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On the construction of optimal cubature formulae which use integrals over hyperspheres
Journal of Complexity 23 (2007) 346 – 358 www.elsevier.com/locate/jco
On the construction of optimal cubature formulae which use integrals over hyperspheres V.F. Babenkoa, b , S.V. Borodachovc,∗ a Department of Mechanics and Mathematics, Dnepropetrovsk National University, vul. Naukova, 13, 49625, Ukraine b Donetsk Institute of Applied Mathematics and Mechanics of NAS of Ukraine, 74 Luxembourg St., Donetsk 340114,
Ukraine c Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
Received 18 June 2004; accepted 3 February 2007 Available online 14 March 2007
Abstract We consider formulae of approximate integration over a d-dimensional ball which use n surface integrals along (d − 1)-dimensional spheres centered at the origin. For a class of functions defined on the ball with gradients satisfying an integral restriction, optimal formulae of this type are obtained. © 2007 Elsevier Inc. All rights reserved. Keywords: Optimal cubature formula; Class of functions; Worst case error
1. Introduction 1.1. Setting of the problem The problem of optimization of approximate integration over a class of functions goes back to the works of A.N. Kolmogorov and S.M. Nikol’skiy published in late forties—early fifties. A large number of results in this direction have been obtained later by many researchers for classes of both univariate and multivariate functions. A review of most of these results can be found, for example, in [13,15]. The majority of works deals with quadrature and cubature formulae using “pointwise” information. During the last decades a number of authors consider formulae which use mean values taken along sufficiently short intervals (in univariate case) or parallelepipeds, balls, etc. (in multidimensional case). This is natural, since results of measurements are mean values of a function along some intervals. Modern applications, such as computer tomography, which ∗ Corresponding author.
E-mail addresses: [email protected] (V.F. Babenko), [email protected] (S.V. Borodachov). 0885-064X/$ - see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jco.2007.02.002
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347
provides information about a three-dimensional object by analysing two-dimensional images of its layers, lead to cubature formulae using as data integrals along sets of smaller dimensionality than the integration domain. In this paper we consider a problem of optimization of cubature formulae for integration over a ball in Rd in the following setting. Let d ∈ N, d > 1, |·| be the Euclidean norm in Rd , Bd [a, r] and Sd [a, r] be the ball and the sphere, respectively, centered at point a ∈ Rd with radius r > 0. Denote by bd the Lebesgue measure of the ball Bd [0, 1], where 0 = (0, . . . , 0) ∈ Rd , and by d —the (d − 1)-dimensional measure of the sphere Sd [0, 1]. It is known that d = dbd = dd/2 / (1 + d/2). Let also C(Bd [0, r]) be the space of continuous functions f : Bd [0, r] → R with the norm f =
max |f (x)|.
x∈Bd [0,r]
For a function u ∈ C Bd [0, r] put 1 H u() = u(x) dS, ∈ (0, r], d d−1 Sd [0,]
H u(0) = u(0),
(1)
where the surface integral taken along the sphere Sd [0, ] is of the first kind (i.e. H u() are mean values of the function u along the spheres Sd [0, ]). Let n ∈ N and Und be the set of all functionals K : C(Bd [0, r]) → R (algorithms of approximate integration of a function f along Bd [0, r]) of the form K(f ) = K(f ; r n , cn ) =
n
ck · Hf (rk ),
(2)
k=1
where cn = (c1 , . . . , cn ), r n = (r1 , . . . , rn ) ∈ Rn , and 0 r1 < · · · < rn r. Let V ⊂ C(Bd [0, r]) be a class of functions. Put R(f ; K) = R (f ; r n , cn ) = f (x) dx − K(f ; r n , cn ), Bd [0,r]
R(V ; K) = R (V ; r n , cn ) = sup |R (f ; r n , cn )|, f ∈V
Rn (V ) = inf R (V ; K). K∈Und
(3)
Problem. It is required to find value (3) and optimal cubature formulae for the class V , that is algorithms K ∗ ∈ Und , if they exist, delivering infinum on the right-hand side of (3). In other words, we look for sets of node hyperspheres and coefficients ck which provide the smallest worst-case error over the considered class. Optimal formulae of the form (2) can also be an intermediate step in construction of optimal multivariate cubatures using as data values of the function at points. Formulae using as information integrals along sets with smaller number of dimensions than the domain, have been considered in earlier mathematical literature. A certain formula of the form (2) exact for all d-variate polynomials of degree 4n − 1 was built in the book of Mysovskih [12]
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(this example is due to Lusternik and Kantorovich). Cubatures which are weighted sums of arbitrary functionals were considered by M¨oller building upon the work of Mysovskih. Formulae using as information integrals along lines were considered by Soler [14]. The formula constructed in [12] has the highest algebraic degree of precision [7]. Its multiple node extensions were built by Bojanov and Dimitrov [6]. Optimization problems for cubature formulae using integrals along intersections of the integration domain and hyperplanes, were considered by Bojanov and Petrova [7] and the authors [3]. The problem set above was considered by authors [4] for classes of functions defined on a ball in Rd and having a given majorant for the modulus of continuity. 2. Main result Let p : (0, r] → (0, ∞) be a continuous function such, that r p(t)t d−1 dt < ∞
(4)
0
and Vd (p) be the closure in C(Bd [0, r]) of the set of continuously differentiable functions f : Bd [0, r] → R, such that |grad f (x)| · p(|x|) dx 1. (5) Bd [0,r]
We solve the previously mentioned problem for classes W 1 Bd [0, r] = Vd (p0 ), where p0 (t) = 1, and V 1 Bd [0, r] = Vd (p1 ), where p1 (t) = t 1−d , t ∈ (0, r]. For classes W1m , m ∈ N, of univariate periodic functions with mth derivative in the unit ball of L1 , problems of optimization of quadrature formulae were considered in [1,8–11]. The classes introduced above can be considered as a generalization of W11 . Denote gp,∞
g(t) . = ess sup p(t)t d−1 t∈[0,r]
With the help of the duality relations, optimization problems considered in [1,5,8–10] were reduced to the problem of minimization of the uniform norm of monosplines corresponding to quadrature formulae. The problem considered in this paper is in general reduced to the problem of the best approximation of a power function by piecewise constant ones in the norm · p,∞ , i.e. in usual uniform norm for the class V 1 Bd [0, r] and in the uniform norm with weight t 1−d —in the case of W 1 Bd [0, r]. Denote by Zn,d the set of functions z such that z(t) = 0, t ∈ [0, r1 ), z(t) = Ck , t ∈ [rk , rk+1 ), k = 1, . . . , n − 1, z(t) = bd r d , t ∈ [rn , r], where Ck ∈ R, k = 1, . . . , n − 1, and 0 r1 < · · · < rn r. Theorem 1. Let d, n ∈ N, d > 1, r > 0. For any positive continuous weight p satisfying (4) the following equality holds: Rn [Vd (p)] =
1 inf z(t) − bd t d p,∞ . d z∈Zn,d
(6)
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349
∗ ), If a function z∗ ∈ Zn,d with nodes 0 r1∗ < · · · < rn∗ r such that z∗ (t) = Ck∗ , t ∈ [rk∗ , rk+1 k = 1, . . . , n − 1, delivers infinum on the right-hand side of (6), then formula n K ∗ (f ) = K f ; r ∗n , c∗n = ck∗ Hf (rk∗ ),
(7)
k=1 ∗ , k = 2, . . . , n − 1, c∗ = b r d − C ∗ , is optimal for the class where c1∗ = C1∗ , ck∗ = Ck∗ − Ck−1 d n n−1 Vd (p) among formulae from Und .
The main results of this paper are contained in the following statement. Theorem 2. Let d, n ∈ N, d > 1, r > 0. Formula (7) with ck∗ = r d bd /n, rk∗ = r, k = 1, . . . , n, is optimal for the class V 1 Bd [0, r]. In addition, rd . Rn V 1 Bd [0, r] = 2nd
√ d
(2k − 1)/(2n) ·
(8)
For the class W 1 B2 [0, r] formula (7) with ck∗ = 2r 2 k(n(n + 1))−1 , rk∗ = rk(n(n + 1))−1/2 , k = 1, . . . , n, is optimal. In addition, r Rn (W 1 B2 [0, r]) = √ . 2 n(n + 1) Let d 3, (t) = (t d − 1)(t d−1 + 1)−1 and numbers x1 > 1,…, xn > 1 be such that (x1 ) = 1, −1 (xk ) = (xk−1 )xk−1 , k = 2, . . . , n. Then formula (7) with r1∗ = (xn )(1 + (xn ))−1/d r, rk∗ = ∗ xk−1 rk−1 , k = 2, . . . , n, ck∗ = 2bd (xk )(rk∗ )d , k = 1, . . . , n, is optimal in the class W 1 Bd [0, r]. In addition, Rn (W 1 Bd [0, r]) = r1∗ /d. For both classes the optimal formula is unique. 3. Proof of Theorem 1 3.1. Reduction to a univariate optimization problem Note, that H defined by (1) can be considered as an operator from C(Bd [0, r]) to C[0, r]—the space of continuous functions on [0, r]. Passing in (1) to the d-dimensional spherical coordinates x (, ) = cos 1 , sin 1 cos 2 , . . . , sin 1 ·. . .· sin d−2 cos d−1 , sin 1 ·. . .· sin d−1 , d−1−j , = 1 , . . . , d−1 ∈ d := [0, ]d−2 ×[0, 2], 0, and setting I () = d−1 j =1 (sin j ) d = d1 . . . dd−1 , we obtain 1 1 f (0) dS = f (x(0, ))I () d, Hf (0) = f (0) = d d d Sd [0,1] 1 1 f (x) dS = f (x(, ))d−1 I () d Hf () = d d−1 Sd [0,] d d−1 d 1 = f (x(, ))I () d, > 0. (9) d d Consider set U of univariate quadrature formulae q(g; t n , cn ) =
n k=1
ck g(tk ),
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where cn = (c1 , . . . , cn ), t n = (t1 , . . . , tn ) ∈ Rn and 0t1 < · · · < tnr. For a class D ⊂ C[0, r] denote r g(t)t d−1 dt − q(g; t n , cn ), r(g; t n , cn ) = d 0
r(D; t n , cn ) = sup r(g; t n , cn ),
rn (D) =
g∈D
inf q(·;t n ,cn )∈U
r(D; t n , cn ).
d (p) the set of continuously differentiable functions on Bd [0, r] which satisfy (5). Denote by V
d (p) we have For every K(·; r n , cn ) ∈ Und and f ∈ V r n R (f ; r n , cn ) = f (x) dx− ck Hf (rk )= f (x (, )) d−1 I () d d Bd [0,r]
−
n
0
k=1
r
ck Hf (rk )=d
k=1
0
d
d−1 Hf () d−
n
ck Hf (rk )=r(Hf ; r n , cn ).
k=1
d (p) is dense in Vd (p), we get Then, since V
d (p); r n , cn ) = r(H (V
d (p)); r n , cn ) R(Vd (p); r n , cn ) = R(V
(10)
d (p))]. Let W 1 (p) be the set of continuously differentiable and hence, Rn [Vd (p)] = rn [H (V functions g : [0, r] → R such that g (0) = 0 and r g (t) p(t)t d−1 dt 1. (11) d 0
d (p)) = W 1 (p). For every f ∈ V
d (p) and 0 ∈ [0, r] we get Show, that H (V * 1 (Hf ) (0 ) = f (x(, ))| = I () d 0 d d * 1 =
grad f | x=x (0 ,) , x (1, ) I () d, d d where a, b is the scalar product of vectors a and b. In particular, 1 1 (Hf ) (0) =
grad f (0), x (1, ) I () d =
grad f (0), x dS = 0. d d d Sd [0,1] Note, that (Hf ) () will be continuous and r (Hf ) () d−1 p() d d 0 r d−1 p() d | =
grad f , x ) I () d (1, x=x(,) d 0 r grad f | x=x(,) p()I () dd−1 d 0 d |grad f (x)| p(|x|) dx 1. = Bd [0,r]
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d (p)) ⊂ W 1 (p). Let g ∈ W 1 (p). We show that f (x) = Thus, Hf ∈ W 1 (p) and hence, H (V
g(|x|) is in Vd (p) and Hf = g. Indeed, by (9) for ∈ [0, r] we get 1 1 1 Hf ()= f (x(, ))I () d= g()I () d= g() dS=g(). d d d d d Sd [0,1] x It is not difficult to see that grad f (x) = g (|x|) |x| , x = 0, grad f (0) = 0 and f is continuously differentiable on Bd [0, r]. In addition, since g satisfies (11), we have g (|x|) · p(|x|) dx |grad f (x)| · p(|x|) dx = Bd [0,r] Brd [0,r] g () p()I () d · d−1 d = d 0 r g () p()d−1 d · I () d = d 0 r g () p()d−1 d · 1 dS = Sd [0,1] 0 r g () p()d−1 d1. = d 0
d (p), Hf = g and H (V
d (p)) = W 1 (p). Then, from (10) for every formula Hence, f ∈ V d K(·; r n , cn ) ∈ Un , R(Vd (p); r n , cn ) = r(W 1 (p); r n , cn )
(12)
and also Rn (Vd (p)) = rn (W 1 (p)). In order to find an optimal formula for Vd (p) we need to obtain an optimal formula for W 1 (p). It is not difficult to see that functions gc (t) = c, c ∈ R, belong to the class W 1 (p). For every formula q(·; r n , cn ) ∈ U we have r n n n r (gc ; r n , cn ) = d ct d−1 dt − ck c = cbd r d − ck c = c bd r d − ck . 0
k=1
k=1
k=1
d Denote by n the set of all pairs of vectors (r n , cn ), such that 0 r1 < · · · < rn r and
n bd r =
n d / n , then, since bd r − k=1 ck = k=1 ck . If formula q(·; r n , cn ) ∈ U is such that (r n , cn ) ∈ 0 and c is arbitrary, value r (gc ; r n , cn ) is unbounded, and r(W 1 (p); r n , cn ) will be infinite. Therefore, it is sufficient to look for optimal formulae only among those with (r n , cn ) ∈ n .
4. Reduction to a problem of the best approximation Let q(·; r n , cn ) ∈ U be such that (r n , cn ) ∈ n . Then for every function g ∈ W 1 (p) taking d into account that d = dbd and bd r = nk=1 ck , we have r r n d−1 d r(g; r n , cn ) = d g() d − ck g (rk ) = bd r g(r) − bd d g () d 0
−
n k=1
k=1
ck g (rk ) =
n k=1
0
r
ck (g(r) − g(rk )) − bd 0
d g () d
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=
n
r
ck 0
k=1 n r
[rk ,r] ()g () d − bd
=
0
ck [rk ,r] () − bd
r
d g () d
0
g () d,
d
(13)
k=1
where [rk ,r] (t) is the characteristic function of the segment [rk , r], k = 1, . . . , n. Denote (t) = (t; r n , cn ) =
n
ck [rk ,r] (t) − bd t d .
k=1
The set of all functions of the form Using (13), we get r(W (p); r n , cn ) = 1
n
k=1 ck [rk ,r] (t)
|r(g; r n , cn )| =
sup g∈W 1 (p)
where (r n , cn ) ∈ n , coincides with Zn,d . sup
g∈W 1 (p) r
(·; r n , cn )p,∞ ·
r 0
(; r n , cn )g () d
g () p()d−1 d
sup
g∈W 1 (p) 0
(·; r n , cn )p,∞ . d
(14)
Equality in (14) can be shown for example in the following way. Let { m }∞ m=1 be a sequence such that m < (·; r n , cn )p,∞ , m ∈ N, and lim m = (·; r n , cn )p,∞ . Note, that since m→∞
(t; r n , cn )/(p(t)t d−1 ) is continuous on [0, r]\{0, r1 , . . . , rn }, for every m ∈ N there is an interval (am , ym ) ⊂ (0, r] where (·; r n , cn ) preserves its sign such that (t; r n , cn )/(p(t)t d−1 ) > ∈ C[0, r] is non-negative and supported m , t ∈ (am , ym ). Let gm be a function such that gm inside (am , ym ) and ym gm (t)p(t)t d−1 dt = 1. d am
Then, gm ∈ W 1 (p) and by (13)
r r(W (p); r n , cn ) |r(gm ; r n , cn )| = (; r n , cn )gm () d 0 ym ym = (; r n , cn )gm () d m gm ()p()d−1 d 1
am
m = , d
am
m ∈ N.
−1 Hence, r(W 1 (p); r n , cn ) −1 d lim m→∞ m = d (·; r n , cn )p,∞ . Taking into account (14) d and (12), for every K(·, r n , cn ) ∈ Un with (r n , cn ) ∈ n we will have 1 (·; r n , cn )p,∞ , (15) R(Vd (p); r n , cn ) = r(W 1 (p); r n , cn ) = d
Rn (Vd (p)) = =
inf
(r n ,cn )∈n
1 d
inf
r(W 1 (p); r n , cn )
(r n ,cn )∈n
(·; r n , cn )p,∞ =
1 inf z(t) − bd t d p,∞ . d z∈Zn,d
V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
353
∗ ), k = 1, . . . , n − 1, Note, that if a function z∗ ∈ Zn,d such that z(t) = Ck∗ , t ∈ [rk∗ , rk+1 ∗ ∗ 0 r1 < · · · < rn r, is the element of the best approximation to y(t) = bd t d , then spline n
∗ ) ∗ (t) − b t d = z∗ (t) − b t d where C ∗ = 0, C ∗ = b r d , has (·; r ∗n , c∗n ) := (Ck∗ − Ck−1 d d d [r ,r] n 0 k
k=1
the least · p,∞ -norm and the corresponding formula K(·; r ∗n , c∗n ) ∈ Und will be optimal for the class Vd (p). Theorem 1 is proved. 5. Proof of Theorem 2 5.1. Best formula for V 1 Bd [0, r] Assume now that p(t) = t 1−d , t ∈ (0, r]. Denote by L∞ [0, r] the space of functions g : [0, r] → R such that g∞ := ess sup |g(t)| < ∞. t∈[0,r]
From (15) for every formula in Und , such that (r n , cn) ∈ n , we have 1 (·; r n , cn )∞ . R V 1 Bd [0, r]; r n , cn = d
(16)
Then 1 Rn V 1 Bd [0, r] = d
inf
(·; r n , cn )∞ =
(r n ,cn )∈n
1 inf z(t) − bd t d ∞ . d z∈Zn,d
(17)
For all vectors r n ,cn , such that (r n , cn ) ∈ n , we get (·; r n , cn )∞ = max {max {|(rk − 0)| , |(rk )|}} k=1,n
k k−1 |(rk )−(rk −0)| 1 d d max ci −bd rk − ci −bd rk = max 2 2 k=1,n k=1,n i=1 i=1 n n 1 1 1 1 |ck | = max |ck | ck = (18) bd r d . 2 k=1,n 2n 2n 2n k=1
Denote rk∗ =
√ d
k=1
(2k − 1)/(2n) · r, ck∗ = bd r d /n, k = 1, . . . , n. Then
k−1 ∗ ∗ d r d bd 2k − 1 d r d bd ∗ ∗ ∗ r − 0; r , c = r = (k − 1) − b r c − b d k d n n k i = 2n , n 2n i=1
k rd b ∗ ∗ ∗ 2k − 1 d r d bd d d ∗ ∗ r ; r , c = k − bd r = , ci − bd rk = n n k n 2n 2n i=1
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V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
k = 1, . . . , n and taking into account estimate (18) we have r d bd (·; r ∗n , c∗n )∞ = max {max {(rk∗ − 0) , (rk∗ )}} = 2n k=1,n = inf (·; r n , cn )∞ . (r n ,cn )∈n
Taking into account (17) and (16) we finally obtain r d bd 1 rd Rn V 1 Bd [0, r] = . (·; r ∗n , c∗n )∞ = R V 1 Bd [0, r]; r ∗n , c∗n = = d 2nd 2nd This means that formula (7) with such rk∗ ’s and ck∗ ’s is optimal in V 1 Bd [0, r] and (8) holds. If we have equality sign everywhere in (18), one can show that ck = ck∗ , k = 1, . . . , n, and then, using equality of absolute values of the one-sided limits of (·; r n , cn ) at nodes, that rk = rk∗ , k = 1, . . . , n. Hence, optimal formula will be unique. 6. Best formula for W 1 Bd [0, r] Let p(t) = 1. Denote
(t) = (t; r n , cn ) = (t; r n , cn ) · t 1−d ,
t ∈ (0, r].
(19)
For every formula K(·; r n , cn ) ∈ Und with (r n , cn ) ∈ n we have R(W 1 Bd [0, r]; r n , cn ) = Rn (W 1 Bd [0, r]) =
1 d
1 (·; r n , cn )∞ , d inf
(r n ,cn )∈n
(·; r n , cn )∞ .
(20) (21)
Denote by An the set of splines (19) with (r n , cn ) ∈ n and r1 > 0. For any ∈ An we have
(t) = −bd t, 0 < t < r1 , and ∞ < ∞. If r1 = 0 and c1 = 0, we get lim (t; r n , cn ) = lim (c1 t 1−d − bd t) = ∞
t→0+
t→0+
and such splines do not deliver the infinum in (21). Below, we will find a unique spline ∗ ∈ An with minimal norm. It will have non-zero coefficients ci∗ . Any spline (19) with r1 = 0 and c1 = 0 coincides with a spline from An having the same ri and ci , i = 2, . . . , n, positive first node and zero first coefficient. Hence, it will not deliver infinum in (21) and ∗ will be the only spline (19) with (r n , cn ) ∈ n for which this infinum is attained. Since every formula in Und with / n has infinite error, formula corresponding to ∗ will be the only optimal formula for (r n , cn ) ∈ 1 W Bd [0, r]. Note, that every ∈ An will be right continuous, if we put (0) = 0. Lemma 1. Given vector r n : 0 < r1 < · · · < rn r, spline (·; r n , c n ) ∈ An with Ci :=
i k=1
ck = bd
ri + ri+1 , 1−d 1−d ri + ri+1
i = 1, . . . , n − 1,
has the least · ∞ -norm among all splines from An with given r n .
V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
355
Proof. Denote C0 = 0, Cn = bd r d , r0 = 0, rn+1 = r. Then (t; r n , c n ) = (Ci − bd t d )t 1−d , t ∈ [ri , ri+1 ), i = 0, . . . , n. It is not difficult to see that (t; r n , c n ) is monotone on (ri , ri+1 ), i = 0, . . . , n, and (0; r n , c n ) = (r; r n , c n ) = 0. Let (t) =
td − 1 , t d−1 + 1
t > 0.
Since 1 + (t) = (1 + t)(1 + t 1−d )−1 , it is not difficult to see that ri+1 ri bd (ri + ri+1 ) d d + 1 = bd ri+1 +1 , = bd ri Ci = 1−d 1−d ri ri+1 ri + ri+1
(22)
i = 1, . . . , n − 1. Using (22) and the fact, that (1/t) = −(t)/t, t > 0, we get (r1 − 0) = −bd r1 < 0,
(ri − 0) =
Ci−1 − bd rid
rid−1
= b d ri
(ri ) = (Ci − bd rid )ri1−d = bd ri
ri−1 ri = −bd ri−1 < 0, ri ri−1
ri+1 ri
i = 2, . . . , n,
> 0,
i = 1, . . . , n − 1,
(23)
(rn ) = bd (r d −rnd )rn1−d 0. Hence, (ri ; r n , c n ) = − (ri+1 −0; r n , c n ) > 0, i = 1, . . . , n−1. Then, since (t; r n , c n ) is monotone between its nodes, we get that (·; r n , c n )∞ = max max{ (ri ; r n , c n ) , (ri − 0; r n , c n )}. i=1,n
Any other spline (·; r n , cn ) with given r n , coincides with (·; r n , c n ) on (0, r1 ) and (rn , r) and has the form (t; r n , cn ) = (Ci − bd t d )t 1−d , t ∈ (ri , ri+1 ), i = 1, . . . , n − 1. Then, for every i = 1, . . . , n−1, either (ri ; r n , cn ) (ri ; r n , c n ) > 0 or (ri+1 −0; r n , cn ) (ri+1 −0; r n , c n ) < 0 and (·; r n , cn )∞ max max{| (ri ; r n , cn )| , | (ri − 0; r n , cn )|} i=1,n max max{ (ri ; r n , c n ) , (ri − 0; r n , c n )} = (·; r n , c n )∞ . i=1,n
Lemma 1 is proved. Denote by Mn the set of all splines (·; r n , cn ) ∈ An with Ci := ik=1 ck = 1−d −1 ) , i = 1, . . . , n − 1. Let numbers x1 > 1, . . . , xn > 1 be defined by bd (ri + ri+1 )(ri1−d + ri+1 recurrence relations (x1 ) = 1,
−1 (xi ) = xi−1 (xi−1 ), i = 2, . . . , n.
(24)
Since for every y > 0 equation (t) = y has exactly one solution in (1, ∞), the sequence of xi ’s is unique. Lemma 2. Let d, n ∈ N, d > 1. There is a unique spline ∗ = (·; r ∗n , c∗n ) ∈ An such that − ∗ (r1∗ − 0) = ∗ (r1∗ ) = − ∗ (r2∗ − 0) = ∗ (r2∗ ) = · · · = − ∗ (rn∗ − 0) = ∗ (rn∗ ). In this case ∗ , i = 2, . . . , n, c∗ = 2b (x )(r ∗ )d , i = 1, . . . , n. r1∗ = (xn )(1 + (xn ))−1/d r, ri∗ = xi−1 ri−1 d i i i
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Proof. Denote Ci∗ = ik=1 ck∗ , i = 1, . . . , n − 1, C0∗ = 0, Cn∗ = bd r d . From equations ∗ (ri∗ ) = ∗ − 0), i = 1, . . . , n − 1, we get (C ∗ − b (r ∗ )d )(r ∗ )1−d = (b (r ∗ )d − C ∗ )(r ∗ )1−d , − ∗ (ri+1 d i d i+1 i i i i+1 Ci∗ =
∗ ) bd (ri∗ + ri+1
∗ )1−d (ri∗ )1−d + (ri+1
i = 1, . . . , n − 1.
,
− ∗ (r1∗ − 0) = ∗ (r1∗ ) is equivalent to bd r1∗ = Then ∗ ∈ Mn and we apply ∗(23). Relation ∗ ∗ ∗ ∗ ∗ bd r1 r2 /r1 , and hence, r2 /r1 = 1, r2 /r1∗ > 1. Then, since x1 is unique, we can only have r2∗ /r1∗ = x1 . For i = 2, . . . , n − 1 relation − ∗ (ri∗ − 0) = ∗ (ri∗ ) is equivalent to ∗ bd ri−1
ri∗ ∗ ri−1
=
bd ri∗
∗ ri+1
ri∗
,
∗ ri+1
ri∗
=
ri∗ ∗ ri−1
−1
r∗ ∗i ri−1
.
∗ /r ∗ > 1, i = 1, . . . , n − 1, satisfy recurrence formulae (24) and the only option is Hence, ri+1 i ∗ ∗ ri+1 /ri = xi , i = 1, . . . , n − 1. Finally, from equality − ∗ (rn∗ − 0) = ∗ (rn∗ ) we get
−bd rn∗ 1+
∗ rn−1
rn∗
(xn−1 ) = xn−1
= (bd r d − bd (rn∗ )d )(rn∗ )1−d ,
r rn∗
d ,
−
1
=
xn−1
r rn∗
d − 1,
rn∗ = (1 + (xn ))−1/d r.
∗ /x , i = n − 1, . . . , 1, are also uniquely deSince rn∗ < r is uniquely determined, ri∗ = ri+1 i ∗ ∗ termined. From equality rn (xn ) = rn (xn−1 )/xn−1 = rn∗ (xn−2 )/(xn−1 · xn−2 ) = · · · = rn∗ /(xn−1 · . . . · x1 ) = r1∗ , we have r1∗ = (xn )(1 + (xn ))−1/d r. For the coefficients ck∗ using (22), we obtain c1∗ = C1∗ = bd (r1∗ )d ((x1 ) + 1) = 2bd (r1∗ )d , −1 ∗ ci∗ = Ci∗ − Ci−1 = bd (ri∗ )d ((xi ) + 1) − bd (ri∗ )d xi−1 +1 −1 ) = 2bd (xi )(ri∗ )d , = bd (ri∗ )d ((xi ) + (xi−1 )xi−1
Using formula for
rn∗
i = 2, . . . , n − 1.
and (22), we also get
−1 ∗ = bd (rn∗ )d (1 + (xn )) − bd (rn∗ )d ((xn−1 ) + 1) cn∗ = bd r d − Cn−1 −1 = bd (rn∗ )d ((xn ) + (xn−1 )xn−1 ) = 2bd (xn )(rn∗ )d .
Lemma 3. For any ∈ Mn , = ∗ , we have ∞ > ∗ ∞ . Proof. Assume, that there is a spline = (·; r n , cn ) ∈ Mn , such that = ∗ and ∞ a := ∗ ∞ = ∗ (ri∗ ) = − ∗ (ri∗ − 0), i = 1, . . . , n. Let ai = − (ri − 0), i = (ri ), i = 1, . . . , n. Then, by relations (23) ai ’s and i ’s are non-negative and ai a, i a, i = 1, . . . , n. In view of Lemma 2 at least one of these inequalities must be strict. Show by induction, that rk rk∗ , k = 1, . . . , n. Since a1 = bd r1 a = bd r1∗ , we get r1 r1∗ = ∗ for some m < n. Then, = (C − b r d )r 1−d a = (C ∗ − a/bd . Assume, that rm rm m d m m m m ∗ d ∗ 1−d 1−d ∗ (r ∗ )1−d b (r −r ∗ ) 0. Hence, C C ∗ . Using equalities and Cm rm −Cm bd (rm ) )(rm ) d m m m m m 1−d d ∗ ∗ )(r ∗ 1−d , we get C = b r d − Cm )rm+1 and a = (bd (rm+1 )d − C m am+1 = (bd rm+1 m d m+1 − m+1 ) d−1 ∗ ∗ ∗ d d−1 am+1 rm+1 Cm = bd (rm+1 ) − a(rm+1 ) . It is not difficult to verify, that for every ∈ [0, a] function (t) = bd t d − t d−1 is strictly increasing on (a/bd , ∞). Then, since am+1 a, we
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357
∗ ∗ ∗ = (r ∗ have a (rm+1 ) am+1 (rm+1 ) = Cm Cm a m+1 ). Since rm+1 > r1 = a/bd , we get that ∗ rm+1 rm+1 . Since ck = (rk ; r n , cn ) − (rk − 0; r n , cn ) = ( (rk ) − (rk − 0))rkd−1 = (ak + k )rkd−1 , k = 1, . . . , n, and ak0 < a or k0 < a for some k0 , we have n
ck =
k=1
n
(ak + k )rkd−1 < 2a
k=1
n k=1
(rk∗ )d−1 =
n
ck∗ = bd r d .
k=1
Hence, spline misses sets An and Mn which contradicts the assumption of the lemma. Lemma 3 is proved. Combined with Lemma 1 it implies, that for any spline (·; r n , cn ) ∈ An with r n = r ∗n we have (·; r n , cn )∞ (·; r n , c n )∞ > ∗ ∞ . For any (·; r ∗n , cn ) ∈ An distinct from
∗ we ∗ ∗ ∗ ∗ ∗ 1 i n − 1, and hence, (ri+1 − 0; r n , cn ) > (ri+1 − 0) = ∗ ∞ have C∗i =∗Ci for some or (ri ; r n , cn ) > ∗ (ri∗ ) = ∗ ∞ . Then, ∗ is the only spline from An with the minimal norm. Taking into consideration our argument before Lemma 1, Eqs. (20) and (21), we obtain Rn (W 1 Bd [0, r]) = −1 d
inf (·; r n , cn )∞ = −1 ∞ d inf ∈An (r n ,cn )∈n ∗ 1 ∗ ∗ ∗ = −1 d ∞ = R(W Bd [0, r]; r n , cn ) = bd r1 /d
= r1∗ /d.
As it was mentioned above, K(·; r ∗n , c∗n ) is unique optimal formula. To find ri∗ ’s, ci∗ ’s and Rn (W 1 Bd [0, r]) for d = 2, it suffices to notice, that in this case (t) = t − 1 and xi = (i + 1)/ i, i = 1, . . . , n. The result of this paper for the class V 1 Bd [0, r] was announced in [2]. Acknowledgments Authors thank one of the reviewers for useful comments which helped to simplify first part of the proof. References [1] V.F. Babenko, On a certain problem of optimal integration, in: Studies on Contemporary Problems of Integration and Approximations of Functions and their Applications, Dnepropetrovsk University Press, Dnepropetrovsk, 1984, pp. 3–13 (in Russian). [2] V.F. Babenko, S.V. Borodachov, On optimization of cubature formulae for certain classes of functions defined on a ball, in: Book of Abstracts of the International Akhiezer Centenary Conference “Theory of Functions and Mathematical Physics”, Kharkiv, Ukraine, 2001, pp. 7–8 (in Russian). [3] V.F. Babenko, S.V. Borodachov, On optimization of cubature formulae for classes of monotone functions of several variables, Vestn. Dnepropetr. Univ. Ser. Mat. 7 (2002) (in Russian). [4] V.F. Babenko, S.V. Borodachov, On optimization of approximate integration over a d-dimensional ball, East J. Approx. 9 (2003) 95–109. [5] B.D. Bojanov, Charachterization and existence of optimal quadrature formulas for a certain class of differentiable functions, Dokl. Akad. Nauk. SSSR 232 (1977) 1233–1236. [6] B. Bojanov, D. Dimitrov, Gaussian extended cubature formula for polyharmonic functions, Math. Comp. 70 (2001) 671–683. [7] B. Bojanov, G. Petrova, Uniqueness of the Gaussian quadrature for a ball, J. Approx. Theory 104 (2000) 21–44. [8] N.P. Korneychuk, N.E. Lushpay, Best quadrature formulae for classes of differentiable functions and piecewise polynomial approximation, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 1416–1437; N.P. Korneychuk, N.E. Lushpay, Math. USSR Izv. 3 (1969) 1335–1355.
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[9] N.E. Lushpay, Best quadrature formulae for classes of differentiable periodic functions, Mat. Zametki 6 (1969) 475–481; N.E. Lushpay, Math. Notes 6 (1969) 740–744. [10] A.A. Lygun, Exact inequalities for spline-functions and best quadrature formulae for some classes of functions, Mat. Zametki 19 (1976) 913–926 (in Russian).
[11] V.P. Motornyi, On the best quadrature formula of the form nk=1 pk f (xk ) for some classes of differentiable periodic functions, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 583–614; V.P. Motornyi, Math. USSR Izv. 8 (1974) 591–620. [12] I.P. Mysovskih, Interpolatory Cubature Formulae, first ed., Nauka, Moscow, 1981 (in Russian). [13] S.M. Nikol’skiy, Quadrature Formulae, fourth ed., Nauka, Moscow, 1988 (in Russian). [14] J. Soler, On cubature with a minimal number of lines, J. Comput. Appl. Math. 19 (1987) 223–230. [15] J.F. Traub, H. Wozniakowski, A General Theory of Optimal Algorithms, first ed., Academic Press, NewYork, London, Toronto, Sydney, San Francisco, 1980.
On the construction of optimal cubature formulae which use integrals over hyperspheres V.F. Babenkoa, b , S.V. Borodachovc,∗ a Department of Mechanics and Mathematics, Dnepropetrovsk National University, vul. Naukova, 13, 49625, Ukraine b Donetsk Institute of Applied Mathematics and Mechanics of NAS of Ukraine, 74 Luxembourg St., Donetsk 340114,
Ukraine c Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
Received 18 June 2004; accepted 3 February 2007 Available online 14 March 2007
Abstract We consider formulae of approximate integration over a d-dimensional ball which use n surface integrals along (d − 1)-dimensional spheres centered at the origin. For a class of functions defined on the ball with gradients satisfying an integral restriction, optimal formulae of this type are obtained. © 2007 Elsevier Inc. All rights reserved. Keywords: Optimal cubature formula; Class of functions; Worst case error
1. Introduction 1.1. Setting of the problem The problem of optimization of approximate integration over a class of functions goes back to the works of A.N. Kolmogorov and S.M. Nikol’skiy published in late forties—early fifties. A large number of results in this direction have been obtained later by many researchers for classes of both univariate and multivariate functions. A review of most of these results can be found, for example, in [13,15]. The majority of works deals with quadrature and cubature formulae using “pointwise” information. During the last decades a number of authors consider formulae which use mean values taken along sufficiently short intervals (in univariate case) or parallelepipeds, balls, etc. (in multidimensional case). This is natural, since results of measurements are mean values of a function along some intervals. Modern applications, such as computer tomography, which ∗ Corresponding author.
E-mail addresses: [email protected] (V.F. Babenko), [email protected] (S.V. Borodachov). 0885-064X/$ - see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jco.2007.02.002
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347
provides information about a three-dimensional object by analysing two-dimensional images of its layers, lead to cubature formulae using as data integrals along sets of smaller dimensionality than the integration domain. In this paper we consider a problem of optimization of cubature formulae for integration over a ball in Rd in the following setting. Let d ∈ N, d > 1, |·| be the Euclidean norm in Rd , Bd [a, r] and Sd [a, r] be the ball and the sphere, respectively, centered at point a ∈ Rd with radius r > 0. Denote by bd the Lebesgue measure of the ball Bd [0, 1], where 0 = (0, . . . , 0) ∈ Rd , and by d —the (d − 1)-dimensional measure of the sphere Sd [0, 1]. It is known that d = dbd = dd/2 / (1 + d/2). Let also C(Bd [0, r]) be the space of continuous functions f : Bd [0, r] → R with the norm f =
max |f (x)|.
x∈Bd [0,r]
For a function u ∈ C Bd [0, r] put 1 H u() = u(x) dS, ∈ (0, r], d d−1 Sd [0,]
H u(0) = u(0),
(1)
where the surface integral taken along the sphere Sd [0, ] is of the first kind (i.e. H u() are mean values of the function u along the spheres Sd [0, ]). Let n ∈ N and Und be the set of all functionals K : C(Bd [0, r]) → R (algorithms of approximate integration of a function f along Bd [0, r]) of the form K(f ) = K(f ; r n , cn ) =
n
ck · Hf (rk ),
(2)
k=1
where cn = (c1 , . . . , cn ), r n = (r1 , . . . , rn ) ∈ Rn , and 0 r1 < · · · < rn r. Let V ⊂ C(Bd [0, r]) be a class of functions. Put R(f ; K) = R (f ; r n , cn ) = f (x) dx − K(f ; r n , cn ), Bd [0,r]
R(V ; K) = R (V ; r n , cn ) = sup |R (f ; r n , cn )|, f ∈V
Rn (V ) = inf R (V ; K). K∈Und
(3)
Problem. It is required to find value (3) and optimal cubature formulae for the class V , that is algorithms K ∗ ∈ Und , if they exist, delivering infinum on the right-hand side of (3). In other words, we look for sets of node hyperspheres and coefficients ck which provide the smallest worst-case error over the considered class. Optimal formulae of the form (2) can also be an intermediate step in construction of optimal multivariate cubatures using as data values of the function at points. Formulae using as information integrals along sets with smaller number of dimensions than the domain, have been considered in earlier mathematical literature. A certain formula of the form (2) exact for all d-variate polynomials of degree 4n − 1 was built in the book of Mysovskih [12]
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V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
(this example is due to Lusternik and Kantorovich). Cubatures which are weighted sums of arbitrary functionals were considered by M¨oller building upon the work of Mysovskih. Formulae using as information integrals along lines were considered by Soler [14]. The formula constructed in [12] has the highest algebraic degree of precision [7]. Its multiple node extensions were built by Bojanov and Dimitrov [6]. Optimization problems for cubature formulae using integrals along intersections of the integration domain and hyperplanes, were considered by Bojanov and Petrova [7] and the authors [3]. The problem set above was considered by authors [4] for classes of functions defined on a ball in Rd and having a given majorant for the modulus of continuity. 2. Main result Let p : (0, r] → (0, ∞) be a continuous function such, that r p(t)t d−1 dt < ∞
(4)
0
and Vd (p) be the closure in C(Bd [0, r]) of the set of continuously differentiable functions f : Bd [0, r] → R, such that |grad f (x)| · p(|x|) dx 1. (5) Bd [0,r]
We solve the previously mentioned problem for classes W 1 Bd [0, r] = Vd (p0 ), where p0 (t) = 1, and V 1 Bd [0, r] = Vd (p1 ), where p1 (t) = t 1−d , t ∈ (0, r]. For classes W1m , m ∈ N, of univariate periodic functions with mth derivative in the unit ball of L1 , problems of optimization of quadrature formulae were considered in [1,8–11]. The classes introduced above can be considered as a generalization of W11 . Denote gp,∞
g(t) . = ess sup p(t)t d−1 t∈[0,r]
With the help of the duality relations, optimization problems considered in [1,5,8–10] were reduced to the problem of minimization of the uniform norm of monosplines corresponding to quadrature formulae. The problem considered in this paper is in general reduced to the problem of the best approximation of a power function by piecewise constant ones in the norm · p,∞ , i.e. in usual uniform norm for the class V 1 Bd [0, r] and in the uniform norm with weight t 1−d —in the case of W 1 Bd [0, r]. Denote by Zn,d the set of functions z such that z(t) = 0, t ∈ [0, r1 ), z(t) = Ck , t ∈ [rk , rk+1 ), k = 1, . . . , n − 1, z(t) = bd r d , t ∈ [rn , r], where Ck ∈ R, k = 1, . . . , n − 1, and 0 r1 < · · · < rn r. Theorem 1. Let d, n ∈ N, d > 1, r > 0. For any positive continuous weight p satisfying (4) the following equality holds: Rn [Vd (p)] =
1 inf z(t) − bd t d p,∞ . d z∈Zn,d
(6)
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349
∗ ), If a function z∗ ∈ Zn,d with nodes 0 r1∗ < · · · < rn∗ r such that z∗ (t) = Ck∗ , t ∈ [rk∗ , rk+1 k = 1, . . . , n − 1, delivers infinum on the right-hand side of (6), then formula n K ∗ (f ) = K f ; r ∗n , c∗n = ck∗ Hf (rk∗ ),
(7)
k=1 ∗ , k = 2, . . . , n − 1, c∗ = b r d − C ∗ , is optimal for the class where c1∗ = C1∗ , ck∗ = Ck∗ − Ck−1 d n n−1 Vd (p) among formulae from Und .
The main results of this paper are contained in the following statement. Theorem 2. Let d, n ∈ N, d > 1, r > 0. Formula (7) with ck∗ = r d bd /n, rk∗ = r, k = 1, . . . , n, is optimal for the class V 1 Bd [0, r]. In addition, rd . Rn V 1 Bd [0, r] = 2nd
√ d
(2k − 1)/(2n) ·
(8)
For the class W 1 B2 [0, r] formula (7) with ck∗ = 2r 2 k(n(n + 1))−1 , rk∗ = rk(n(n + 1))−1/2 , k = 1, . . . , n, is optimal. In addition, r Rn (W 1 B2 [0, r]) = √ . 2 n(n + 1) Let d 3, (t) = (t d − 1)(t d−1 + 1)−1 and numbers x1 > 1,…, xn > 1 be such that (x1 ) = 1, −1 (xk ) = (xk−1 )xk−1 , k = 2, . . . , n. Then formula (7) with r1∗ = (xn )(1 + (xn ))−1/d r, rk∗ = ∗ xk−1 rk−1 , k = 2, . . . , n, ck∗ = 2bd (xk )(rk∗ )d , k = 1, . . . , n, is optimal in the class W 1 Bd [0, r]. In addition, Rn (W 1 Bd [0, r]) = r1∗ /d. For both classes the optimal formula is unique. 3. Proof of Theorem 1 3.1. Reduction to a univariate optimization problem Note, that H defined by (1) can be considered as an operator from C(Bd [0, r]) to C[0, r]—the space of continuous functions on [0, r]. Passing in (1) to the d-dimensional spherical coordinates x (, ) = cos 1 , sin 1 cos 2 , . . . , sin 1 ·. . .· sin d−2 cos d−1 , sin 1 ·. . .· sin d−1 , d−1−j , = 1 , . . . , d−1 ∈ d := [0, ]d−2 ×[0, 2], 0, and setting I () = d−1 j =1 (sin j ) d = d1 . . . dd−1 , we obtain 1 1 f (0) dS = f (x(0, ))I () d, Hf (0) = f (0) = d d d Sd [0,1] 1 1 f (x) dS = f (x(, ))d−1 I () d Hf () = d d−1 Sd [0,] d d−1 d 1 = f (x(, ))I () d, > 0. (9) d d Consider set U of univariate quadrature formulae q(g; t n , cn ) =
n k=1
ck g(tk ),
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where cn = (c1 , . . . , cn ), t n = (t1 , . . . , tn ) ∈ Rn and 0t1 < · · · < tnr. For a class D ⊂ C[0, r] denote r g(t)t d−1 dt − q(g; t n , cn ), r(g; t n , cn ) = d 0
r(D; t n , cn ) = sup r(g; t n , cn ),
rn (D) =
g∈D
inf q(·;t n ,cn )∈U
r(D; t n , cn ).
d (p) the set of continuously differentiable functions on Bd [0, r] which satisfy (5). Denote by V
d (p) we have For every K(·; r n , cn ) ∈ Und and f ∈ V r n R (f ; r n , cn ) = f (x) dx− ck Hf (rk )= f (x (, )) d−1 I () d d Bd [0,r]
−
n
0
k=1
r
ck Hf (rk )=d
k=1
0
d
d−1 Hf () d−
n
ck Hf (rk )=r(Hf ; r n , cn ).
k=1
d (p) is dense in Vd (p), we get Then, since V
d (p); r n , cn ) = r(H (V
d (p)); r n , cn ) R(Vd (p); r n , cn ) = R(V
(10)
d (p))]. Let W 1 (p) be the set of continuously differentiable and hence, Rn [Vd (p)] = rn [H (V functions g : [0, r] → R such that g (0) = 0 and r g (t) p(t)t d−1 dt 1. (11) d 0
d (p)) = W 1 (p). For every f ∈ V
d (p) and 0 ∈ [0, r] we get Show, that H (V * 1 (Hf ) (0 ) = f (x(, ))| = I () d 0 d d * 1 =
grad f | x=x (0 ,) , x (1, ) I () d, d d where a, b is the scalar product of vectors a and b. In particular, 1 1 (Hf ) (0) =
grad f (0), x (1, ) I () d =
grad f (0), x dS = 0. d d d Sd [0,1] Note, that (Hf ) () will be continuous and r (Hf ) () d−1 p() d d 0 r d−1 p() d | =
grad f , x ) I () d (1, x=x(,) d 0 r grad f | x=x(,) p()I () dd−1 d 0 d |grad f (x)| p(|x|) dx 1. = Bd [0,r]
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351
d (p)) ⊂ W 1 (p). Let g ∈ W 1 (p). We show that f (x) = Thus, Hf ∈ W 1 (p) and hence, H (V
g(|x|) is in Vd (p) and Hf = g. Indeed, by (9) for ∈ [0, r] we get 1 1 1 Hf ()= f (x(, ))I () d= g()I () d= g() dS=g(). d d d d d Sd [0,1] x It is not difficult to see that grad f (x) = g (|x|) |x| , x = 0, grad f (0) = 0 and f is continuously differentiable on Bd [0, r]. In addition, since g satisfies (11), we have g (|x|) · p(|x|) dx |grad f (x)| · p(|x|) dx = Bd [0,r] Brd [0,r] g () p()I () d · d−1 d = d 0 r g () p()d−1 d · I () d = d 0 r g () p()d−1 d · 1 dS = Sd [0,1] 0 r g () p()d−1 d1. = d 0
d (p), Hf = g and H (V
d (p)) = W 1 (p). Then, from (10) for every formula Hence, f ∈ V d K(·; r n , cn ) ∈ Un , R(Vd (p); r n , cn ) = r(W 1 (p); r n , cn )
(12)
and also Rn (Vd (p)) = rn (W 1 (p)). In order to find an optimal formula for Vd (p) we need to obtain an optimal formula for W 1 (p). It is not difficult to see that functions gc (t) = c, c ∈ R, belong to the class W 1 (p). For every formula q(·; r n , cn ) ∈ U we have r n n n r (gc ; r n , cn ) = d ct d−1 dt − ck c = cbd r d − ck c = c bd r d − ck . 0
k=1
k=1
k=1
d Denote by n the set of all pairs of vectors (r n , cn ), such that 0 r1 < · · · < rn r and
n bd r =
n d / n , then, since bd r − k=1 ck = k=1 ck . If formula q(·; r n , cn ) ∈ U is such that (r n , cn ) ∈ 0 and c is arbitrary, value r (gc ; r n , cn ) is unbounded, and r(W 1 (p); r n , cn ) will be infinite. Therefore, it is sufficient to look for optimal formulae only among those with (r n , cn ) ∈ n .
4. Reduction to a problem of the best approximation Let q(·; r n , cn ) ∈ U be such that (r n , cn ) ∈ n . Then for every function g ∈ W 1 (p) taking d into account that d = dbd and bd r = nk=1 ck , we have r r n d−1 d r(g; r n , cn ) = d g() d − ck g (rk ) = bd r g(r) − bd d g () d 0
−
n k=1
k=1
ck g (rk ) =
n k=1
0
r
ck (g(r) − g(rk )) − bd 0
d g () d
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V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
=
n
r
ck 0
k=1 n r
[rk ,r] ()g () d − bd
=
0
ck [rk ,r] () − bd
r
d g () d
0
g () d,
d
(13)
k=1
where [rk ,r] (t) is the characteristic function of the segment [rk , r], k = 1, . . . , n. Denote (t) = (t; r n , cn ) =
n
ck [rk ,r] (t) − bd t d .
k=1
The set of all functions of the form Using (13), we get r(W (p); r n , cn ) = 1
n
k=1 ck [rk ,r] (t)
|r(g; r n , cn )| =
sup g∈W 1 (p)
where (r n , cn ) ∈ n , coincides with Zn,d . sup
g∈W 1 (p) r
(·; r n , cn )p,∞ ·
r 0
(; r n , cn )g () d
g () p()d−1 d
sup
g∈W 1 (p) 0
(·; r n , cn )p,∞ . d
(14)
Equality in (14) can be shown for example in the following way. Let { m }∞ m=1 be a sequence such that m < (·; r n , cn )p,∞ , m ∈ N, and lim m = (·; r n , cn )p,∞ . Note, that since m→∞
(t; r n , cn )/(p(t)t d−1 ) is continuous on [0, r]\{0, r1 , . . . , rn }, for every m ∈ N there is an interval (am , ym ) ⊂ (0, r] where (·; r n , cn ) preserves its sign such that (t; r n , cn )/(p(t)t d−1 ) > ∈ C[0, r] is non-negative and supported m , t ∈ (am , ym ). Let gm be a function such that gm inside (am , ym ) and ym gm (t)p(t)t d−1 dt = 1. d am
Then, gm ∈ W 1 (p) and by (13)
r r(W (p); r n , cn ) |r(gm ; r n , cn )| = (; r n , cn )gm () d 0 ym ym = (; r n , cn )gm () d m gm ()p()d−1 d 1
am
m = , d
am
m ∈ N.
−1 Hence, r(W 1 (p); r n , cn ) −1 d lim m→∞ m = d (·; r n , cn )p,∞ . Taking into account (14) d and (12), for every K(·, r n , cn ) ∈ Un with (r n , cn ) ∈ n we will have 1 (·; r n , cn )p,∞ , (15) R(Vd (p); r n , cn ) = r(W 1 (p); r n , cn ) = d
Rn (Vd (p)) = =
inf
(r n ,cn )∈n
1 d
inf
r(W 1 (p); r n , cn )
(r n ,cn )∈n
(·; r n , cn )p,∞ =
1 inf z(t) − bd t d p,∞ . d z∈Zn,d
V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
353
∗ ), k = 1, . . . , n − 1, Note, that if a function z∗ ∈ Zn,d such that z(t) = Ck∗ , t ∈ [rk∗ , rk+1 ∗ ∗ 0 r1 < · · · < rn r, is the element of the best approximation to y(t) = bd t d , then spline n
∗ ) ∗ (t) − b t d = z∗ (t) − b t d where C ∗ = 0, C ∗ = b r d , has (·; r ∗n , c∗n ) := (Ck∗ − Ck−1 d d d [r ,r] n 0 k
k=1
the least · p,∞ -norm and the corresponding formula K(·; r ∗n , c∗n ) ∈ Und will be optimal for the class Vd (p). Theorem 1 is proved. 5. Proof of Theorem 2 5.1. Best formula for V 1 Bd [0, r] Assume now that p(t) = t 1−d , t ∈ (0, r]. Denote by L∞ [0, r] the space of functions g : [0, r] → R such that g∞ := ess sup |g(t)| < ∞. t∈[0,r]
From (15) for every formula in Und , such that (r n , cn) ∈ n , we have 1 (·; r n , cn )∞ . R V 1 Bd [0, r]; r n , cn = d
(16)
Then 1 Rn V 1 Bd [0, r] = d
inf
(·; r n , cn )∞ =
(r n ,cn )∈n
1 inf z(t) − bd t d ∞ . d z∈Zn,d
(17)
For all vectors r n ,cn , such that (r n , cn ) ∈ n , we get (·; r n , cn )∞ = max {max {|(rk − 0)| , |(rk )|}} k=1,n
k k−1 |(rk )−(rk −0)| 1 d d max ci −bd rk − ci −bd rk = max 2 2 k=1,n k=1,n i=1 i=1 n n 1 1 1 1 |ck | = max |ck | ck = (18) bd r d . 2 k=1,n 2n 2n 2n k=1
Denote rk∗ =
√ d
k=1
(2k − 1)/(2n) · r, ck∗ = bd r d /n, k = 1, . . . , n. Then
k−1 ∗ ∗ d r d bd 2k − 1 d r d bd ∗ ∗ ∗ r − 0; r , c = r = (k − 1) − b r c − b d k d n n k i = 2n , n 2n i=1
k rd b ∗ ∗ ∗ 2k − 1 d r d bd d d ∗ ∗ r ; r , c = k − bd r = , ci − bd rk = n n k n 2n 2n i=1
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V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
k = 1, . . . , n and taking into account estimate (18) we have r d bd (·; r ∗n , c∗n )∞ = max {max {(rk∗ − 0) , (rk∗ )}} = 2n k=1,n = inf (·; r n , cn )∞ . (r n ,cn )∈n
Taking into account (17) and (16) we finally obtain r d bd 1 rd Rn V 1 Bd [0, r] = . (·; r ∗n , c∗n )∞ = R V 1 Bd [0, r]; r ∗n , c∗n = = d 2nd 2nd This means that formula (7) with such rk∗ ’s and ck∗ ’s is optimal in V 1 Bd [0, r] and (8) holds. If we have equality sign everywhere in (18), one can show that ck = ck∗ , k = 1, . . . , n, and then, using equality of absolute values of the one-sided limits of (·; r n , cn ) at nodes, that rk = rk∗ , k = 1, . . . , n. Hence, optimal formula will be unique. 6. Best formula for W 1 Bd [0, r] Let p(t) = 1. Denote
(t) = (t; r n , cn ) = (t; r n , cn ) · t 1−d ,
t ∈ (0, r].
(19)
For every formula K(·; r n , cn ) ∈ Und with (r n , cn ) ∈ n we have R(W 1 Bd [0, r]; r n , cn ) = Rn (W 1 Bd [0, r]) =
1 d
1 (·; r n , cn )∞ , d inf
(r n ,cn )∈n
(·; r n , cn )∞ .
(20) (21)
Denote by An the set of splines (19) with (r n , cn ) ∈ n and r1 > 0. For any ∈ An we have
(t) = −bd t, 0 < t < r1 , and ∞ < ∞. If r1 = 0 and c1 = 0, we get lim (t; r n , cn ) = lim (c1 t 1−d − bd t) = ∞
t→0+
t→0+
and such splines do not deliver the infinum in (21). Below, we will find a unique spline ∗ ∈ An with minimal norm. It will have non-zero coefficients ci∗ . Any spline (19) with r1 = 0 and c1 = 0 coincides with a spline from An having the same ri and ci , i = 2, . . . , n, positive first node and zero first coefficient. Hence, it will not deliver infinum in (21) and ∗ will be the only spline (19) with (r n , cn ) ∈ n for which this infinum is attained. Since every formula in Und with / n has infinite error, formula corresponding to ∗ will be the only optimal formula for (r n , cn ) ∈ 1 W Bd [0, r]. Note, that every ∈ An will be right continuous, if we put (0) = 0. Lemma 1. Given vector r n : 0 < r1 < · · · < rn r, spline (·; r n , c n ) ∈ An with Ci :=
i k=1
ck = bd
ri + ri+1 , 1−d 1−d ri + ri+1
i = 1, . . . , n − 1,
has the least · ∞ -norm among all splines from An with given r n .
V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
355
Proof. Denote C0 = 0, Cn = bd r d , r0 = 0, rn+1 = r. Then (t; r n , c n ) = (Ci − bd t d )t 1−d , t ∈ [ri , ri+1 ), i = 0, . . . , n. It is not difficult to see that (t; r n , c n ) is monotone on (ri , ri+1 ), i = 0, . . . , n, and (0; r n , c n ) = (r; r n , c n ) = 0. Let (t) =
td − 1 , t d−1 + 1
t > 0.
Since 1 + (t) = (1 + t)(1 + t 1−d )−1 , it is not difficult to see that ri+1 ri bd (ri + ri+1 ) d d + 1 = bd ri+1 +1 , = bd ri Ci = 1−d 1−d ri ri+1 ri + ri+1
(22)
i = 1, . . . , n − 1. Using (22) and the fact, that (1/t) = −(t)/t, t > 0, we get (r1 − 0) = −bd r1 < 0,
(ri − 0) =
Ci−1 − bd rid
rid−1
= b d ri
(ri ) = (Ci − bd rid )ri1−d = bd ri
ri−1 ri = −bd ri−1 < 0, ri ri−1
ri+1 ri
i = 2, . . . , n,
> 0,
i = 1, . . . , n − 1,
(23)
(rn ) = bd (r d −rnd )rn1−d 0. Hence, (ri ; r n , c n ) = − (ri+1 −0; r n , c n ) > 0, i = 1, . . . , n−1. Then, since (t; r n , c n ) is monotone between its nodes, we get that (·; r n , c n )∞ = max max{ (ri ; r n , c n ) , (ri − 0; r n , c n )}. i=1,n
Any other spline (·; r n , cn ) with given r n , coincides with (·; r n , c n ) on (0, r1 ) and (rn , r) and has the form (t; r n , cn ) = (Ci − bd t d )t 1−d , t ∈ (ri , ri+1 ), i = 1, . . . , n − 1. Then, for every i = 1, . . . , n−1, either (ri ; r n , cn ) (ri ; r n , c n ) > 0 or (ri+1 −0; r n , cn ) (ri+1 −0; r n , c n ) < 0 and (·; r n , cn )∞ max max{| (ri ; r n , cn )| , | (ri − 0; r n , cn )|} i=1,n max max{ (ri ; r n , c n ) , (ri − 0; r n , c n )} = (·; r n , c n )∞ . i=1,n
Lemma 1 is proved. Denote by Mn the set of all splines (·; r n , cn ) ∈ An with Ci := ik=1 ck = 1−d −1 ) , i = 1, . . . , n − 1. Let numbers x1 > 1, . . . , xn > 1 be defined by bd (ri + ri+1 )(ri1−d + ri+1 recurrence relations (x1 ) = 1,
−1 (xi ) = xi−1 (xi−1 ), i = 2, . . . , n.
(24)
Since for every y > 0 equation (t) = y has exactly one solution in (1, ∞), the sequence of xi ’s is unique. Lemma 2. Let d, n ∈ N, d > 1. There is a unique spline ∗ = (·; r ∗n , c∗n ) ∈ An such that − ∗ (r1∗ − 0) = ∗ (r1∗ ) = − ∗ (r2∗ − 0) = ∗ (r2∗ ) = · · · = − ∗ (rn∗ − 0) = ∗ (rn∗ ). In this case ∗ , i = 2, . . . , n, c∗ = 2b (x )(r ∗ )d , i = 1, . . . , n. r1∗ = (xn )(1 + (xn ))−1/d r, ri∗ = xi−1 ri−1 d i i i
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V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
Proof. Denote Ci∗ = ik=1 ck∗ , i = 1, . . . , n − 1, C0∗ = 0, Cn∗ = bd r d . From equations ∗ (ri∗ ) = ∗ − 0), i = 1, . . . , n − 1, we get (C ∗ − b (r ∗ )d )(r ∗ )1−d = (b (r ∗ )d − C ∗ )(r ∗ )1−d , − ∗ (ri+1 d i d i+1 i i i i+1 Ci∗ =
∗ ) bd (ri∗ + ri+1
∗ )1−d (ri∗ )1−d + (ri+1
i = 1, . . . , n − 1.
,
− ∗ (r1∗ − 0) = ∗ (r1∗ ) is equivalent to bd r1∗ = Then ∗ ∈ Mn and we apply ∗(23). Relation ∗ ∗ ∗ ∗ ∗ bd r1 r2 /r1 , and hence, r2 /r1 = 1, r2 /r1∗ > 1. Then, since x1 is unique, we can only have r2∗ /r1∗ = x1 . For i = 2, . . . , n − 1 relation − ∗ (ri∗ − 0) = ∗ (ri∗ ) is equivalent to ∗ bd ri−1
ri∗ ∗ ri−1
=
bd ri∗
∗ ri+1
ri∗
,
∗ ri+1
ri∗
=
ri∗ ∗ ri−1
−1
r∗ ∗i ri−1
.
∗ /r ∗ > 1, i = 1, . . . , n − 1, satisfy recurrence formulae (24) and the only option is Hence, ri+1 i ∗ ∗ ri+1 /ri = xi , i = 1, . . . , n − 1. Finally, from equality − ∗ (rn∗ − 0) = ∗ (rn∗ ) we get
−bd rn∗ 1+
∗ rn−1
rn∗
(xn−1 ) = xn−1
= (bd r d − bd (rn∗ )d )(rn∗ )1−d ,
r rn∗
d ,
−
1
=
xn−1
r rn∗
d − 1,
rn∗ = (1 + (xn ))−1/d r.
∗ /x , i = n − 1, . . . , 1, are also uniquely deSince rn∗ < r is uniquely determined, ri∗ = ri+1 i ∗ ∗ termined. From equality rn (xn ) = rn (xn−1 )/xn−1 = rn∗ (xn−2 )/(xn−1 · xn−2 ) = · · · = rn∗ /(xn−1 · . . . · x1 ) = r1∗ , we have r1∗ = (xn )(1 + (xn ))−1/d r. For the coefficients ck∗ using (22), we obtain c1∗ = C1∗ = bd (r1∗ )d ((x1 ) + 1) = 2bd (r1∗ )d , −1 ∗ ci∗ = Ci∗ − Ci−1 = bd (ri∗ )d ((xi ) + 1) − bd (ri∗ )d xi−1 +1 −1 ) = 2bd (xi )(ri∗ )d , = bd (ri∗ )d ((xi ) + (xi−1 )xi−1
Using formula for
rn∗
i = 2, . . . , n − 1.
and (22), we also get
−1 ∗ = bd (rn∗ )d (1 + (xn )) − bd (rn∗ )d ((xn−1 ) + 1) cn∗ = bd r d − Cn−1 −1 = bd (rn∗ )d ((xn ) + (xn−1 )xn−1 ) = 2bd (xn )(rn∗ )d .
Lemma 3. For any ∈ Mn , = ∗ , we have ∞ > ∗ ∞ . Proof. Assume, that there is a spline = (·; r n , cn ) ∈ Mn , such that = ∗ and ∞ a := ∗ ∞ = ∗ (ri∗ ) = − ∗ (ri∗ − 0), i = 1, . . . , n. Let ai = − (ri − 0), i = (ri ), i = 1, . . . , n. Then, by relations (23) ai ’s and i ’s are non-negative and ai a, i a, i = 1, . . . , n. In view of Lemma 2 at least one of these inequalities must be strict. Show by induction, that rk rk∗ , k = 1, . . . , n. Since a1 = bd r1 a = bd r1∗ , we get r1 r1∗ = ∗ for some m < n. Then, = (C − b r d )r 1−d a = (C ∗ − a/bd . Assume, that rm rm m d m m m m ∗ d ∗ 1−d 1−d ∗ (r ∗ )1−d b (r −r ∗ ) 0. Hence, C C ∗ . Using equalities and Cm rm −Cm bd (rm ) )(rm ) d m m m m m 1−d d ∗ ∗ )(r ∗ 1−d , we get C = b r d − Cm )rm+1 and a = (bd (rm+1 )d − C m am+1 = (bd rm+1 m d m+1 − m+1 ) d−1 ∗ ∗ ∗ d d−1 am+1 rm+1 Cm = bd (rm+1 ) − a(rm+1 ) . It is not difficult to verify, that for every ∈ [0, a] function (t) = bd t d − t d−1 is strictly increasing on (a/bd , ∞). Then, since am+1 a, we
V.F. Babenko, S.V. Borodachov / Journal of Complexity 23 (2007) 346 – 358
357
∗ ∗ ∗ = (r ∗ have a (rm+1 ) am+1 (rm+1 ) = Cm Cm a m+1 ). Since rm+1 > r1 = a/bd , we get that ∗ rm+1 rm+1 . Since ck = (rk ; r n , cn ) − (rk − 0; r n , cn ) = ( (rk ) − (rk − 0))rkd−1 = (ak + k )rkd−1 , k = 1, . . . , n, and ak0 < a or k0 < a for some k0 , we have n
ck =
k=1
n
(ak + k )rkd−1 < 2a
k=1
n k=1
(rk∗ )d−1 =
n
ck∗ = bd r d .
k=1
Hence, spline misses sets An and Mn which contradicts the assumption of the lemma. Lemma 3 is proved. Combined with Lemma 1 it implies, that for any spline (·; r n , cn ) ∈ An with r n = r ∗n we have (·; r n , cn )∞ (·; r n , c n )∞ > ∗ ∞ . For any (·; r ∗n , cn ) ∈ An distinct from
∗ we ∗ ∗ ∗ ∗ ∗ 1 i n − 1, and hence, (ri+1 − 0; r n , cn ) > (ri+1 − 0) = ∗ ∞ have C∗i =∗Ci for some or (ri ; r n , cn ) > ∗ (ri∗ ) = ∗ ∞ . Then, ∗ is the only spline from An with the minimal norm. Taking into consideration our argument before Lemma 1, Eqs. (20) and (21), we obtain Rn (W 1 Bd [0, r]) = −1 d
inf (·; r n , cn )∞ = −1 ∞ d inf ∈An (r n ,cn )∈n ∗ 1 ∗ ∗ ∗ = −1 d ∞ = R(W Bd [0, r]; r n , cn ) = bd r1 /d
= r1∗ /d.
As it was mentioned above, K(·; r ∗n , c∗n ) is unique optimal formula. To find ri∗ ’s, ci∗ ’s and Rn (W 1 Bd [0, r]) for d = 2, it suffices to notice, that in this case (t) = t − 1 and xi = (i + 1)/ i, i = 1, . . . , n. The result of this paper for the class V 1 Bd [0, r] was announced in [2]. Acknowledgments Authors thank one of the reviewers for useful comments which helped to simplify first part of the proof. References [1] V.F. Babenko, On a certain problem of optimal integration, in: Studies on Contemporary Problems of Integration and Approximations of Functions and their Applications, Dnepropetrovsk University Press, Dnepropetrovsk, 1984, pp. 3–13 (in Russian). [2] V.F. Babenko, S.V. Borodachov, On optimization of cubature formulae for certain classes of functions defined on a ball, in: Book of Abstracts of the International Akhiezer Centenary Conference “Theory of Functions and Mathematical Physics”, Kharkiv, Ukraine, 2001, pp. 7–8 (in Russian). [3] V.F. Babenko, S.V. Borodachov, On optimization of cubature formulae for classes of monotone functions of several variables, Vestn. Dnepropetr. Univ. Ser. Mat. 7 (2002) (in Russian). [4] V.F. Babenko, S.V. Borodachov, On optimization of approximate integration over a d-dimensional ball, East J. Approx. 9 (2003) 95–109. [5] B.D. Bojanov, Charachterization and existence of optimal quadrature formulas for a certain class of differentiable functions, Dokl. Akad. Nauk. SSSR 232 (1977) 1233–1236. [6] B. Bojanov, D. Dimitrov, Gaussian extended cubature formula for polyharmonic functions, Math. Comp. 70 (2001) 671–683. [7] B. Bojanov, G. Petrova, Uniqueness of the Gaussian quadrature for a ball, J. Approx. Theory 104 (2000) 21–44. [8] N.P. Korneychuk, N.E. Lushpay, Best quadrature formulae for classes of differentiable functions and piecewise polynomial approximation, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 1416–1437; N.P. Korneychuk, N.E. Lushpay, Math. USSR Izv. 3 (1969) 1335–1355.
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[9] N.E. Lushpay, Best quadrature formulae for classes of differentiable periodic functions, Mat. Zametki 6 (1969) 475–481; N.E. Lushpay, Math. Notes 6 (1969) 740–744. [10] A.A. Lygun, Exact inequalities for spline-functions and best quadrature formulae for some classes of functions, Mat. Zametki 19 (1976) 913–926 (in Russian).
[11] V.P. Motornyi, On the best quadrature formula of the form nk=1 pk f (xk ) for some classes of differentiable periodic functions, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 583–614; V.P. Motornyi, Math. USSR Izv. 8 (1974) 591–620. [12] I.P. Mysovskih, Interpolatory Cubature Formulae, first ed., Nauka, Moscow, 1981 (in Russian). [13] S.M. Nikol’skiy, Quadrature Formulae, fourth ed., Nauka, Moscow, 1988 (in Russian). [14] J. Soler, On cubature with a minimal number of lines, J. Comput. Appl. Math. 19 (1987) 223–230. [15] J.F. Traub, H. Wozniakowski, A General Theory of Optimal Algorithms, first ed., Academic Press, NewYork, London, Toronto, Sydney, San Francisco, 1980.