λk,s (f )|p ≤ Auxiliary lemmas For m, n ∈ N with m ≥ n, we introduce the function Fm,n : (0, 1) → R by ∞ Fm,n (t) := s=0 m+s s t . n From the definition it follows that Fm,n (t) = m n + tFm+1,n (t) and consequently, tFm+1,n (t) < Fm,n (t) < Fm+1,n (t), t ∈ (0, 1). (2.4) We will need following equation proven in [3, (3.67), p.29]. 1 Fm,n (t) = 1−t n s=0 m s t 1−t n−s . (2.5) 7 For nonnegative integer n, we define the function (1 − 2t)−1 , t < 1/2, bn (t) := 2(n + 1), t = 1/2, (2t − 1)−1 [t/(1 − t)]n+1 , t > 1/2. (2.6) Lemma 2.2 Let m, n ∈ N, m ≥ n, and 0 < t < 1. Then we have Fm,n (t) ≤ m bn (t), n∗ (2.7) where n∗ := min{n, m/2 }. Proof. We have n s=0 m s ≤ (n + 1) m . n∗ (2.8) Hence, by (2.5) the case t = 1/2 of the inequality (2.7) is proven. Next, we have for x > 0 and x = 1, n s=0 m s x ≤ s n m n∗ xs = s=0 xn+1 − 1 , x−1 m n∗ (2.9) and (x − 1)−1 xn+1 , xn+1 −1 ≤ (1 − x)−1 , x−1 x > 1, (2.10) x < 1. By using the last two inequalities for x = (1 − t)/t from (2.5) we prove the cases t < 1/2 and t > 1/2 of the inequality (2.7). Lemma 2.3 Let m, n ∈ N, m ≥ n, and 0 < t < 1. Then we have Fm,n (t) ≤ 1 1−t t 1−t n exp 1−t t mn (2.11) Proof. We have for x > 0, n s=0 m s x = s n s=0 m! xs ≤ s!(m − s)! n ms s=0 xs ≤ mn ex . s! Using this estimate from (2.5) we deduce the lemma. 8 (2.12) 2.3 Upper bounds By the definition we can see that α α ˚ U∞ = f ∈ U∞ :f = qk (f ) . d k∈N ˚m for f ∈ U ˚α by For m ∈ Z+ , we define the operator R ∞ ˚m (f ) := R λk,s (f )ϕk,s . qk (f ) = k∈Nd : |k| k∈Nd : |k| 1 ≤m 1 ≤m s∈Z d (k) ˚m defines a linear sampling algorithm S ˚m (Ψ, f ) on the grid G ˚d (m) by The operator R ˚m (f ) = S ˚m (Ψ, f ) = R f (ξ)ψξ , ˚d (m) ξ∈G where Ψ∗ := {ψξ }ξ∈G ˚d (m) , d ψ2−ki si (xi ), k ∈ Nd , s ∈ I d (k), ψ2−k s (x) = i=1 and the univariate functions ψk,s , k ∈ N, s ∈ I 1 (k), are defined by 1 − ϕ0,0 , k = 1, s = 0 ϕ0,0 , k = 1, s = 1 ψ2−k s = − 12 (ϕk,j + ϕk,j−1 ), k > 1, s = 2j, 0 ≤ j < 2k−1 − 1, ϕk,j , k > 1, s = 2j + 1, 0 < j < 2k−1 − 1. In what follows, for notational convenience we write x0 = 1 for x ∈ [0, ∞]. For 0 < p ≤ ∞, α > 0 and m ≥ l − 1, put b(α, p) := 2(2α − 1)(p + 1)1/p , and l−1 α −l β(α, l, m) := (2 − 1) s=0 m (2α − 1)s , s β(α, 0, m) := 1. ˚α and and every Theorem 2.2 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U ∞ m ≥ d − 1, ˚m (f ) f −R p ≤ [2(p + 1)1/p ]−d 2−αm β(α, d, m) ≤ exp(2α − 1) [b(α, p)]−d 2−αm md−1 . (2.13) 9 Moreover, if in addition, m ≥ 2(d − 1), ˚m (f ) f −R p ≤ a◦ (α, p, d) 2−αm m , d−1 (2.14) where [2(p + 1)1/p ]−d , a◦ (α, p, d) := |2α − 2|−| sgn(α−1)| × d [2(p + 1)1/p ]−d , [2(p + 1)1/p (2α − 1)]−d , Further, we have ˚d (f ) f −R α > 1, α = 1, α < 1. ≤ [b(α, p)]−d . p (2.15) (2.16) Proof. Let us prove the lemma for 1 ≤ p ≤ ∞. It can be proven in a similar way with a slight modification for 0 < p < 1. Put t = 2−α . From Theorem 2.1 and (2.5) it follows that for every ˚α and and every m ≥ d − 1, f ∈U ∞ f − Rm (f ) p ≤ qk (f ) p k∈Nd : |k|1 >m 2−d (p + 1)−d/p 2−α|k|1 ≤ k∈Nd : |k|1 >m = 2−d (p + 1)−d/p 2−α|k|1 (2.17) k∈Nd : |k|1 >m ∞ = 2−d (p + 1)−d/p j=1 m + j − 1 −α(m+j) 2 d−1 = 2−α(m+1) 2−d (p + 1)−d/p Fm,d−1 (t). Hence, replacing t = 2−α we obtain the first inequality in (2.13) and the second one by applying Lemma 2.2. The inequality (2.14) can be derived from (2.17) by applying Lemma 2.3 for t = 2−α . Let us prove (2.16) where m = d. From (2.17), and (2.5) we have f − Rd (f ) p ≤ 2 −d(α+1) (p + 1) −d/p 1 1−t = 2−d(α+1) (p + 1)−d/p d−1 s=0 1 1−t = [2(2α − 1)(p + 1)1/p ]−d 10 d d−1 s t 1−t d−1−s (2.18) For m ∈ Z+ , we define the operator Rm by Rm (f ) := qk (f ) = k∈Zd+ : |k|1 ≤m λk,s (f )ϕk,s . k∈Zd+ : |k|1 ≤m s∈Z d (k) For functions f on Td , Rm defines a linear sampling algorithm Sm (Ψ, f ) on the Smolyak grid by Gd (m) f (ξ)ψξ , Rm (f ) = Sm (Ψ, f ) = ξ∈Gd (m) where Ψ := {ψξ }ξ∈Gd (m) . Put d γ(α, d, p, m) := l=0 d −l 2 (p + 1)−l/p β(α, l, m). l α and every Theorem 2.3 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U∞ m ≥ d − 1, f − Rm (f ) p ≤ 2−αm γ(α, d, p, m) ≤ exp(2α − 1) [1 + 1/b(α, p)]d 2−αm md−1 . (2.19) Moreover, if in addition, m ≥ 2(d − 1), f − Rm (f ) p ≤ a(α, p, d) 2−αm m , d−1 (2.20) where [1 + 1/2(p + 1)1/p ]d , a(α, p, d) := |2α − 2|−| sgn(α−1)| × d [1 + 1/2(p + 1)1/p ]d , [1 + 1/2(p + 1)1/p (2α − 1)]d , α > 1, α = 1, (2.21) α < 1. Further, we have f − Rd (f ) p ≤ [1 + 1/(2α − 1)]d [1 + 1/2(p + 1)1/p ]d . (2.22) Proof. Let us prove the lemma for 1 ≤ p ≤ ∞. It can be proven in a similar way with a slight modification for 0 < p < 1. Put t = 2−α . From Theorem 2.1 and Lemma 2.2 it follows that for 11 α and and every m ≥ d − 1, every f ∈ U∞ f − Rm (f ) p ≤ qk (f ) p |u|≤d k∈Zd+ (u): |k|1 >m 2−|u| (p + 1)−|u|/p 2−α|k|1 ≤ |u|≤d k∈Zd+ (u): |k|1 >m d = l=0 d = l=0 d −l 2 (p + 1)−l/p l d −l 2 (p + 1)−l/p l d = 2−α(m+1) l=0 2−α|k|1 (2.23) k∈Nl : |k|1 >m ∞ j=1 m + j − 1 −α(m+j) 2 l−1 d −l 2 (p + 1)−l/p Fm,l−1 (t) l Hence, replacing t = 2−α we proves the first inequality in (2.19). This inequality easily implies (2.22) by considering m = d. Next, applying Lemma 2.2 to Fm,l−1 (t) in (2.23) we get d f − Rm (f ) p ≤ 2−α(m+1) l=0 ≤ 2−α(m+1) exp 1−t t d t−1 md−1 l=0 d = 2−αm exp (2α − 1) md−1 l=0 1−t t d −l 2 (p + 1)−l/p l t 1−t 12 t 1−t d −l 2 (p + 1)−l/p (2α − 1)−l l = exp(2α − 1) [1 + 1/b(α, p)]d 2−αm md−1 . The second inequality in (2.19) is proven. l−1 exp d −l 1 2 (p + 1)−l/p l 1−t ml−1 l (2.24) Next, let us verify (2.20). Applying Lemma 2.3 to Fm,l−1 (t) in (2.23) we have for m ≥ 2(d − 1), d f − Rm (f ) p ≤ 2−α(m+1) l=0 d l−1 s=0 m s t 1−t l−1−s d −l 2 (p + 1)−l/p β(α, l, m) l = 2−αm l=0 d = 2−α(m+1) l=0 ≤ 2−α(m+1) d l m d−1 = a(α, p, d) 2−αm 2.4 d −l 1 2 (p + 1)−l/p l 1−t m 2−l (p + 1)−l/p bl−1 (2−α ) l−1 d (2.25) d −l 2 (p + 1)−l/p bl−1 (2−α ) l l=0 m . d−1 Lower bounds ˚α and and every Theorem 2.4 Let 1 ≤ p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U ∞ m ≥ d − 1, α ˚∞ ˚ sm (U )p ≥ 2−7d 2−αm β(α, d, m) ≥ (2α − 1)−1 2−7d 2−αm m , d−1 (2.26) and therefore, ˚α )p ≥ (2α − 1)−1 2−7d (d − 1)−(d−1) 2−αm md−1 . ˚ sm (U ∞ (2.27) Further, we have α ˚∞ ˚ sd (U )p ≥ (2α − 1)−d 2−7d . (2.28) Proof. Let M4 be the cubic cardinal B-spline with support [0, 4] and knots at the points 0, 1, 2, 3, 4. Since the support of functions M4 (2k+2 ·) for k ∈ Z+ is the interval [0, 2−k ], we can extend these functions to an 1-periodic function on the whole R. Denote this periodic extension by µk . Define the univariate nonnegative function gk,s and gk on T by gk,s (x) := µk (x − 4s), k ∈ Z+ , s ∈ I 1 (k); (2.29) gk (x) := (2.30) gk,s (x). s∈I 1 (k) 13 (2) From the identity M4 (x) = M2 (x) − 2M2 (x − 1) + M2 (x − 2), x ∈ R, it follows that (2) |M4 (x)| ≤ 2M2 (2−1 x), x ∈ R, and consequently, (2) |gk,s (x)| ≤ 22k+5 ϕk,s (x), x ∈ T. (2.31) One can also verify that supp gk,s = Ik,s =: [2−k s, 2−k (s + 1)], 1 0 M4 (x) dx and by the equation 1 int Ik,s ∩ int Ik,s = ∅, s = s , (2.32) = 1, gk,s (x) dx = 2−k−2 . (2.33) 0 Let d gkj ,sj , k ∈ Zd+ , s ∈ I d (k). gk,s := j=1 Put for u ⊂ [d], Du2 := ∂ 2|u| 2. j∈u ∂xj Then we have for k ∈ Zd+ , s, s ∈ I d (k) and u ⊂ [d], |Du2 gk,s (x)| = 25|u| 22|ku |1 gkj ,sj (xj ), x ∈ Td , ϕkj ,sj (xj ) j∈u (2.34) j∈u d Ikj ,sj , supp gk,s = Ik,s =: int Ik,s ∩ int Ik,s = ∅, s = s , j=1 and gk,s (x) dx = 2−2d 2−|k|1 . Td Take m, n ∈ N with n > m and we define the function fm,n by n fm,n := 2−5d 2−αl l=m gk,s . |k|1 =l (2.35) s∈I d (k) Let us prove α ˚∞ fm,n ∈ U , (2.36) 14 and ˚d (m). fm,n (ξ) = 0, ξ ∈ G (2.37) Let us first prove (2.37). Let k, k ∈ Nd with |k|1 ≥ m and |k |1 = m. Then there is an index j ∈ [d] such that kj ≥ kj . Hence, 2−kj sj ∈ int Ik1j ,sj for every s ∈ ˚ I d (k) and s ∈ ˚ I d (k ). This means that gkj ,sj (2−kj sj ) = 0, and consequently, gk,s (2−k s ) = 0 for every s ∈ ˚ I d (k), s ∈ ˚ I d (k ). This proves (2.37). We now prove (2.46). Observe that n fm,n = 2−5d 2−αl gk , |k|1 =l l=m where d gkj , k ∈ Nd . gk := (2.38) j=1 To prove (2.46) it is sufficient to show that ∆2,u h (fm,n , x) ≤ |hj |α , x ∈ Td , h ∈ Rd . (2.39) j∈u Let us prove this inequality for u = [d] and h ∈ Rd+ , the general case of u can be proven in a similar way with a slight modification. For h ∈ R and k ∈ N, let h(k) ∈ [0, 2−k ) is the number defined by h = h(k) + s2−k for some s ∈ Z. For h ∈ Rd and k ∈ Nd , put d (kj ) h(k) := hj . j=1 Since gk is 2−kj -periodic in variable xj , we have 2,[d] ∆h 2,[d] (gk , x) = ∆h(k) (gk , x), x ∈ Td , h ∈ Rd . By using the formula (x + y)[h−1 M2 (h−1 y)] dy ∆2h (f, x) = h2 R for twice-differentiable function f on R (see, e.g., [7, p.45]), and (2.31) we have that d 2,[d] ∆h d h2j (gk,s , x) = j=1 Rd −1 h−1 j M2 (hj (yj − xj )) dy. 2 D[d] gk,s (y) j=1 15 2,[d] ∆h 2,[d] ∆h(k) (gk,s , x) (gk , x) = s∈I d (k) d d (k) (hj )2 s∈I d (k) j=1 = (k) Rd (k) (hj )−1 M2 ((hj )−1 (yj − xj )) dy. 2 gk,s (y) D[d] j=1 Hence, d 2,[d] |∆h d (k) 5d 2|k|1 ϕk,s (y) (gk , x)| ≤ s∈I d (k) Rd (k) j=1 j=1 d ≤ 25d 2|k|1 Rd d j=1 s∈I d (k) d (k) (k) j=1 d (k) (2−kj hj )2−α (hj )α ϕk,s (y) (k) (hj )−1 M2 ((hj )−1 (yj − xj )) dy (hj )2 (k) (hj )−1 M2 ((hj )−1 (yj − xj )) dy j=1 d ≤ 25d 2|k|1 hαj (k) Rd j=1 (k) (hj )−1 M2 ((hj )−1 (yj − xj )) dy. ϕk,s (y) j=1 s∈I d (k) Hence, by the inequalities (2.34) and n ϕk,s (y) ≤ 1, y ∈ Rd , l=m |k|1 =l s∈I d (k) we derive that n 2,[d] |∆h (fm,n , x)| ≤ 2 −5d 2,[d] 2−αl |∆h (gk , x)| |k|1 =l l=m n d ≤ 2−5d d j=1 j=1 (k) (k) j=1 (k) Rd j=1 d hαj . M2 (yj ) dy ≤ Rd j=1 j=1 j=1 16 (k) (hj )−1 M2 ((hj )−1 (yj − xj )) dy (hj )−1 M2 ((hj )−1 (yj − xj )) dy d hαj (k) d Rd l=m |k| =l s∈I d (k) 1 d j=1 s∈I d (k) d hαj = Rd ϕk,s (y) d (k) (hj )−1 M2 ((hj )−1 (yj − xj )) dy ϕk,s (y) n hαj ≤ hαj j=1 l=m |k|1 =l ≤ d 2−αl 25d 2|k|1 The inequality (2.39) is proven for u = [d] and h ∈ Rd+ . From (2.38), (2.32), (2.33) we can also verify that if m is given then for arbitrary n ≥ m, n fm,n p ≥ fm,n = 2−5d 1 n 2−αl gk = 2−7d 1 |k|1 =l l=m 2−αl l=m l−1 . d−1 (2.40) ˚m (Φ, fm,n ) = 0 for arbitrary Φ, and consequently, by (2.40) for arbitrary By (2.37) we have S n ≥ m, n ˚α )p ≥ ˚ sm (U ∞ ˚m (Φ, fm,n ) fm,n − S p = fm,n p ≥ 2 −7d 2−αl l=m l−1 d−1 (2.41) This means that for t = 2−α , ∞ α ˚∞ ˚ sm (U )p ≥ 2−7d 2−αl l=m l−1 d−1 = 2−7d 2−αm Fm−1,d−1 (t). (2.42) m (2α − 1)d−1 d−1 m d−1 (2.43) Therefore, applying (2.4) gives ˚α )p > 2−7d 2−αm tFm,d−1 (t) ˚ sm (U ∞ = 2−7d 2−αm β(α, d, m) > 2−7d 2−αm (2α − 1)−d = (2α − 1)−1 2−7d 2−αm which proves (2.41). The inequality (2.28) can be deduced from the first inequality in (2.52) in a way similar to (2.18). Put d γ (α, d, m) := l=0 d −7l 2 (p + 1)−l/p β(α, l, m). l α and and every Theorem 2.5 Let 1 ≤ p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U∞ m ≥ d − 1, α sm (U∞ )p ≥ 2−αm γ (α, d, m) > [2−7 (2α − 1)]−1 2α − 127 128 d−1 2−αm m , d−1 (2.44) and α sd (U∞ )p > [2−7 (2α − 1)]−1 [1 + 2−7−α ]d−1 . 17 (2.45) Proof. We take the univariate functions gk,s , gk as in (2.29), (2.30), and for every u ⊂ [d], define the functions u gk,s := gkj ,sj , k ∈ Zd+ , s ∈ I d (k), j∈u and gku := gkj . j∈u If m, n ∈ N with n > m, and u ⊂ [d], we define the |u|-variate function fm,n (u) by n u := 2−5|u| fm,n 2−αl u gk,s , |k|1 =l s∈I ( k) l=m and the d-variate function φm,n by u fm,n . φm,n := u⊂[d] From the proof of Theorem 2.4 we can see that α φm,n ∈ U∞ , (2.46) φm,n (ξ) = 0, ξ ∈ Gd (m). (2.47) and Observe that n 2−5|u| φm,n := 2−αl |k|1 =l l=m u⊂[d] gku From (2.38), (2.32), (2.33) we can also verify that if m is given then for arbitrary n > m, φm,n p ≥ φm,n 1 n = 2 −5|u| 2−αl n 2−7|u| = u⊂[d] = s=1 2−αl l=m d −7s 2 s 1 |k|1 =l l=m u⊂[d] gku l−1 |u| − 1 n 2−αl l=m (2.48) l−1 s−1 18 By (2.37) we have Sm (Φ, φm,n ) = 0 for arbitrary Φ, and consequently, by (2.40) for arbitrary n > m, d α sm (U∞ )p ≥ φm,n p d −7s 2 s ≥ s=1 n 2−αl l=m l−1 . s−1 (2.49) This means that for t = 2−α , d α sm (U∞ )p ≥ s=1 d −7s 2 s ∞ 2 −αl l=m d l−1 s−1 = s=1 d −7s 2 Fm−1,s−1 (t). s (2.50) Therefore, applying (2.4) gives d α sm (U∞ )p > s=1 −αm = 2 d −7s −αm 2 2 tFm,s−1 (t) s γ (α, d, m) d > 2−αm s=1 d−1 = 2 −αm s=0 d−1 = 2−αm l=0 = [2 α s−1 l=0 m (2α − 1)l l d 2−7(s+1) (2α − 1)−(s+1) s+1 m (2α − 1)l l l (2 − 1)] −1 −αm 2 s l=0 m (2α − 1)l l d 2−7(s+1) (2α − 1)−(s+1) s+1 s=0 m (2α − 1)d−1 d−1 > 2−αm −7 d −7s α 2 (2 − 1)−s s l s=0 d − 1 −7(s+1) α 2 (2 − 1)−(s+1) s m (2α − 1)d−1 d−1 d−1 s=0 d−1 [2−7 (2α − 1)]−s s = [2−7 (2α − 1)]−1 (2α − 1)d−1 [1 + 2−7 (2α − 1)−1 ]d−1 2−αm = [2−7 (2α − 1)]−1 2α − 127 128 d−1 2−αm which proves (2.44). 19 m d−1 m d−1 (2.51) From the fifth inequality in (2.52) we have d−1 d−1 (2α − 1)l l α sd−1 (U∞ )p > 2−α(d−1) l=0 d−1 −7 > [2 α −1 −α(d−1) (2 − 1)] 2 l=0 d−1 = [2−7 (2α − 1)]−1 2−α(d−1) = [2 −7 α −1 −α(d−1) (2 − 1)] 2 l=0 α l s=0 d − 1 −7(s+1) α 2 (2 − 1)−(s+1) s d−1 (2α − 1)l l l s=0 l [2−7 (2α − 1)]−s s (2.52) d−1 (2α − 1)l [1 + 2−7 (2α − 1)]l l [2 + 2−7 ]d−1 = [2−7 (2α − 1)]−1 [1 + 2−7−α ]d−1 . This proves (2.45). 2.5 Cubature We are interested in cubature formulas on Smolyak grid for approximately computing of the integral I(f ) := f (x) dx. (2.53) [0,1]d α. for a f ∈ U∞ If Λm = (λξ )ξ∈Gd (m) , we consider the cubature formula Λsm (f ) := Λm (Gd (m), f ) on grids Gd (m) given by Λsm (f ) = λξ f (ξ). ξ∈Gd (m) The quantity of optimal cubature Intsm (Fd ) on Smolyak grids Gd (m) is introduced by Intsm (Fd ) := inf sup |f − Λsm (f )|. (2.54) Λm f ∈Fd For a family Φ = {ϕξ }ξ∈Gd (m) of functions on Td , the linear sampling algorithm Sm (Φ, ·) generates the cubature formula Λsm (f ) on Smolyak grid Gd (m) by Λsm (f ) = λξ f (ξ), (2.55) ξ∈Gd (m) where the integration weights Λm = (λξ )ξ∈Gd (m) are given by λξ = ϕξ (x) dx. (2.56) Id 20 Hence, it is easy to see that |I(f ) − Λsm (f )| ≤ f − Sm (Φ, f ) 1 , and, as a consequence of (1.8) and (2.54), Intsm (Fd ) ≤ sm (Fd )1 . (2.57) ˚α , based on the grids G ˚d (m), we can define the cubature formula ˚ For functions f ∈ U Λsm (f ) := ∞ s d d ˚ ˚ (m), f ) on grids G ˚ (m), and the quantity of optimal cubature Int ˚ (Fd ) on Smolyak grids Λ(G m d ˚ G (m). Moreover, thereholds true the inequality ˚ sm (Fd ) ≤ ˚ Int sm (Fd )1 . (2.58) Theorem 2.6 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every every m ≥ 2d−1 , ˚ s (U ˚α ) ≤ 4−d 2−αm β(α, d, m), 2−7d 2−αm β(α, d, m) ≤ Int m ∞ (2.59) and therefore, ˚ s (U ˚α ) ≤ exp(2α − 1) 4−d 2−αm md−1 . (2.60) (2α − 1)−1 2−7d (d − 1)−(d−1) 2−αm md−1 ≤ Int m ∞ Moreover, if in addition, m ≥ 2(d − 1), (2α − 1)−1 7−d 2−αm m d−1 α ˚ sm (U ˚∞ ≤ Int ) ≤ a◦ (α, d) 2−αm m , d−1 (2.61) where −d 4 , a◦ (α, d) := |2α − 2|−| sgn(α−1)| × d 4−d , [4(2α − 1)]−d , α > 1, (2.62) α = 1, α < 1, and α ˚ sd (U ˚∞ (2α − 1)−d 2−7d ≤ Int ) ≤ 4−d . (2.63) Proof. The upper bounds in (2.61)–(2.63) follow from (2.58) and Theorem 2.2. α as in (2.35) with the property as ˚∞ To prove the lower bounds, we take the function fm,n ∈ U in (2.37): ˚d (m). fm,n (ξ) = 0, ξ ∈ G Notice that fm,n is a nonnegative function. Hence, we have ˚ Λsm (Φ, fm,n ) = 0 for arbitrary Φ, and consequently, by (2.40) for arbitrary n ≥ m, α ˚ sm (U ˚∞ Int ) ≥ |fm,n − ˚ Λsm (Φ, fm,n )| = n fm,n 1 ≥ 2−7d 2−αl l=m 21 l−1 . d−1 (2.64) ˚ s (U ˚α ) can be estimated from below as in the proof of Comparing with (2.41), we can see that Int m ∞ ˚α ). This proves the lower bounds in in (2.61)–(2.63). Theorem (2.4) for ˚ ssm (U ∞ Put d γ(α, d, m) := l=0 d −l 4 β(α, l, m). l In a similar way we can prove Theorem 2.7 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every every m ≥ 2d−1 , α ) ≤ 2−αm γ(α, d, m), 2−αm γ (α, d, m) ≤ Intsm (U∞ (2.65) and therefore, α (2α − 1)−1 2−7d (d − 1)−(d−1) 2−αm md−1 ≤ Intsm (U∞ ) ≤ exp(2α − 1) 4−d 2−αm md−1 . (2.66) Moreover, if in addition, m ≥ 2(d − 1), [2−7 (2α − 1)]−1 2α − 127 128 d−1 2−αm m d−1 α ≤ Intsm (U∞ ) ≤ a(α, d) 2−αm m , (2.67) d−1 where d (5/4) , a(α, d) := |2α − 2|−| sgn(α−1)| × d (5/4)d , [1 + 1/4(2α − 1)]d , α > 1, α = 1, (2.68) α < 1. Further, we have α ) ≤ (5/4)d [1 + 1/(2α − 1)]d . [2−7 (2α − 1)]−1 [1 + 2−7−α ]d−1 ≤ Intsd (U∞ 3 3.1 (2.69) Sampling recovery based on B-spine quasi-interpolation representations Periodic B-spline quasi-interpolation representations We introduce quasi-interpolation operators for functions on Rd . For a given natural number , denote by M the cardinal B-spline of order with support [0, ] and knots at the points 0, 1, ..., . We introduce quasi-interpolation operators for functions on Rd . In what follows we fixed r ∈ N and consider the cardinal B-spline M := M2r of even order 2r. Let Λ = {λ(s)}|j|≤µ be a given 22 finite even sequence, i.e., λ(−j) = λ(j) for some µ ≥ r − 1. We define the linear operator Q for functions f on R by Q(f, x) := Λ(f, s)M (x − s), (3.1) λ(j)f (s − j + r). (3.2) s∈Z where Λ(f, s) := |j|≤µ The operator Q is local and bounded in C(R) (see [5, p. 100–109]). An operator Q of the form (3.1)–(3.2)is called a quasi-interpolation operator in C(R) if it reproduces P2r−1 , i.e., Q(f ) = f for every f ∈ P2r−1 , where Pl denotes the set of d-variate polynomials of degree at most l − 1 in each variable. If Q is a quasi-interpolation operator of the form (3.1)–(3.2), for h > 0 and a function f on R, we define the operator Q(·; h) by Q(f ; h) := σh ◦ Q ◦ σ1/h (f ), where σh (f, x) = f (x/h). Let Q be a quasi-interpolation operator of the form (3.1)–(3.2) in C(R). If k ∈ Z+ , we introduce the operator Qk by Qk (f, x) := Q(f, x; h(k) ), x ∈ R, h(k) := (2r)−1 2−k . We define the integer translated dilation Mk,s of M by Mk,s (x) := M (2r2k x − s), k ∈ Z+ , s ∈ Z, where Z+ := {s ∈ Z : s ≥ 0}. Then we have for k ∈ Z+ , ak,s (f )Mk,s (x), ∀x ∈ R, Qk (f )(x) = s∈Z where the coefficient functional ak,s is defined by ak,s (f ) := Λ(f, s; h(k) ) = λ(j)f (h(k) (s − j + r)). (3.3) |j|≤µ Notice that Qk (f ) can be written in the form: f (h(k) (s + r))Lk (x − s), ∀x ∈ R, Qk (f )(x) = (3.4) s∈Z where the function Lk is defined by Lk := = λ(j)Mk,j . (3.5) |j|≤µ 23 From (3.4) and (3.5) we get for a function f on R, Qk (f ) ≤ C(R) LΛ C(R) f C(R) ≤ Λ f C(R) , (3.6) where λ(j)M (x − j − s), LΛ (x) := = |λ(j)|. Λ = s∈Z |j|≤µ (3.7) |j|≤µ For k ∈ Zd+ , let the mixed operator Qk be defined by d Qk := Qki , (3.8) i=1 where the univariate operator Qki is applied to the univariate function f by considering f as a function of variable xi with the other variables held fixed. We define the d-variable B-spline Mk,s by d Mki ,si (xi ), k ∈ Zd+ , s ∈ Zd , Mk,s (x) := (3.9) i=1 where Zd+ := {s ∈ Zd : si ≥ 0, i ∈ [d]}. Then we have Qk (f, x) = ak,s (f )Mk,s (x), ∀x ∈ Rd , s∈Zd where Mk,s is the mixed B-spline defined in (3.9), and d ak,s (f ) = akj ,sj (f ), (3.10) j=1 and the univariate coefficient functional aki ,si is applied to the univariate function f by considering f as a function of variable xi with the other variables held fixed. Since M (2r2k x) = 0 for every k ∈ Z+ and x ∈ / (0, 1), we can extend the univariate B-spline M (2r.2k ·) to an 1-periodic function on the whole R. Denote this periodic extension by Nk and define Nk,s (x) := Nk (x − s), k ∈ Z+ , s ∈ I(k), where I(k) := {0, 1, ..., 2r2k − 1}. We define the d-variable B-spline Nk,s by d Nki ,si (xi ), k ∈ Zd+ , s ∈ I d (k), Nk,s (x) := i=1 24 (3.11) d i=1 I(ki ). where I d (k) := Qk (f, x) = Then we have for functions f on Td , ak,s (f )Nk,s (x), ∀x ∈ Td . (3.12) s∈I d (k) Since the function LΛ defined in (3.7) is 1-periodic, from (3.6) it follows that for a function f on T, Qk (f ) C(T) ≤ LΛ C(T) f ≤ C(T) Λ f C(T) , (3.13) For k ∈ Zd+ , we write k → ∞ if ki → ∞ for i ∈ [d]). The following lemma can be proven in a way similar to the proof of [15, Lemma 2.2]. Lemma 3.1 We have for every f ∈ C(Td ), f − Qk (f ) C(Td ) u ω2r (f, 2−k ), ≤ C (3.14) u⊂[d], u=∅ and, consequently, f − Qk (f ) C(Td ) → 0, k → ∞. (3.15) For convenience we define the univariate operator Q−1 by putting Q−1 (f ) = 0 for all f on I. Let the operators qk and qu,k be defined in the manner of the definition (3.8) by d (Qki − Qki −1 ) , k ∈ Zd+ . qk := (3.16) i=1 We have Qk = qk . (3.17) k ≤k From (3.17) and (3.15) it is easy to see that a continuous function f has the decomposition f = qk (f ) k∈Zd+ with the convergence in the norm of C(Td ). From the definition of (3.16) and the refinement equation for the B-spline M , in the univariate case, we can represent the component functions qk (f ) as qk (f ) = ck,s (f )Nk,s , (3.18) s∈I d (k) 25 where ck,s are certain coefficient functionals of f . In the multivariate case, the representation (3.18) holds true with the ck,s which are defined in the manner of the definition (3.10) by d ck,s (f ) = ckj ,sj (f ). (3.19) j=1 See [14] for details. Thus, we have proven the following Lemma 3.2 Every continuous function f on Td is represented as B-spline series f = qk (f ) = k∈Zd+ ck,s (f )Nk,s , k∈Zd+ (3.20) s∈I d (k) converging in the norm of C(Td ), where the coefficient functionals ck,s (f ) are explicitly constructed as linear combinations of at most m0 function values of f for some m0 ∈ N which is independent of k, s and f . The formula (3.12) with the coefficients valued functionals ak,s (f ) given as in (3.3) and (3.10), defines a multivariate periodic quasi-interpolation operator for functions on Td . While the formula (3.20) with the coefficients valued functionals ak,s (f ) given as in (3.16), defines a multivariate periodic B-spline quasi-interpolation representation for functions on Td . They are completely generated from the initial quasi-interpolation operator Q given as in (3.1) and (3.2). For this reason we will call Q the generating quasi-interpolation for Qk and for the representation (3.20). 3.2 A formula for the coefficients in quasi-interpolation representations If h ∈ Rd , we define the shift operator Ths for functions f on Td by Th (f ) := f (· + h). Recall that a d-variate Laurent polynomial is call a function P of the form cs z s , P (z) = (3.21) s∈A where A is a finite subset in Zd and z s := generates the operator [P ] Th (f ) = [P ] Th sj d j=1 zj . A d-variate Laurent polynomial P as (3.21) by cs Tsh (f ), (3.22) s∈A [P ] where sh := (s1 h1 , ..., sd hd ). Sometimes we also write Th [P (z)] = Th . Notice that any operation over polynomials generates a corresponding operation over operators Thus, in particular, we have [P ] Th . [a1 P1 +a2 P2 ] Th [P1 ] (f ) = a1 Th [P2 ] (f ) + a2 Th (f ), 26 [P1 .P2 ] Th [P1 ] (f ) = Th [P2 ] ◦ Th (f ). By definitions we have d [Dl ] ∆lh = Th (zj − 1)l , , Dl := [Dl,u ] ∆l,u h = Th j=1 , Dl,u := (zj − 1)l . j∈u We say that a d-variate polynomial is a tensor product polynomial if it is of the form d P (z) = Pj (zj ), j=1 where Pj (zj ) are univariate polynomial in variable zj . Lemma 3.3 Let P be a tensor product Laurent polynomial, h ∈ Rd with hj = 0, and l ∈ N. [P ] Assume that Th (g) = 0 for every polynomial g ∈ Pl Then P has a factor Dl and consequently, [P ] Th [P ∗ ] = Th ◦ ∆lh , P (z) = Dl P ∗ (z), where P ∗ is a tensor product Laurent polynomial. Proof. By the tensor product argument it is enough to prove the lemma for the case d = 1. We prove this case by induction on l. Let P (z) = ns=−m cs z s for some m, n ∈ Z+ . Consider first the [P ] case l = 1. Assume that Th (g) = 0 for every constant functions g. Then replacing by g0 = 1 in (3.22) we get n [P ] Th (g0 ) = cs = 0. (3.23) s=−m By B´ezout’s theorem P has a factor (z − 1). This proves the lemma for l = 1. Assume it is true for [P ] l − 1 and Th (g) = 0 for every polynomial g of degree at most l − 1. By the induction assumption we have [P ] Th [P1 ] = Th ◦ ∆l−1 h , P (z) := (z − 1)l−1 P1 (z). (3.24) We take a proper polynomial gl of degree l − 1 (with the nonzero eldest coefficient). Hence ψl = ∆l−1 h (gl ) = a where a is a nonzero constant. Similarly to the case l = 1, from the equations [P ] [P ] 0 = Th (gl ) = Th 1 (ψl ) we conclude that P1 has a factor (z − 1). Hence, by (3.24) we can see that P has a factor (z − 1)l . The lemma is proved. Let us return to the definition of quasi-interpolation operator Q of the form (3.1) induced by the sequence Λ as in (3.2) which can be uniquely characterized by the univariate symmetric Laurent polynomial PΛ (z) := z r λ(s)z s . (3.25) |s|≤µ 27 Let the d-variate symmetric tensor product Laurent polynomial PΛ be given by d PΛ (z) := z r λ(sj )z j . (3.26) j=1 |sj |≤µ Re call that the periodic quasi-interpolation operator Qk is given in (3.12) by ak,s (f )Nk,s (x), Qk (f, x) = ∀x ∈ Td . (3.27) s∈I d (k) From (3.3) we get [P ] Λ ak,s (f ) = Th(k) (f )(sh(k) ), (3.28) where h(k) := (2r)−1 2−k . Let us find an explicit formula for the univariate operator qk (f ). We have for k > 0, [P ] Λ Th(k) (f )(sh(k) ))Nk,s Qk (f ) = s∈I(k) [P ] [P ] Λ Th(k) (f )(2sh(k) ))Nk,2s + = s∈I(k−1) Λ Th(k) (f )((2s + 1)h(k) ))Nk,2s+1 . s∈I(k−1) From (3.28) and the refinement equation for M , we deduce that 2r 2r [PΛ ] Qk−1 (f ) = Th(k−1) (f )(sh(k−1) )) 2−2r+1 Nk,2s+j j j=0 s∈I(k−1) r = 2−2r+1 j=0 r−1 + 2−2r+1 j=0 2r 2j [P ] Λ Th(k−1) (f )(sh(k−1) ))Nk,2s+2j s∈I(k−1) 2r 2j + 1 [P ] Λ Th(k−1) (f )(sh(k−1) ))Nk,2s+2j+1 s∈I(k−1) odd =: Qeven k−1 (f ) + Qk−1 (f ). By the identities h(k−1) = 2h(k) , Nk,2r2k +m = Nk,m and f (h(k) )(2r2k + m) = f (h(k) m) for k ∈ Z+ 28 and m ∈ Z, we have r 2r 2j −2r+1 Qeven k−1 (f ) = 2 j=0 r [P ] Λ Th(k) (f )(2(s − j)h(k) ))Nk,2s s∈j+I(k−1) 2r 2j = 2−2r+1 j=0 [P [P ] Λ Th(k) (f )(2(s − j)h(k) ))Nk,2s s∈I(k−1) ] even Th(k) (f )(2sh(k) )Nk,2s , = s∈I(k−1) where r Peven (z) := 2−2r+1 PΛ (z 2 ) j=0 2r −2j z 2j In a similar way we obtain [P ] odd Th(k) (f )((2s + 1)h(k) )Nk,2s+1 , Qodd k−1 (f ) = s∈I(k−1) where r−1 −2r+1 Podd (z) := 2 2 PΛ (z ) j=0 2r z −2j−1 . 2j + 1 We define Peven := PΛ − Peven , Podd := PΛ − Podd Then from the definition qk (f ) = Qk (f ) − Qk−1 (f ) we receive the following representation for qk (f ), [P ] Th(0)Λ (f )(sh(0) ))N0,s , q0 (f ) = (3.29) s∈I(0) and for k > 0, qk (f ) = qkeven (f ) + qkodd (f ) (3.30) with [P qkeven (f ) = ] even Th(k) (f )(2sh(k) )Nk,2s , s∈I(k−1) (3.31) qkodd (f ) [Podd ] Th(k) (f )((2s = + 1)h (k) )Nk,2s+1 . s∈I(k−1) 29 From the definitions of Qk and qk it follows that [P ] even Th(k) (g)(2sh(k) ) = 0 and [P ] odd (g)((2s + 1)h(k) ) = 0 Th(k) for every g ∈ Pr . Hence, by Lemma 3.3 we prove the following lemma for the univariate operators qk . Theorem 3.1 We have ∗ Peven (z) = Dr (z)Peven (z) (3.32) ∗ Podd (z) = Dr (z)Podd (z). ∗ , P∗ where Peven odd are a symmetric Laurent polynomial. Therefore, in the representation (3.29)– (3.30) of qk (f ), we have for k > 0, [P ∗ ] even ◦ ∆rh(k) (f )(2sh(k) )Nk,2s , Th(k) qkeven (f ) = s∈I(k−1) [P ∗ ] odd Th(k) ◦ ∆rh(k) (f )((2s + 1)h(k) )Nk,2s+1 . qkodd (f ) = s∈I(k−1) Equivalently, in the representation (3.18) of qk (f ), we have for s ∈ I(0) [P ] c0,s (f ) = Th(0)Λ (f )(sh(0) ), (3.33) and for k > 0 and s ∈ I(k), [P ∗ ] even ◦ ∆rh(k) (f )(sh(k) ), Th(k) ck,s (f ) = ∗ ] [Podd Th(k) ◦ ∆rh(k) (f )(sh(k) ), s even (3.34) s odd. Proof. Consider the representation (3.18) for qk (f ) and d = 1. If g is arbitrary polynomial of degree at most r−1, then since Qk reproduces g we have qk (g) = 0 and consequently, ck,s (g) = 0 for k > 0. [Peven ] The equations (3.29)–(3.30) give an explicit formula for the coefficient ck,s (g) as Th(k) (g)(2sh(k) ) [P ] odd and Th(k) (g)((2s + 1)h(k) ). Hence, by Lemma 3.3 we get (3.32). Theorem 3.2 In the representation (3.18) of qk (f ), we have for every k ∈ Zd+ (u) and s ∈ I d (k), [P ] k,s ck,s (f ) = Th(k) (f )(sh(k) ), (3.35) where Pk,s (z) = Pk∗j ,sj (zj ) PΛ (zj ) j∈u Pk∗j ,sj (zj ) = j∈u D2r (zj ) (3.36) j∈u ∗ (z ), Peven j s even P ∗ (z ), odd j s odd. (3.37) 30 Proof. Indeed, from the definition of ck,s (f ) and Theorem 3.1 we have for every k ∈ Zd+ (u) and s ∈ I d (k), d ck,s (f ) = T j=1 [Pkj ,sj ] (k) hj k,s ] (f )(sh(k) ), = T [P(k) (f )(sh(k) ), h where PΛ (zj ), ∗ (z )D (z ), Pkj ,sj (zj ) = Peven j 2r j ∗ Podd (zj )D2r (zj ), kj = 0, (3.38) kj > 0, s even kj > 0, s odd. For a Laurent polynomial P as in (3.21) we introduce the norm |cs |. P := s∈A Notice that LΛ C(T) ≤ PΛ . (3.39) Recall that for notational convenience we write x0 = 1 for any x ∈ [0, ∞]. For given Λ, 1 ≤ p ≤ ∞ and 0 < α ≤ 2r, we define a = a(p, α, Λ, r) := (2r)−α [r(p + 1)]−1/p max b = b(Λ) := LΛ ∗ ∗ Peven , Podd , (3.40) C(T). α . Then f can be represented by the Theorem 3.3 Let 1 ≤ p ≤ ∞, and 0 < α ≤ 2r. Let f ∈ H∞ series (3.20) converging in the norm of C(Td ). Moreover, we have for every k ∈ Zd+ (u), qk (f ) p α (u) . ≤ a|u| bd−|u| 2−α|k|1 |f |H∞ (3.41) Proof. The first part of the lemma on representation and convergence is in Lemma 3.2. Let us first prove (3.41) for p = ∞. For convenience, we temporarily use the notation a = ap . In this case, (3.41) is as qk (f ) ∞ ≤ d−|u| −α|k|1 α (u) . sup |ck,s (f )| ≤ a|u| 2 |f |H∞ ∞ b s∈I d (k) By definition we have qk (f ) = j∈u qkj (g), g := qkj (f ). j∈u 31 (3.42) Hence, by (3.13) we derive qk (f ) ≤ bd−|u| g ∞ ∞, (3.43) Similarly to the proof of Theorem 2.1 we can show that for every k ∈ Zd+ (u), g = ∞ ≤ qkj (f ) j∈u ∞ sup s∈I d (k) ckj ,sj (f ) . (3.44) j∈u Observe that for every Laurent polynomial and every h ∈ R, [P ] Th (f ) ∞ ≤ P f ∞. (3.45) Setting Pk∗j ,sj (zj ). Pk,s (z) := (3.46) j∈u we get [P ] k,s ckj ,sj (f ) = Th(k) ∆l,u (f, sh(k) ) h(k) (3.47) j∈u Hence, by (3.47)–(3.38) and (3.45) sup s∈I d (k) ckj ,sj (f ) = j∈u ≤ sup s∈I d (k) sup s∈I d (k) ≤ max [P ] k,s Th(k) ∆l,u (f, sh(k) )) h(k) Pk,s (f, sh(k) )) ∆l,u h(k) ∗ ∗ Peven , Podd |u| (k) α (u) |hj |α |f |H∞ j∈u −α|k|1 α (u) . ≤ a|u| |f |H∞ ∞ 2 This together with (3.43) and (3.44) proves (3.42). 32 Let us prove (3.41) for the case 1 ≤ p < ∞. We have for every k ∈ Zd+ (u), qk (f ) p p p ck,s (f )Mk,s (x) dx = Td ≤ s∈I d (k) Mk,s (x) dx Td s∈I d (k) = p sup |ck,s (f )|p s∈I d (k) |Mk,0 (x)|p dx sup |ck,s (f )|p |I d (k)| Td s∈I d (k) (2r)−1 2−kj = |M (2kj xj )|p dxj sup |ck,s (f )|p |I d (k)| s∈I d (k) j∈u 0 |u| = sup |ck,s (f )|p |I d (k)| (2r)−|u| 2−|k|1 s∈I d (k) |M2r (t)|p dt . R By employing (3.42), Young’s inequality M2r Lp (R) = M2r−2 ∗ M2 and the equations M2r−2 M2 p Lp (R) 1 =2 Lp (R) ≤ M2r−2 L1 (R) M2 Lp (R) , = 1 and L1 (R) tp dt = 2(p + 1)−1 , 0 we complete the estimation as follows qk (f ) 3.3 p p ≤ d−|u| −α|k|1 α (u) a|u| 2 |f |H∞ ∞ b = r(p + 1) = d−|u| a|u| p b −|u| p d−|u| a|u| ∞ b p p 2|k|1 (2r)−|u| 2−|k|1 2−pα|k|1 |f |pH α (u) ∞ 2−pα|k|1 |f |pH α (u) . ∞ Sampling recovery For m ∈ Z+ , we define the operator Rm by Rm (f ) := qk (f ) = k∈Zd+ : |k|1 ≤m ck,s (f )Nk,s . k∈Zd+ : |k|1 ≤m 33 s∈I d (k) 2|u| (p + 1)−|u| For functions f on Td , Rm defines a linear sampling algorithm Ln (f ) on the Smolyak grid Gd (m) More precisely, f (ξ)ψξ , Rm (f ) = Sn (Ψ, f ) = ξ∈Gd (m) where n := |Gd (m)|, Ψ := {ψξ }ξ∈Gd (m) and for ξ = 2−k s, ψξ are explicitly constructed as linear combinations of at most at most N B-splines Nk,j for some N ∈ N which is independent of k, s, m and f . Put d δ(α, d, a, b, m) := l=0 d l d−l a b β(α, l, m), l and [a + b]d , c(α, d, a, b) := |2α − 2|−| sgn(α−1)| × d [a + b]d , [a/(2α − 1) + b]d , α > 1, α = 1, (3.48) α < 1, where a, b are as in (3.40). α and every Theorem 3.4 Let 1 ≤ p ≤ ∞ and 0 < α ≤ 2r. Then we have for every f ∈ U∞ m ≥ d − 1, f − Rm (f ) p ≤ 2−αm δ(α, d, a, b, m) ≤ exp(2α − 1) [a/(2α − 1) + b]d 2−αm md−1 . (3.49) Moreover, if in addition, m ≥ 2(d − 1), f − Rm (f ) p ≤ c(α, d, a, b) 2−αm m . d−1 (3.50) Further, we have f − Rd (f ) p ≤ [1 + 1/(2α − 1)]d [a + b]d . (3.51) α . Put t = 2−α . From Theorem 3.3 and and (2.5) it follows that for every Proof. Let f ∈ H∞ 34 m ≥ d − 1, f − Rm (f ) p ≤ qk (f ) |u|≤d p k∈Zd+ (u): |k|1 >m a|u| bd−|u| 2−α|k|1 ≤ |u|≤d k∈Zd+ (u): |k|1 >m d = l=0 d = l=0 d l d−|u| a b l d l d−l a b l d = 2 −α(m+1) l=0 2−α|k|1 k∈Nl : |k|1 >m ∞ j=1 (3.52) m + j − 1 −α(m+j) 2 l−1 d l d−l a b Fm,l−1 (t) l = 2−αm δ(α, d, a, b, m). The first inequality in (3.52) is proven. Hence, applying Lemma 2.3 for t = 2−α gives d f − Rm (f ) p ≤ exp(2α − 1) 2−αm l=0 d l d−l α a b [2 − 1]−l ml−1 l (3.53) ≤ exp(2α − 1) [a/(2α − 1) + b]d 2−αm md−1 which proves the second inequality in (3.52). The first inequality in (3.52) easily implies (3.51) by considering m = d. If m ≥ 2(d − 1), by (3.52) and Lemma 2.2 for t = 2−α we get d f − Rm (f ) p ≤ 2 −α(m+1) l=0 ≤ 2 −α(m+1) d l d−l m a b bl−1 (2−α ) l l−1 m d−1 = c(α, d, a, b) 2−αm d l=0 d l d−l a b bl−1 (2−α ) l m . d−1 35 (3.54) 3.4 Cubature Put d δ1 (α, d, a1 , b, m) := l=0 d l d−l a1 b β(α, l, m), l and [a1 + b]d , c1 (α, d, a1 , b) := |2α − 2|−| sgn(α−1)| × d [a1 + b]d , [a1 /(2α − 1) + b]d , α > 1, α = 1, (3.55) α < 1, where b is as in (3.40) and a1 = a1 (α, Λ, r) := (2r)−α−1 max ∗ ∗ Peven , Podd . From Theorem (3.4) and (2.57) we obtain α and every Theorem 3.5 Let 0 < p ≤ ∞ and 0 < α ≤ 2r. Then we have for every f ∈ U∞ m ≥ d − 1, α Intsm (U∞ ) ≤ 2−αm δ1 (α, d, a1 , b, m) ≤ exp(2α − 1) [a1 /(2α − 1) + b]d 2−αm md−1 . (3.56) Moreover, if in addition, m ≥ 2(d − 1), α Intsm (U∞ ) ≤ c1 (α, d, a1 , b) 2−αm m . d−1 (3.57) Further, we have α Intsd (U∞ ) ≤ [1 + 1/(2α − 1)]d [a1 + b]d . 4 4.1 (3.58) Examples Faber-Schauder series (revisit) Let us consider the case r = 1 when M (x) = (1 − |x − 1|)+ is the piece-wise linear cardinal B-spine with knot at 0, 1, 2. Let Λ = {λ(s)}j=0 (µ = 0) be a given by λ(0) = 1. If Nk is the periodic extension of M (2k+1 ·), then Nk,s (x) := Nk (x − s), k ∈ Z+ , s ∈ I(k), 36 where I(k) := {0, 1, ..., 2k+1 − 1}. Consider the related periodic quasi-interpolation operator for functions f on T and k ∈ Z+ , f (2−(k+1) (s + 1))Nk,s (x) Qk (f, x) = (4.1) s∈I(k) We have PΛ (z) = z, 1 Peven (z) = − (z − 1)2 , 2 1 ∗ Peven (z) = − , 2 (4.2) ∗ Podd (z) = Podd (z) = 0, and LΛ = 1, ∗ Peven = 1 , 2 ∗ Podd = 0. Hence, 1 [P ] T2−1Λ (f )(2−1 s)N0,s = f (0), q0 (f ) = (4.3) s=0 and for k > 0, 2k −1 qk (f ) = qkeven (f ) = s=0 1 − ∆22−(k+1) f (2−k s) Nk,2s . 2 (4.4) With these formulas for qk (f ) after redefining Nk,2s as ϕk,s , the quasi-interpolation representation (3.20) becomes the Faber-Schauder series. 4.2 Cubic B-spline quasi-interpolation representation For r = 2 in the definition of the generating quasi-interpolation operator Q we take PΛ (z) = z2 1 8 1 (−z + 8 − z −1 ), = − z 3 + z 2 − z. 6 6 6 6 (4.5) Then, ∗ Peven (z) = (z − 1)4 Peven (z), 4 Podd (z) := (z − 1) ∗ Podd (z), ∗ Peven (z) = ∗ Podd (z) 1 −2 4 z z + 4z 3 + 8z 2 + 4z + 1 , 48 1 2 z + 4z + 1 , := 12 and LΛ = 11 , 9 ∗ Peven = 3 , 8 ∗ Podd = 37 1 . 2 (4.6) 4.3 Quintic B-spline quasi-interpolation representation For r = 3 in the definition of the generating quasi-interpolation operator Q we take z3 [25150 − 5876(z + z −1 ) + 448(z 2 + z −2 ) + 52(z 3 + z −3 ) + (z 4 + z −4 )]. (4.7) 14400 PΛ (z) = Then ∗ Peven (z) = (z − 1)6 Peven (z) = z3 372312 − 188032(z + z −1 ) − 21732(z 2 + z −2 ) + 1664(z 3 + z −3 ) 460800 + 26016(z 4 + z −4 ) − 3734(z 6 + z −6 ) − 332(z 8 + z −8 ) − 6(z 10 + z −10 ) , ∗ Podd (z) := (z − 1)6 Podd (z) = z4 804800 − 471714(z + z −1 ) + 14336(z 2 + z −2 ) + 57882(z 3 + z −3 ) 460800 + 32(z 4 + z −4 ) − 1625(z 5 + z −5 ) − 1243(z 7 + z −7 ) − 67(z 9 + z −9 ) − (z 11 + z −11 ) , (4.8) 1 139760 + 97002(z + z −1 ) + 42508(z 2 + z −2 ) + 11462(z 3 + z −3 ) 460800 ∗ Peven (z) = + 2328(z 4 + z −4 ) + 458(z 5 + z −5 ) + 36(z 6 + z −6 ) + 6(z 7 + z −7 ) , ∗ (z) Podd z = 164910 + 97002(z + z −1 ) + 36632(z 2 + z −2 ) + 11462(z 3 + z −3 ) 460800 (4.9) + 2776(z 4 + z −4 ) + 458(z 5 + z −5 ) + 88(z 6 + z −6 ) + 6(z 7 + z −7 ) + (z 8 + z −8 ) . PΛ = 37904 ≈ 2.63, 14400 ∗ Peven = 447360 ≈ 0.97, 460800 ∗ Podd = 467160 ≈ 1.00. 460800 Acknowledgments. 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Hackbusch, ed.), Vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg, Braunschweig/Wiesbaden, 1991. 41 [...]... quasi- interpolation representations Periodic < /b> B- spline quasi- interpolation representations We introduce quasi- interpolation operators for functions on < /b> Rd For a given natural number , denote by M the cardinal B- spline of order with support [0, ] and < /b> knots at the points 0, 1, , We introduce quasi- interpolation operators for functions on < /b> Rd In what follows we fixed r ∈ N and < /b> consider the cardinal B- spline M := M2r... function f by considering f as a function of variable xi with the other variables held fixed Since M (2r2k x) = 0 for every k ∈ Z+ and < /b> x ∈ / (0, 1), we can extend the univariate B- spline M (2r.2k ·) to an 1 -periodic < /b> function on < /b> the whole R Denote this periodic < /b> extension by Nk and < /b> define Nk,s (x) := Nk (x − s), k ∈ Z+ , s ∈ I(k), where I(k) := {0, 1, , 2r2k − 1} We define the d-variable B- spline Nk,s by... independent of k, s and < /b> f The formula (3.12) with the coefficients valued functionals ak,s (f ) given as in (3.3) and < /b> (3.10), defines a multivariate periodic < /b> quasi- interpolation operator for functions on < /b> Td While the formula (3.20) with the coefficients valued functionals ak,s (f ) given as in (3.16), defines a multivariate periodic < /b> B- spline quasi- interpolation representation for functions on < /b> Td They are... proves (2.45) 2.5 Cubature < /b> We are interested in cubature < /b> formulas on < /b> Smolyak < /b> grid for approximately computing of the integral I(f ) := f (x) dx (2.53) [0,1]d α for a f ∈ U∞ If Λm = (λξ )ξ∈Gd (m) , we consider the cubature < /b> formula Λsm (f ) := Λm (Gd (m), f ) on < /b> grids < /b> Gd (m) given by Λsm (f ) = λξ f (ξ) ξ∈Gd (m) The quantity of optimal cubature < /b> Intsm (Fd ) on < /b> Smolyak < /b> grids < /b> Gd (m) is introduced by Intsm (Fd... 1)−d 2−7d (2.28) Proof Let M4 be the cubic cardinal B- spline with support [0, 4] and < /b> knots at the points 0, 1, 2, 3, 4 Since the support of functions M4 (2k+2 ·) for k ∈ Z+ is the interval [0, 2−k ], we can extend these functions to an 1 -periodic < /b> function on < /b> the whole R Denote this periodic < /b> extension by µk Define the univariate nonnegative function gk,s and < /b> gk on < /b> T by gk,s (x) := µk (x − 4s), k ∈... initial quasi- interpolation operator Q given as in (3.1) and < /b> (3.2) For this reason we will call Q the generating quasi- interpolation for Qk and < /b> for the representation (3.20) 3.2 A formula for the coefficients in quasi- interpolation representations If h ∈ Rd , we define the shift operator Ths for functions f on < /b> Td by Th (f ) := f (· + h) Recall that a d-variate Laurent polynomial is call a function P of... −α(m+1) l=0 ≤ 2 −α(m+1) d l d−l m a b bl−1 (2−α ) l l−1 m d−1 = c(α, d, a, b) 2−αm d l=0 d l d−l a b bl−1 (2−α ) l m d−1 35 (3.54) 3.4 Cubature < /b> Put d δ1 (α, d, a1 , b, m) := l=0 d l d−l a1 b β(α, l, m), l and < /b> [a1 + b] d , c1 (α, d, a1 , b) := |2α − 2|−| sgn(α−1)| × d [a1 + b] d , [a1 /(2α − 1) + b] d , α > 1, α = 1, (3.55) α < 1, where b is as in (3.40) and < /b> a1 = a1 (α, Λ, r) := (2r)−α−1... functions on < /b> Td , the linear sampling < /b> algorithm Sm (Φ, ·) generates the cubature < /b> formula Λsm (f ) on < /b> Smolyak < /b> grid Gd (m) by Λsm (f ) = λξ f (ξ), (2.55) ξ∈Gd (m) where the integration weights Λm = (λξ )ξ∈Gd (m) are given by λξ = ϕξ (x) dx (2.56) Id 20 Hence, it is easy to see that |I(f ) − Λsm (f )| ≤ f − Sm (Φ, f ) 1 , and,< /b> as a consequence of (1.8) and < /b> (2.54), Intsm (Fd ) ≤ sm (Fd )1 (2.57) ˚α , based.< /b> .. Intsm (Fd ) ≤ sm (Fd )1 (2.57) ˚α , based < /b> on < /b> the grids < /b> G ˚d (m), we can define the cubature < /b> formula ˚ For functions f ∈ U Λsm (f ) := ∞ s d d ˚ ˚ (m), f ) on < /b> grids < /b> G ˚ (m), and < /b> the quantity of optimal cubature < /b> Int ˚ (Fd ) on < /b> Smolyak < /b> grids < /b> Λ(G m d ˚ G (m) Moreover, thereholds true the inequality ˚ sm (Fd ) ≤ ˚ Int sm (Fd )1 (2.58) Theorem 2.6 Let 0 < p ≤ ∞ and < /b> 0 < α ≤ 2 Then we have for every every... polynomials of degree at most l − 1 in each variable If Q is a quasi- interpolation operator of the form (3.1)–(3.2), for h > 0 and < /b> a function f on < /b> R, we define the operator Q(·; h) by Q(f ; h) := σh ◦ Q ◦ σ1/h (f ), where σh (f, x) = f (x/h) Let Q be a quasi- interpolation operator of the form (3.1)–(3.2) in C(R) If k ∈ Z+ , we introduce the operator Qk by Qk (f, x) := Q(f, x; h(k) ), x ∈ R, h(k) := (2r)−1 ... (2.69) Sampling recovery based on B- spine quasi- interpolation representations Periodic B- spline quasi- interpolation representations We introduce quasi- interpolation operators for functions on Rd... upper bounds for the constant C(d, α, p) and C (d, α, p) in (1.9) To obtain these upper and lower bounds we construct linear sampling algorithms on Smolyak grids of the form (1.5) based on a B- spline. .. , the quasi- interpolation representation (3.20) becomes the Faber-Schauder series 4.2 Cubic B- spline quasi- interpolation representation For r = in the definition of the generating quasi- interpolation