Optimal cubature formulas for tensor products of certain classes of functions

Journal of Complexity 27 (2011) 519–530

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Optimal cubature formulas for tensor products of certain classes of functions V.F. Babenko a,c , S.V. Borodachov b,∗ , D.S. Skorokhodov a a

Dnepropetrovsk National University, Dnepropetrovsk, 49050, Ukraine

b

Towson University, Baltimore, MD, 21252, USA

c

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Donetsk, 83114, Ukraine

article

info

Article history: Received 6 September 2010 Accepted 5 January 2011 Available online 27 January 2011 Keywords: Optimal cubature in the worst-case setting Radon transform Modulus of continuity Tensor product of classes

abstract We study the problem of constructing an optimal formula of approximate integration along a d-dimensional parallelepiped. Our construction utilizes mean values along intersections of the integration domain with n hyperplanes of dimension (d − 1), each of which is perpendicular to some coordinate axis. We find an optimal cubature formula of this type for two classes of functions. The first class controls the moduli of continuity with respect to all variables, whereas the second class is the intersection of certain periodic multivariate Sobolev classes. We prove that all node hyperplanes of the optimal formula in each case are perpendicular to a certain coordinate axis and are equally spaced and the weights are equal. For specific moduli of continuity and for sufficiently large n, the formula remains optimal for the first class among cubature formulas with arbitrary positions of hyperplanes. Published by Elsevier Inc.

1. Setting of the problem The problem of constructing an optimal quadrature formula for a class of functions was first posed by Kolmogorov in 1940s. The idea is to find a formula using n function values or more generally n information pieces about a function such that the error is minimized. First results on special cases of this problem were obtained by Nikol’skii [45] and Sard [48] during 1949–50. Later, a large number of results concerning optimal quadrature and cubature formulas were obtained for classes of univariate and multivariate functions. For a detailed review we refer the reader to books [42,44,43,50,53–55,58], and papers [57,38,23].

∗

Corresponding author. E-mail address: [email protected] (S.V. Borodachov).

0885-064X/$ – see front matter. Published by Elsevier Inc. doi:10.1016/j.jco.2011.01.001

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Optimal cubature formulas are mostly known for classes of univariate functions or for classes of multivariate functions which can be analyzed by reduction to the univariate case. For the multivariate case, we usually know cubature formulas which are only optimal within a factor and enjoy the best rate of convergence sometimes modulo a power of logarithm of n. For many standard classes of multivariate functions, we are still waiting to find optimal cubature formulas. The majority of known results for univariate classes of functions deal with formulas that use the information given by values of the function at some points or mean values over some intervals. Similarly, for multivariate classes of functions, cubature formulas are constructed using data from values at some points or from mean values along parallelepipeds. In some modern applications such as computer tomography and thermoacoustic tomography another type of information about multivariate functions is available. It is given by some integrals over sets whose dimension is strictly greater than zero and less than the dimension of the integration domain. More information on computational methods arising in tomography can be found, for example, in [19,41]. In this paper we continue studying the problem about optimal cubature formulas by sampling mean values along intersections of the domain with n hyperplanes. We now give a rigorous setting of the problem. Let a = (a1 , . . . , ad ) ∈ (0, ∞)d , Πd = [0, a1 ] × · · · × [0, ad ]. We want to approximate the integrals

∫ Πd

f (x) dx

for integrable functions from some class. For i = 1, . . . , d, denote bi =

d ∏

ak ,

and

Pi = [0, a1 ] × · · · × [0, ai−1 ] × [0, ai+1 ] × · · · × [0, ad ].

k=1 k̸=i

For every x ∈ [0, aj ], let Ij (f ; x) =

∫

1 bj

f (t1 , . . . , tj−1 , x, tj+1 , . . . , td )dtd . . . dtj+1 dtj−1 . . . dt1 . Pj

Denote by Qna the set of all cubature formulas of the form q(f ) =

ni d − −

ci,k Ii (f ; xik ),

(1)

i=1 k=1

where ni ’s are non-negative integers such that n1 + · · · + nd = n, real weights ci,k are arbitrary, and numbers 0 ≤ xi1 < · · · < xini ≤ ai , i = 1, . . . , d, are also arbitrary. Formulas from Qna recover the integral over Πd from known integrals of f over intersections of Πd with shifts of (d − 1)-dimensional coordinate subspaces. We also consider cubature formulas with arbitrarily oriented node hyperplanes. They are defined as the class Gan , n ∈ N, being the set of all cubature formulas for approximate integration over Πd of the form q(f ) =

n −

ck

k=1

|Πd ∩ Lk |

∫ Πd ∩Lk

f (x)dx.

Here L1 , . . . , Ln are arbitrary (d − 1)-dimensional hyperplanes having non-empty intersection with Πd and ck , k = 1, . . . , n, are arbitrary real weights. If the dimension of Πd ∩ Lk is less than d − 1, then |Πd ∩ Lk | is the area measure of the corresponding dimension and integration over Πd ∩ Lk takes place with respect to the area measure of that dimension. If Πd ∩ Lk consists of one point, the average value of f along Πd ∩ Lk is replaced by the value of f at that point. Let Q = Qna or Gan . For every formula q ∈ Q , denote the integration error as R(f ; q) =

∫ Πd

f (x)dx − q(f ).

V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

521

Let C (Πd ) be the space of continuous functions defined on the parallelepiped Πd . Given a class of functions Y ⊆ C (Πd ), we define R(Y ; q) = sup |R(f ; q)| . f ∈Y

Problem 1. Find the value of R(Y ; Q ) = inf R(Y ; q)

(2)

q∈Q

and find at least one optimal cubature formula (if it exists), i.e. a functional q∗ ∈ Q , which attains the infimum on the right-hand side of (2). The problem stated above will be reduced to the following problem of optimization of quadrature formulas: for n ∈ N and a > 0, denote by Tna , the set of all formulas for approximate integration along the interval [0, a] of the form S (g ) =

n −

ck g (xk ),

k =1

where weights ck , k = 1, . . . , n, and nodes 0 ≤ x1 < · · · < xn ≤ a are arbitrary. For a quadrature formula S ∈ Tna , let

ρ(g ; S ) =

a

∫

g (t )dt − S (g ). 0

For a class of functions K ⊆ C [0, a], define

ρ(K ; S ) = sup |ρ(g ; S )| g ∈K

and let

ρ(K ; Tna ) = infa ρ(K ; S ).

(3)

S ∈Tn

A formula S ∗ ∈ Tna is called optimal for the class K , if it attains the infimum on the right-hand side of (3). 2. Definition of the classes and review of known results Let ω be a modulus of continuity (i.e. a non-decreasing, continuous, sub-additive function defined on [0, ∞) such that ω(0) = 0). Denote by H ω [0, a] the class of functions g : [0, a] → R such that

|g (x) − g (y)| ≤ ω (|x − y|) , x, y ∈ [0, a]. For ω(t ) = t, the class H ω [0, a] becomes the Lipschitz class of functions, whereas for ω(t ) = t α , with α ∈ (0, 1), we obtain the Hölder class. Definition 1. Given an ordered collection ω = (ω1 , . . . , ωd ) of moduli of continuity, denote by H ω Πd = H ω1 ,...,ωd Πd the class of functions f : Πd → R such that

|f (x) − f (y)| ≤ ω1 (|x1 − y1 |) + · · · + ωd (|xd − yd |)

(4)

for every x = (x1 , . . . , xd ), y = (y1 , . . . , yd ) ∈ Πd . Let ω be a given modulus of continuity. Denote by Hpω Πd , 1 ≤ p ≤ ∞, the class of functions f ∈ C (Πd ) such that

  |f (x) − f (y)| ≤ ω ‖x − y‖p ,

x, y ∈ Πd ,

where

 1/p |x1 |p + · · · + |xd |p , 1 ≤ p < ∞, max{|x1 | , . . . , |xd |}, p = ∞. r  Denote by W∞ , r ∈ N, the classof 2π -periodic functions of the form g : R → R such that g (r −1) is  (r )   locally absolutely continuous and g (t ) ≤ 1 for almost all t ∈ R. ‖x‖p =

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V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530 r ,...,r

r ∞ ∞1 d the class of continuous Definition 2. Given vector r = (r1 , . . . , rd ) ∈ Nd , denote by W =W functions f : Rd → R having period 2π along each variable and such that for every i = 1, . . . , d and for almost every vector (x1 , . . . , xi−1 , xi+1 , . . . , xd ) ∈ Rd−1 , the function

h(t ) =

∂ r i −1 f r −1

∂ xi i

(x1 , . . . , xi−1 , t , xi+1 , . . . , xd )

is locally absolutely continuous, and

  r  ∂ if  r (x) ≤ 1,  ∂x i  i

for a.e. x ∈ Rd , i = 1, . . . , d.

This class of functions is quite often considered in numerical analysis (see e.g. [20, Chapter III], [5]). r ∞ Smoothness of functions is defined in a natural way for this class. The class W is also interesting due to the fact that the existence of partial derivatives partial derivatives

∂ i1 +···+id f i

i

∂ x11 ...∂ xdd

, where i1 , . . . , id ≥ 0 and

∂ r1 f r ∂ x11

i1 r1

,..., ∂

rd f r

∂ xdd

implies the existence of all mixed

+· · ·+ ridd < 1 (see e.g. [20, Theorem III.10.1]).

The optimality of the formula Sn∗ (g ) =

n − a k=1

n

 ·g

2k − 1 2n

 a

(5)

for the class H ω [0, a] (among formulas from Tna ) was shown by Nikol’skii [45] for the case ω(t ) = t and by Korneichuk [30] for arbitrary ω. Motornyi [36] proved the optimality of formula (5) with a = 2π r ∞ for the class W and for the class of 2π -periodic functions whose rth derivatives (with odd r) have a given concave majorant for the modulus of continuity. See also [39], where an analogous result for the latter class is obtained when r = 2. For Sobolev classes of periodic functions whose rth derivative is bounded in the Lp -norm, 1 ≤ p < ∞, the optimality of formula (5) was shown by Motornyi [36], Ligun [34] and Žensykbaev [56]. For classes of periodic functions whose rth derivative belongs to a rearrangement-invariant set, the optimality of formula (5) was proved by Babenko in [6]. In the general case involving classes defined using convolutions he obtained a similar result in [7]. The optimality of the ‘‘interval’’ analogue of formula (5) was shown in [4] for the class of periodic r ∞ functions whose rth derivative has a bounded L1 -norm, in [37] for the class W , in [26] for the nonω symmetric analogue of H [0, a], in [16] for classes of periodic functions whose rth derivative belongs to a rearrangement-invariant set, and in [17] for classes defined with the help of convolutions. The optimal cubature formula with nodes at rectangular grids was found by Korneichuk in [30] for e ∞ the classes H2ω Πd and H ω Πd . In particular, it was done for the class W , where e = (1, . . . , 1) ∈ Rd . Cubature formulas which use function values at number-theoretical nets were, in particular, studied by Korobov [32] and Temlyakov (see [51,52] and the references therein). The problem about asymptotically optimal cubature formulas that use function values at arbitrary node configurations and achieve both the optimal rate of the error as n gets large and the best constant for classes Hpω Πd , d ≥ 1, was studied in papers by Babenko [1–3,8], and Chernaya [27]. A formula for approximate integration over Πd with a positive weight, which has fixed nodes and is optimal for the class Hpω Πd in the exact sense, was found in [33] in the one-dimensional case and in [1] in the multivariate case. r ∞ An optimal cubature formula with nodes at rectangular grids was obtained in [11] for the class W and an optimal ‘interval’ cubature formula with nodes at rectangular grids was obtained in [11] for r ∞ classes H ω Πd and W . An optimal cubature formula from the set Qne was found for the class of monotone and bounded ω functions on the cube [0, 1]d (see [9]) and for the class H∞ [0, 1]d (see [14]). An optimal cubature ω a formula from the set Gn was found for the class H1 Πd in [15]. For the class H2ω Bd defined on the unit

V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

523

ball Bd in Rd , the first two authors of the current paper describe an optimal formula of the form

∫

f (x)dx ≈ Bd

∫ n − c  k f (x)dx, σr  σr k k=1 k

(6)

where σrk is the sphere of radius rk ∈ (0, 1] centered at the center of Bd , k = 1, . . . , n, (see [10]). An optimal formula of the form (6) was also found by them for the class of differentiable functions on Bd whose gradient is bounded in the L1 -norm (cf. [12]). Finally, these authors found an optimal formula of the same form for the class of twice differentiable functions on Bd , whose Laplacian is bounded in the L1 -norm (cf. [13]). Cubature formulas of the form (6) were considered in mathematical literature starting as early as in 1940s by Lusternik [35], Kantorovich [29] and later by Mysovskih [40], who obtained the algebraic degree of precision of certain formulas of this type. Bojanov and Petrova found the Gaussian formula with arbitrary positions of node hyperplanes (cf. [25]) for approximate integration along Bd . Extended Gaussian formulas of the form (6), were completely characterized by Bojanov and Dimitrov (cf. [21,22,24,28]; see also [46]). Petrova also studied the problem about Gaussian formulas of the type (6) (cf. [47] and the references therein). For more results on optimal quadrature and cubature formulas see books [42,44,53–55,58], and papers [57,38,23]. r ∞ In this paper we present a solution to Problem 1 for the classes H ω Πd and W for cubature formulas a 2π e from Qn and Qn respectively (see Theorems 1 and 2). In addition, for certain classes H ω Πd we solved Problem 1 for formulas from Gan and sufficiently large n; see Theorem 3 and Corollary 1. 3. Main results Let  H ω Πd =  H ω1 ,...,ωd Πd be the class of functions f : Rd → R, which are ai -periodic along the ith variable, i = 1, . . . , d, and satisfy (4) for every x, y ∈ Rd . Theorem 1. Let n, d ∈ N, a = (a1 , . . . , ad ) ∈ (0, ∞)d , ω = (ω1 , . . . , ωd ) be an ordered collection of moduli of continuity, and let the index j0 be such that 1

aj 0 2n

∫

aj 0

ωj0 (t )dt = min

1

i=1,...,d

0

ai 2n

∫

ai

ωi (t )dt .

(7)

0

Then the cubature formula qn (f ) = ∗

n − |Πd | k=1

n

 · Ij 0 f ;

2k − 1 2n

 aj 0

is optimal for the classes H ω Πd and  H ω Πd among all formulas from Qna . Moreover, we have the following equalities: R(H

ω

Πd Qna

;

ω

) = R(H

Πd Qna

;

) = 2nbj0

aj 0 2n

∫

ωj0 (t )dt .

(8)

0

To formulate the next result we will need to recall Favard’s constant, Kr =

∞ 4 − (−1)k(r +1)

π

k=0

(2k + 1)r +1

,

r ∈ N.

For approximate integration of differentiable periodic multivariate functions we obtain the following result. Theorem 2. Let r1 , . . . , rd ∈ N, and the indexing number j be such that rj = maxi=1,...,d ri . Then the cubature formula qn (f ) = ∗

n − (2π)d k=1

n

 · Ij f ;

2π k n



,

n ≥ 2,

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V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

r ∞ is optimal for the class W among all formulas from Qn2π e , where 2π e denotes (2π , . . . , 2π ). In addition, r ∞ R(W ; Qn2π e ) =

(2π )d Krj nrj

,

n ≥ 2.

(9)

Below are the cases where we were able to solve the general problem for the classes H ω Πd . Theorem 3. Let n, d ∈ N, a = (a1 , . . . , ad ) ∈ (0, ∞)d , ω = (ω1 , . . . , ωd ) be an ordered collection of moduli of continuity such that the quotients ωi (t )/t , i = 1, . . . , d, are monotonically non-increasing on (0, ∞) (in particular, all ωi ’s are concave). Assume that there exists an index j0 such that aj0 = mini=1,...,d ai and

ωj0 (t ) = min ωi (t )

(10)

i=1,...,d

aj

for every t from some interval [0, ϵ), ϵ > 0. Then for every n > 2ϵ0 , the cubature formula q∗n (f ) =

n − |Πd |

n

k=1

  2k − 1 · Ij0 f ; aj 0 2n

is optimal for the class H ω Πd among formulas from Gan . Moreover, R(H

ω

Πd Gan

;

) = 2bj0 n

aj 0 2n

∫

ωj0 (t )dt ,

n>

0

aj0 2ϵ

.

(11)

An important case when assumptions of Theorem 3 hold is the case

ωi (t ) = Ci t αi , with Ci > 0, αi ∈ (0, 1], i = 1, . . . , d, and Πd = [0, 1]d (i.e. a = (1, . . . , 1)). We let j0 be an index such that αj0 = maxi=1,...,d αi and Cj0 ≤ Ci for every i such that αi = αj0 . Denote   α 1−α j0 i Ci ϵ0 = min . i=1,...,d αi <αj 0

C j0

It is not difficult to see that ωj0 (t ) ≤ ωi (t ) for every t ∈ [0, ϵ0 ) and i = 1, . . . , d. The following statement is an immediate corollary of Theorem 3. Corollary 1. Let ω = (ω1 , . . . , ωd ), where ωi (t ) = Ci t αi , Ci > 0, αi ∈ (0, 1], i = 1, . . . , d. For every n > 21ϵ , the cubature formula 0

q∗n (f ) =

n − 1 k=1

n

  2k − 1 · Ij0 f ; 2n

is optimal for the class H ω [0, 1]d among formulas from Gen . In addition, R(H ω [0, 1]d ; Gen ) =

Cj0

αj0

(αj0 + 1)(2n)

,

n>

1 2ϵ0

.

4. The general auxiliary result A class K ⊆ C [0, a] is called convex if for any functions g , h ∈ K and any α ∈ [0, 1], we have α g + (1 − α)h ∈ K . A class K ⊆ C [0, a] is called centrally symmetric, if whenever g ∈ K , we also have −g ∈ K . Recall that Pi = [0, a1 ] × · · · × [0, ai−1 ] × [0, ai+1 ] × · · · × [0, ad ]. Given univariate classes Y1 ⊆ C [0, a1 ], . . . , Yd ⊆ C [0, ad ], denote by Y1 ×· · ·× Yd the set of all functions f ∈ C (Πd ) such that for every i = 1, . . . , d, the set of points (x1 , . . . , xi−1 , xi+1 , . . . , xd ) ∈ Pi , for which the univariate function ψ(·) = f (x1 , . . . , xi−1 , ·, xi+1 , . . . , xd ) belongs to the class Yi , is dense in Pi .

V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

525

Let also Y1 + · · · + Yd be the class of all functions f : Πd → R of the form f (t1 , . . . , td ) = ϕ1 (t1 ) + · · · + ϕd (td ), where ϕ1 ∈ Y1 , . . . , ϕd ∈ Yd . Denote by B the closure of the set B ⊆ C [0, a] in the uniform norm. We obtain the following general result. Theorem 4. Let d, n ∈ N, d > 1, Y1 ⊆ C [0, a1 ], . . . , Yd ⊆ C [0, ad ] be convex and centrally symmetric classes of functions, each of which contains all constant functions. Denote by j0 such an index that 1 aj 0

aj

ρ(Yj0 ; Tn 0 ) = min

i=1,...,d

and let the formula Sn∗ (g ) = the formula q∗n (f ) =

n −

1 ai

ρ(Yi ; Tnai )

∑n

k=1

(12) aj

ck∗ g (x∗k ) be optimal for the class Yj0 among formulas from Tn 0 . Then

bj0 ck∗ · Ij0 (f ; x∗k )

k=1

is optimal among formulas from Qna for any class Y ⊆ C (Πd ) such that Y1 +· · ·+ Yd ⊆ Y ⊆ Y 1 ×· · ·× Y d . Moreover, aj

R(Y ; Qna ) = bj0 ρ(Yj0 ; Tn 0 ).

(13)

Remark. Since under assumptions of Theorem 4 the closures of classes Y1 , . . . , Yd are invariant with respect to adding a constant, there holds Y1 + · · · + Yd ⊆ Y 1 × · · · × Y d and hence, there exists at least one class Y ⊆ C (Πd ) satisfying the assumptions of the theorem. 4.1. Proof of Theorem 4. Upper estimate Denote u = (t1 , . . . , tj0 −1 ), v = (tj0 +1 , . . . , td ), and d(v, u) = dtd . . . dtj0 +1 dtj0 −1 . . . dt1 . For every function f ∈ Y , we will have

∫  n   −     R(f ; q∗ ) =  f (t)dt − bj0 ck∗ · Ij0 (f ; x∗k ) n  Πd  k =1 ∫ ∫  ∫ n   aj − 0   ∗ ∗ f (u, tj0 , v)dtj0 d(v, u) − = ck f (u, xk , v)d(v, u)  Pj 0  Pj k=1 0  ∫ 0 ∫   n   aj − 0   = f (u, tj0 , v)dtj0 − ck∗ f (u, x∗k , v) d(v, u)  Pj  0 k = 1 0   ∫ ∫ aj n  −  0  ≤ f (u, t , v)dt − ck∗ f (u, x∗k , v) d(v, u)   Pj  0 k=1 ∫ 0   ρ(f (u, ·, v); S ∗ ) d(v, u). = n Pj

0

By assumption, the set D0 of vectors (u, v) ∈ Pj0 , for which function ψ(t ) = f (u, t , v) belongs to the class Y j0 , is dense in Pj0 . Then for every (u, v) ∈ Pj0 , there is a sequence {(um , vm )}∞ m=1 ⊂ D0 converging to (u, v). Since f is uniformly continuous on Πd , the sequence of functions gm (·) = f (um , ·, vm ) ∈ Y j0 , m ∈ N, converges uniformly to ψ on [0, aj0 ], which implies that ψ ∈ Y j0 . Taking into account the optimality of the quadrature formula Sn∗ for the class Yj0 , we have

  R(f ; q∗ ) ≤ n

∫

  ρ(ψ; S ∗ ) d(v, u) ≤ n Pj

0

∫

ρ(Y j0 ; Sn∗ )d(v, u) Pj

0

aj

= bj0 ρ(Y j0 ; Sn∗ ) = bj0 ρ(Yj0 ; Sn∗ ) = bj0 ρ(Yj0 ; Tn 0 ).

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V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

In view of arbitrariness of the function f ∈ Y , we obtain aj

R(Y ; q∗n ) ≤ bj0 ρ(Yj0 ; Tn 0 ).

(14)

4.2. Lower estimate It was shown by Smolyak [49] (the proof is also cited in [18]) that for every convex and centrally symmetric class of functions K ⊆ C [0, a] and every fixed set of nodes Xn = {x1 , . . . , xn } ⊂ [0, a], there holds

 ∫ n   a −   g (t )dt − ck g (xk ) = ρ(K ; Xn ) := inf sup  c1 ,...,cn ∈R g ∈K  0  k=1

a

∫

g (t )dt .

sup g ∈K g (xk )=0,k=1,n

(15)

0

Let q(f ) =

ni d − −

ci,k Ii (f ; xik )

(16)

i=1 k=1

be the arbitrary formula from Qna . Assume first that relation ni d − −

ci,k = |Πd |

(17)

i=1 k=1

does not hold. Since each class Yi contains constant functions, the function hc (t1 , . . . , td ) = c belongs to Y1 + · · · + Yd and hence, to Y . Then

 ∫ ni d −   −   cdx − ci,k c  R(Y ; q) ≥ sup |R(hc ; q)| = sup   c ∈R c ∈R  Πd i=1 k=1   ni d −   −   = sup |c |  |Πd | − ci,k  = ∞.   c ∈R i=1 k=1

(18)

Thus, it remains to compare the upper estimate (14) with the error of formulas of ∑ the form (16), ni where relation (17) holds. Equality (17) implies that for at least one index i, we have k= 1 ci,k > 0. ∑ni ϵ Choose any ϵ > 0 and for every i such that c > 0, let ϕ ∈ Y be a function such that i , k i i k=1 ϕiϵ (xik ) = 0, k = 1, . . . , ni , and ai

∫ 0

ϕiϵ (t )dt > ρ(Yi ; {xi1 , . . . , xini }) − ϵ

(19)

(such a function ϕiϵ exists due to relation (15)). For every index i such that

ϕiϵ (t ) ≡ 0. Then the function fϵ (t1 , . . . , td ) =

∑ni

k=1 ci,k ≤ 0, we let ϵ ϕ ( t ) belongs to the class Y and taking into k k=1 k

∑d

account (17), we will have R(Y ; q) ≥ R(fϵ ; q) =

∫

 Πd

ni d − −  ϕ1ϵ (t1 ) + · · · + ϕdϵ (td ) dtd . . . dt1 − ci,k Ii (fϵ ; xik ) i=1 k=1

a1

∫ = b1

ϕ1ϵ (t )dt + · · · + bd

0

0

 ∫ × Pi

ad

∫

ϕdϵ (t )dt −

ni d − −

ci,k

i=1 k=1

 d

−   ϵ i ϕjϵ (tj ) dtd . . . dti+1 dti−1 . . . dt1 ϕi (xk ) + j=1 j̸=i

ai

|Πd |

V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

527



 =

d −

aj

∫

ϕjϵ (t )dt −

bj 0

j =1

∫ aj ni d d − − ci,k −  ϕjϵ (t )dt  bj  | | Π d 0 j=1 i=1 k=1 j̸=i

 =

ni d − − ci,k 1− | Π d| i=1 k=1

+

∫ ni d − − bi ci,k i=1 k=1

=

d −

 ′

ni −

i=1

|Πd |  ci,k

k=1

 d −

bj

 ϵ

ϕj (t )dt

0

j=1 ai

aj

∫

ϕiϵ (t )dt

0

1

ai

∫

ai

ϕiϵ (t )dt ,

0

∑n

i where means that the summation takes place only over those i’s, for which k=1 ci,k > 0 (the ϵ equality remains true because ϕi is identically zero for the rest of indices i). Then taking into account (19) we obtain

∑′

R(Y ; q) ≥

d −

 ′

i =1

≥

d −

ni −

 ci,k

ai

k=1

 ′

i =1

ni −

1 

 ci,k



1  ai

k=1

ρ(Yi ; {xi1 , . . . , xini }) − ϵ  ρ(Yi ; Tnai i ) − ϵ .

In view of arbitrariness of ϵ , and taking into account relations (12) and (17) and the fact that a a ρ(Yi ; Tni i ) ≥ ρ(Yi ; Tn i ), i = 1, . . . , d, we will have R(Y ; q) ≥

d −

 ′

i =1

≥

|Πd | aj 0

ni −

 ci,k

k=1

1 ai

ρ( ;

Yi Tnai

)≥

aj

1 aj 0

ρ(Yj0 ;

aj Tn 0

)

d −

′

 n i −

i=1

aj

ρ(Yj0 ; Tn 0 ) = bj0 ρ(Yj0 ; Tn 0 ).

 ci,k

k=1

(20)

Taking into account relations (14), (18) and (20), we obtain the optimality of the formula q∗n for the class Y . Since relation (20) also holds for the formula q∗n , we obtain (13). Theorem 4 is proved. 5. Proof of the main results 5.1. Proof of Theorem 1 We recall that formula (5) is optimal for the class H ω [0, a] (cf. [30]) and is also optimal for its periodic analogue  H ω [0, a] (of period a) with

ρ(H ω [0, a]; Tna ) = ρ( H ω [0, a]; Tna ) = 2n

a 2n

∫

ω(t )dt .

(21)

0

In view of relations (7) and (21), we have 1 aj 0

aj

ρ(H ωj0 [0, aj0 ]; Tn 0 ) ≤

1 ai

ρ(H ωi [0, ai ]; Tnai ),

i = 1, . . . , d.

It is not difficult to see that H ω Πd ⊆ H ω1 [0, a1 ] × · · · × H ωd [0, ad ] and whenever ϕ1 ∈ H ω1 [0, a1 ], . . . , ϕd ∈ H ωd [0, ad ], we have

528

V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

  d d d −  − −   |ϕi (xi ) − ϕi (yi )| ϕi (xi ) − ϕi (yi ) ≤   i=1  i =1 i =1 ≤

d −

ωi (|xi − yi |),

(x1 , . . . , xd ), (y1 , . . . , yd ) ∈ Πd .

i =1

Then by Theorem 4, the formula q∗n is optimal for the class H ω Πd . In view of relation (13) and relation (21) for ω = ωj0 and a = aj0 , we have (8) for the non-periodic case. The assertion of Theorem 1 for the periodic class  H ω Πd is shown analogously. Theorem 1 is proved. 5.2. Proof of Theorem 2 Recall (cf. [36]) that the quadrature formula Sn∗ (g ) =

  n − 2π k 2π k=1

n

g

n

r ∞ is optimal for the class W , r ∈ N, among formulas from Tn2π with r ∞ ρ(W ; Tn2π ) =

2π Kr

.

nr π Since 1 ≤ Kr ≤ 2 , r ∈ N (cf. e.g. [31, page 105]), for every i such that ri < rj and n ≥ 2, we have 1

r

∞j ; Tn2π ) = ρ(W

Krj

2π n It is not difficult to verify that rj

≤

(22)

π Kr i Kr 1 ri ∞ ≤ ri = ρ(W ; Tn2π ). rj −1 i n 2π 2n · n

r1 rd r r1 rd ∞ ∞ ∞ ∞ ∞ W × ··· × W ⊃W ⊃W + ··· + W .

Then by Theorem 4, we obtain the optimality of the formula q∗n , n ≥ 2, and from relation (13) and relation (22) with r = rj we obtain (9). Theorem 2 is proved. 5.3. Proof of Theorem 3 Let

v(t ) = min ωi (t ), i=1,...,d

t ≥ 0.

Then the ratio v(t )/t is monotonically non-increasing on (0, ∞). It was shown in [31, page 255] that this property implies sub-additivity. For completeness, we give this argument below. For a pair of numbers z , p ∈ (0, ∞), we have

v(z + p) v(z + p) v(z ) v(p) +p ≤z +p = v(z ) + v(p) z+p z+p z p which shows sub-additivity of v . All other properties of the modulus of continuity for v follow v(z + p) = z

immediately. It was proved in [15, Theorem 1] that the cubature formula q∗n is optimal for the class H1v Πd among formulas from Gan and v

R(H1 Πd ; qn ) = 2bj0 n ∗

aj 0 2n

∫

v(t )dt .

0

For every f ∈ H1v Πd , we have

|f (x1 , . . . , xd ) − f (y1 , . . . , yd )| ≤ v(|x1 − y1 | + · · · + |xd − yd |) ≤ v(|x1 − y1 |) + · · · + v(|xd − yd |) ≤ ω1 (|x1 − y1 |) + · · · + ωd (|xd − yd |).

(23)

V.F. Babenko et al. / Journal of Complexity 27 (2011) 519–530

529

Hence, H1v Πd ⊆ H ω Πd , where ω = (ω1 , . . . , ωd ). Then for every cubature formula q ∈ Gan , taking into aj

account (23), (10) and (8), for n > 2ϵ0 , we obtain R(H ω Πd ; q) ≥ R(H1v Πd ; q) ≥ R(H1v Πd ; q∗n ) aj 0 2n

∫ = 2bj0 n

v(t )dt

0 aj 0 2n

∫ = 2bj0 n

ωj0 (t )dt = R(H ω Πd ; q∗n ),

0

which implies the optimality of q∗n and relation (11). Theorem 3 is proved. Acknowledgments The authors would like to thank both referees and Professor Elizabeth Goode for their comments which helped in improving the presentation and the language in this paper. References [1] V.F. Babenko, Asymptotically exact estimate of the remainder of the best cubature formulae for certain classes of functions, Math. Notes 19 (1976) 187–193. [2] V.F. Babenko, Faithful asymptotics of remainders optimal for some glasses of functions with cubic weight formulas, Math. Notes 20 (4) (1976) 887–890 (in Russian). [3] V.F. Babenko, On the optimal error bound for cubature formulae on certain classes of continuous functions, Anal. Math. 3 (1977) 3–9. [4] V.F. Babenko, On a certain problem of optimal integration. Investigations in current problems in summation and approximation of functions and their applications, Dnepropetrovsk State Univ., Dnepropetrovsk, 1984, pp. 3–13 (in Russian). [5] K.I. Babenko, Some problems in approximation theory and numerical analysis, Russian Math. Surveys 40 (1) (1985) 1–30. [6] V.F. Babenko, Approximations, widths and optimal quadrature formulae for classes of periodic functions with rearrangement invariant sets of derivatives, Anal. Math. 13 (4) (1987) 281–306. [7] V.F. Babenko, Widths and optimal quadrature formulas for convolution classes, Ukrainian Math. J. 43 (9) (1991) 1064–1076. [8] V.F. Babenko, On the optimization of weighted quadrature formulas, Ukrainian Math. J. 47 (8) (1995) 1157–1168. [9] V.F. Babenko, S.V. Borodachov, On optimization of cubature formulae for classes of monotone functions of several variables, Vestn. Dnepropet. Univ. Math. 7 (2002) 3–7 (in Russian). [10] V.F. Babenko, S.V. Borodachov, On optimization of approximate integration over a d-dimensional ball, East J. Approx. 9 (1) (2003) 95–109. [11] V.F. Babenko, S.V. Borodachov, On optimization of approximate integration of multivariate periodic functions, NorthHolland Math. Stud. 197 (2004) 13–22. [12] V.F. Babenko, S.V. Borodachov, On the construction of optimal cubature formulae which use integrals over hyperspheres, J. Complexity 23 (3) (2007) 346–358. [13] V.F. Babenko, S.V. Borodachov, Optimal integration of functions with bounded L1 -norm of the Laplacian on a ball (submitted for publication). [14] V.F. Babenko, S.V. Borodachov, D.S. Skorokhodov, Optimal cubature formulae for certain classes of functions defined on a unit cube (submitted for publication). [15] V.F. Babenko, S.V. Borodachov, D.S. Skorokhodov, Construction of optimal cubature formulas related to computer tomography, Constr. Approx. (in press) (doi:10.1007/s00365-010-9095-6). [16] V.F. Babenko, D.S. Skorokhodov, On the best interval quadrature formulae for classes of differentiable periodic functions, J. Complexity 23 (4–6) (2007) 890–917. [17] V.F. Babenko, D.S. Skorokhodov, On the best interval quadrature formulae for convolution classes, East J. Approx. 14 (3) (2008) 353–376. [18] N.S. Bahvalov, The optimality of linear operator approximation methods on convex function classes, Zh. Vychisl. Mat. Mat. Fiz. 11 (1971) 1014–1018 (in Russian). [19] J.J. Benedetto, A.I. Zayed (Eds.), Sampling, Wavelets, and Tomography, Birkhäuser, 2004. [20] O.V. Besov, V.P. Il’in, S.M. Nikol’skii, Integral Representations of Functions and Embedding Theoems, Nauka Fizmatlit, Moscow, 1996, 480 pp. English translation: John Wiley and Sons Inc, 1979. [21] B.D. Bojanov, An extension of the Pizzetti formula for polyharmonic functions, Acta Math. Hungar. 91 (1–2) (2001) 99–113. [22] B.D. Bojanov, Cubature formulae for polyharmonic functions, recent progress in multivariate approximation, WittenBommerholz, 2000, in: Internat. Ser. Numer. Math., vol. 137, Birkhauser, Basel, 2001, pp. 49–74. [23] B.D. Bojanov, Optimal quadrature formulas, Russian Math. Surveys 60 (6) (2005) 1035–1055. [24] B.D. Bojanov, D.K. Dimitrov, Guassian extended cubature formulae for polyharmonic functions, Math. Comp. 70 (234) (2001) 671–683. [25] B. Bojanov, G. Petrova, Uniqueness of the Gaussian quadrature for a ball, J. Approx. Theory 104 (2000) 21–44.

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