A PIECEWISE CONSTANT ALGORITHM FOR WEIGHTED L 1 APPROXIMATION OVER BOUNDED OR UNBOUNDED REGIONS IN R s

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1 A PIECEWISE CONSTANT ALGORITHM FOR WEIGHTED L 1 APPROXIMATION OVER BONDED OR NBONDED REGIONS IN R s FRED J. HICKERNELL, IAN H. SLOAN, AND GRZEGORZ W. WASILKOWSKI Abstract. sing Smolyak s construction [5], we derive a new algorithm for approximating multivariate functions over bounded or unbounded regions in R s with the error measured in a weighted L 1 -norm. We provide upper bounds for the algorithm s cost and error for a class of functions whose mixed first order partial derivatives are bounded in L 1 -norm. In particular, we prove that the error and the cost (measured in terms of the number of function evaluations) satisfy the following relation: error s (s 1)π ( e ln(cost) (s 1) 2 ln(2) ) 2(s 1) 1 cost whenever the cost is sufficiently large relative to the number s of variables. More specifically, the inequality holds when q 2(s 1), where q is a special parameter defining the refinement level in the Smolyak algorithm, and hence the number of function evaluations used by the algorithm. We also discuss extensions of the results to the spaces with the derivatives bounded in L p -norms. 1. Introduction In this paper, we derive a relatively simple to use, piecewise constant algorithm for approximating functions in a weighted L 1 sense. Function approximation has been studied quite extensively, see e.g., [3, 4, 6, 7] and the papers cited there. However, such problems were considered mainly for functions with a bounded domain D, say D = [0, 1] s. The worst case complexity of weighted approximation over unbounded domains D has recently been studied in, e.g., [2, 9], assuming that the corresponding function classes F are isotropic. The analysis of the approximation problem for tensor product spaces F is quite straightforward if F is a Hilbert space, since then desirable properties of Smolyak s construction could be used, see e.g., [8]. In this paper we study a weighted approximation problem with an emphasis on unbounded domains D and tensor product function classes F in a non-hilbert-space setting. More specifically, we study a ρ-weighted L 1 approximation problem with the error between f and the approximation A(f) measured in the following semi-norm (1) (f A(f))ρ L1 (D) = ρ(x) f(x) A(f)(x) dx. D We have chosen L 1 -norm here (as opposed to L r -norm for r > 1) since then the approximation problem is related to weighted integration. This relation is briefly discussed in Section 6; here we only mention that any algorithm A for L 1 approximation yields an Date: May

2 2 HICKERNELL, SLOAN, AND WASILKOWSKI integration algorithm with the error at least as small as the error of A. Another reason for choosing r = 1 is the simplicity of analysis; for r 1, 2 the corresponding weighted L r approximation problem is more difficult to analyze. Throughout this article, it is assumed that ρ(x) = s k=1 ρ k(x k ) is a given integrable weight function. The domain D of functions can be an arbitrary box, including D = R s. The class F is a space of functions whose dominating mixed first order derivatives are bounded in L 1 -norm. The case of derivatives bounded in L p -norms seems to be harder than the case with L 1 -norm (especially for unbounded D), and is briefly addressed in Section 5. We stress that such classes F are commonly assumed in the context of integration problems over D = [0, 1] s. Moreover, for bounded D, these classes are related to the classes MWp r (with r = 1) of periodic functions considered, e.g., in [6, Chapt.4]. The main result of the paper is the derivation and analysis of a family {A q,s } q=s of algorithms that provide approximations that are special piecewise constant functions. They are obtained by applying Smolyak s construction (see [5]) to scalar piecewise constant interpolation methods. As we shall see, given a number s of variables and a parameter q s, the algorithm has the worst case error bounded by { ( ) s 2 2q+2s 3 2q 2s+3 if q < 2(s 1), error(a q,s ) s 2 q ( q s 1 ) q s+1 if q 2(s 1). Here q is the refinement parameter in Smolyak s algorithm. nder an additional symmetry assumption (3) that is stated later, we get ( ) q s error(a q,s ) s 2 q+s 1 a, a where { q s+1 if q < 4(s 1), a = 3 s 1 otherwise. Let n = card(a q,s ) denote the number of function evaluations used by A q,s. sing n 2 q s+1( q 1 s 1) (see [8]), we conclude that for every s 2 and q 2(s 1), error(a q,s ) s (s 1)π ( e ln(n) (s 1) 2 ln(2) ) 2(s 1) 1 n. This is the best known rate of convergence for this problem. It is as good as the rate from [6, Thm ] for D = [0, 1] s, ρ 1, and F = MW 1 1. Moreover, it shows in an explicit way the dependence of the errors on the dimension s. Since A q,s (f) is a piecewise constant function, the algorithm is easy to implement. Its only drawback is in exponential gaps between consecutive numbers card(a q,s ) of function evaluations. However, we believe that it is of a practical interest, especially for small to moderate values of s. Implementation of the algorithm and numerical tests will be reported later. We summarize the content of this paper. Section 2 provides some basic definitions and assumptions, as well as an error bound for an arbitrary algorithm A. Since Smolyak s construction depends on the specific choice of scalar algorithms, Section 3 considers

3 PIECEWISE CONSTANT ALGORITHM 3 very special scalar algorithms based on a piecewise constant interpolation. The corresponding algorithm and its properties are presented in Section 4. An extension to functions with derivatives bounded in L p -norm is provided in Section 5. Section 6 briefly explains why the error bounds obtained for the ρ-weighted L 1 approximation also hold for the corresponding ρ-weighted integration problem. 2. Basic Definitions In this section, we briefly present some definitions and basic facts concerning the worst case setting. A more detailed discussion can be found, e.g., in [3, 7]. We consider a weighted L 1 approximation of functions of s variables whose domain D is a box, D = (a 1,b 1 ) (a s,b s ). The values a i and b i might be infinite; this is why we write (a i,b i ) instead of [a i,b i ]. Let F be a Banach space of functions f : D R that will be specified later. The approximation problem depends on a weight function ρ which is assumed to have the following properties: s (2) ρ 0, ρ(x) = ρ k (x k ). k=1 For simplicity of presentation, we also assume that bk a k ρ k (t)dt = 1, k = 1,...,s. However, it is enough to assume that the integrals of ρ k are finite; in such a case, all error bounds derived in this paper should be multiplied by the constant c = ρ(x)dx. D Functions from F are approximated by an algorithm A, n f A(f) = f(x i )g i for some points x i and functions g i, with the error between f and A(f) measured in the ρ-weighted L 1 -norm, see (1). The worst case error (with respect to F) of A is defined by i=1 error(a) := sup (f A(f))ρ L1 (D), f 1 where f denotes the norm of f in the space F. The importance of this definition is that (f A(f))ρ L1 f error(a), f F. Each algorithm uses a finite number n of function evaluations. That number is called the cardinality and is denoted by card(a). With the exception of Sections 5 and 6, the following space F = F 1,s is considered. Let H k be the space of absolutely continuous functions on (a k,b k ) whose first derivative

4 4 HICKERNELL, SLOAN, AND WASILKOWSKI is in L 1 ((a k,b k )). Let H s = s k=1 H k be the space consisting of linear combinations of functions f of the tensor product form s f : D R and f(x) = h k (x k ) with h k H k. The space F 1,s is the completion of H s with respect to the following norm k=1 f 1,s := f(c) + f L1 ( ). Here c = [c 1,...,c s ] D is a fixed point, called an anchor. The summation is with respect to subsets of {1,...,s}, and f (x ) := k x f(x,c), k where (x,c) denotes the s-dimensional vector whose kth component is x k if k, and c k otherwise. By x we mean the -dimensional vector obtained from x by removing all components x k with k /. This means that f is a function defined on := k (a k,b k ) and x. To simplify the notation, we will also write f and f L p to denote f(c) and f(c), respectively; and we often drop by writing L1 instead of L1 ( ). This allows the more concise formula: f 1,s = f L1. We illustrate this for s = 2 f 1,2 = f(c 1,c 2 ) + + b1 a 1 b1 b2 a 1 b2 f(x 1,c 2 ) x 1 dx 1 + f(c 1,x 2 ) a 2 x 2 dx 2 2 f(x 1,x 2 ) x 1 x 2 dx 2 dx 1. a 2 Although in general the anchor c and the weight ρ are not related, we shall obtain stronger results under the following symmetry condition: (3) ck ρ k (x)dx = 1, k = 1,...,s. a k 2 Of course, this condition is satisfied if (a k,b k ) and ρ k are symmetric with respect to c k (k = 1,...,s), e.g., if a k = b k, c k = 0 and ρ k (x) = ρ k ( x). In general, for a given ρ k one may always choose c k to satisfy (3). The following fact will play an important role. Let M(x,t) := M k (x,t) := s 1 if c k < t < x, 1 if x < t < c k, 0 otherwise, k=1 M k (x k,t k ) and M (x,t ) := k M k (x k,t k )

5 PIECEWISE CONSTANT ALGORITHM 5 with the convention that M 1. Then for every f F 1,s and every x D, (4) f(x) = f (t )M (x,t )dt. The representation (4) has been used, at least implicitly, in a number of papers, and its short proof can be found in [1]. From it we have f ρ L1 f (t ) ρ(x)m (x,t ) dxdt f 1,s. D This means that the approximation problem is well defined since the corresponding embedding operator is bounded. Actually, the following theorem, when applied to the zero algorithm A 0, implies that the norm of the embedding is equal to one, i.e., f ρ L1 sup = 1. f F f 1,s Theorem 1. The error of any A is bounded by (5) error(a) sup max h (x,t )dx, t where h (x,t ) = ρ (x ) M (x,t ) A(M (,t ))(x ). If A is based on piecewise constant interpolation then (5) holds with equality. Proof. The proof is deferred to Section 5, where a more general result is proven. 3. Scalar Functions Since Smolyak s construction depends on specific algorithms for the scalar cases, we consider now approximating univariate functions whose domain is (a k,b k ) and weight function is ρ k. To simplify the notation in this section, we write a,b and ω instead of a k,b k and ρ k. For i = 1, 2,..., consider the set of points x k i,j (j = 0,...,2 i ) such that and (6) a = x k i,0 < x k i,1 < < x k i,2 i = b x k i,j x k i,j 1 ω(t)dt = 2 i. For simplicity, we will write in this section x i,j instead x k i,j. Of course, x i 1,j = x i,2j. We take the following family of algorithms A i (i = 1, 2...) based on piecewise constant interpolation: A i (f)(x) = f(x i,j ) if c k x i,j x < x i,j+1 or x i,j 1 < x x i,j c k. Moreover, when x,c k (x i,j,x i,j+1 ) then A i (f)(x) equals f(x i,j ) or f(x i,j+1 ) depending on whether x c k or not. Note that under the symmetry condition (3), x 1,1 = x i,2 i 1 = c k and A i (f)(x) = f(c k ) for x (x i,2 i 1 1,x i,2 i 1 +1). For given k, define δ k,1 (x,t) := A 1 (M k (,t))(x)

6 6 HICKERNELL, SLOAN, AND WASILKOWSKI and (7) δ k,i (x,t) := A i (M k (,t))(x) A i 1 (M k (,t))(x), i 2. The following result is needed in Section 4. Proposition 1. For every t, we have Moreover, if (3) holds then ω δ k,i (,t) L1 2 i, i 1. A 1 (M k (,t)) = 0. The first part of the proposition is an immediate consequence of the following lemma that will be used in Section 5. For simplicity of presentation, we state the lemma only for arguments x and t greater than c k. The analogous result is true for arguments smaller than c k with only difference being that δ i,k (x,t) { 1, 0}. Moreover, δ k,i (x,t) = 0 when c k is between x and t. Lemma 1. The following statements hold for any x,t > c k. (i) δ k,i (x,t) {0, 1}. (ii) If δ k,i (x,t) = 1 then there exists j 2 i 1 1 such that t (x i,2j,x i,2j+1 ] and x (x i,2j+1,x i,2j+2 ]. (iii) There is at most one i such that δ k,i (x,t) = 1. Proof. (i) follows immediately from the facts that M k (x,t) {0, 1}, is nondecreasing, and A i uses the points used by A i 1. (ii) Let x (x i,l,x i,l+1 ] for some l. Then A i (M k (,t))(x) = 1 only if t < x i,l. However, for A i 1 (M k (,t))(x) = 0, x i,l has to be different than any point x i 1,j used by A i 1. This means that l has to be odd. (iii) follows from (ii). Indeed, if δ k,i (x,t) = 1 then t and x are in two neighboring subintervals with the evaluation point x i,l between them and l = 2j + 1. Hence δ k,i+n (x,t) cannot be equal to 1 since x n+i,m = x i,l is between t and x for even m = 2 n l. This completes the proof. 4. The Algorithm Let {A k,i } be the families of algorithms from the previous section, each for ω = ρ k and (a,b) = (a k,b k ), respectively. Recall that A k,i uses function values at points x k i,1,...,x k i,2 i 1. Define and k,1 := A k,1, k,i := A k,i A k,i 1 for i 2, (8) A q,s := i q s k=1 k,ik for q s. Here and elsewhere, i = [i 1,...,i s ] N s + is a multi-index with i k 1 and i = s k=1 i k.

7 PIECEWISE CONSTANT ALGORITHM 7 Theorem 2. Let s 2 and q s. In general, { ( s 2 q q ) if q < 2(s 1), error(a q,s ) q+1 2 s 2 ( ) q q if q 2(s 1). s 1 If, additionally, (3) holds then where error(a q,s ) s 2 q+s 1 a ( q s a a = { q s+1 3 if q < 4(s 1), s 1 if q 4(s 1). To prove this theorem, we need the following lemma. We think it is known; however, we have not seen it in the literature. For q s 1, define (9) B(q,s) := 2 i. j=0 i q+1 Lemma 2. For every q s 1, s 1 ( ) q (10) B(q,s) = 2 q B(q,s), j where (11) B(q,s) := s 2 q Proof of Lemma 2. Indeed, and ( ) j 1 2 j s 1 B(q,s) = ), ( q q+1 2 ) if 2s q + 3, ( q s 1 j=q+1 ) otherwise. ( ) j 1 2 j s 1 = 2 s 2 (j s) (j 1) (j s + 1) (s 1)! x s = (s 1)! (x j 1) (s 1) x=1/2. Taking the summation with respect to j inside the differentiation, we get ( x s ) (s 1) ( x=1/2 B(q, s) = x j 1 x s x q (s 1) = x=1/2 (s 1)! (s 1)! 1 x) = x s s 1 (s 1)! j=0 j=q+1 ( s 1 j ) (x q ) (j) ( (1 x) 1) (s 1 j) x=1/2 s 1 = x s 1 j!(s 1 j)! (s 1 j)! xq j q (q j + 1) x=1/2, (1 x) s j j=0

8 8 HICKERNELL, SLOAN, AND WASILKOWSKI which, after some elementary manipulation, can be shown to be equal to 2 q s 1 j=0 ( q j). This completes the proof of (10). The upper bound B(q,s) follows from this and the well-known fact that ( q j 1) ( q j) iff 2j q + 1. This completes the proof of Lemma 2. Proof of Theorem 2. Note that for any scalar function f H k, A k,n (f) converges pointwise to f with n and that n i=1 k,i = A k,n. Therefore, for any x and any f F 1,s. Hence f(x) = i N s + k=1 s k,ik (f)(x) f A q,s (f) = E q,s (f) with E q,s (f) := i q+1 k=1 Due to (4) and the fact that E q,s vanishes on constant functions, s k,ik (f). E q,s (f) = f (t ) E q,s (M (,t )) dt which yields (f A q,s (f))ρ L1 f 1,s sup t D max ρ E q,s (M (,t )) L1. Hence, to complete the proof, we only need to estimate the above supremum. For that purpose, note that s k,ik (M (,t )) 0 if i k 2 for some k /. k=1 This follows from the fact that k,i (1) 0 for any i 2 which, in turn, is a consequence of the fact that A k,i (1) 1 for any i 1. Otherwise, i.e., when i k = 1 for all k /, due to Proposition 1, s ρ k,ik (M (,t )) ρ k k,ik (M k (,t k )) L1 2 i k, k k k=1 independently of t. Therefore L 1 = (12) ρ E q,s (M (,t )) L1 i 2 i B(q s +, ) B(q s +, ), with the summation in (12) over i N such that i q + 1 (s ). The first reult in the theorem is proved by showing that B(q s +, ) is increasing with, i.e. max B(q s +, ) = B(q,s).

9 PIECEWISE CONSTANT ALGORITHM 9 Consider first q 2(s 1). To see that B(q s +, ) is increasing with, note that B(q s + + 1, + 1) B(q s +, ) q s , 2 = ( + 1)(q s + + 1) 2 2 with the last inequality due to the fact that +1 s and hence s 1 q s+1. Suppose now q < 2(s 1). To show that B(q s +, ) increases with also in this case, we need to consider three different cases. Consider first > q s + 3 (i.e., 2 > q s + + 3). Then B(q s + + 1, + 1) B(q s +, ) = + 1 2l + 1 2l + 2 > 1 = ( q s q s ) / ( ) q s + q s with the last equality due to an extra assumption that q s + = 2l (the proof for odd q s + is very similar). Consider next the case of = q s + 2. Then B(q s + + 1, + 1) B(q s +, ) Consider now + 1 < q s + 3. Then B(q s + + 1, + 1) = = = B(q s +, ) ( + 1)(q s + + 1) (q s + 3)(2q 2s + 3) (q s + 2)(2q 2s + 4) 1. ( ) ( ) q s q s + / 1 This completes the proof of the first part of the theorem. Assume now that (3) holds. Then s k,ik (M (,t )) 0 if i k = 1 for some k, k=1 as follows directly from the second part of Proposition 1. This means that the sum in (12) is now over i N such that i max{q+1 s+, 2 } and i 2. Replacing i by j = i 1, the sum becomes (13) 2 2 j = 2 B(max{q s, 1}, ). j max{q+1 s, } Therefore, for < q s + 1 we have 1 (14) ρ E q,s (M (,t )) L1 2 q +s j=0 ( q s j ),

10 10 HICKERNELL, SLOAN, AND WASILKOWSKI and for q s + 1 we have (15) ρ E q,s (M (,t )) L1 2 2 q+s 1. Since s, we have to have q 2s 1 for the latter case to happen. To estimate the maximum of the upper bound with respect to, we consider the following cases. Case q 3(s 1): Then, because q s 2 3, the right-hand side of (14) is bounded from above by s 2 ( q +s q s 1), which can be shown to increase with as long as 1 + (q s+1)/3. Since s, this yields the following upper bounds ( ) q s ρ E q,s (M (,t )) L1 s 2 q s 1 if q 4(s 1), and if q/(s 1) [3, 4). q s+1 q+s 1 ρ E q,s (M (,t )) L1 s 2 3 ( ) q s (q s + 1)/3 Case 2(s 1) < q < 3(s 1): First note that the right-hand side of (14) decreases with when > (q s)/2. This follows from the fact that the value of the righthand side for minus the value for + 1 equals 2 q 1+s 1 ( ) q s j j=0 ( ) q s 0, as claimed. Moreover, for (q s)/2, the right-hand side of (14) is bounded by s 2 ( q +s q s 1) which attains its maximum for = 1+ (q s+1)/3. Hence again, ( ) q s+1 q+s 1 ρ E q,s (M (,t )) L1 s 2 3 q s (q s + 1)/3 Case q 2(s 1): Due to (15), we need only to estimate the right-hand side of (14) for < q s+1. However, as in the previous case, it is bounded by s 2 ( ) q +s q s 1 with = 1 + (q s + 1)/3. Since ( ) q s+1 q+s 1 q s s q+s 1, (q s + 1)/3 the left-hand side of the above inequality is an upper bound on ρ E q,s (M (,t )) L1 also in this case. This completes the proof.. We end this section by relating the error of A q,s to its cardinality card(a q,s ). Theorem 3. For every s 2 and q 2(s 1), ( ) 2(s 1) s e ln(card(aq,s )) (16) error(a q,s ) (s 1)π (s 1) 1 2 ln(2) card(a q,s )..

11 PIECEWISE CONSTANT ALGORITHM 11 Proof. Since the information used by A q,s is nested, we know from [8] that ( ) q 1 card(a q,s ) 2 q s+1. s 1 Let s represent q as q = (t + 1)(s 1) with t 1. Then applying Stirling s formula to the error bound from Theorem 2, one can show that ( ) (t + 1)(s 1) error(a q,s ) s 2 (t+1)(s 1) s 1 s 2 (t+1)(s 1) ((t + 1)e) s 1 t + 1 t2π(s 1) s 2 (t+1)(s 1) ((t + 1)e) s 1 1 π(s 1), with the last inequality due to the fact that t 1. Similarly, card(a q,s ) 2 t(s 1) (e(t + 1)) s 1 1 π(s 1). Since x/(ln(x)) 2(s 1) increases for x e 2(s 1), we can replace card(a q,s ) by the righthand side of the above inequality in the following estimation: L := card(a q,s) error(a q,s ) (ln(card(a q,s ))) ( 2(s 1) s 2 (t+1)(s 1) ((t + 1)e) s 1 / ) ( π(s 1) 2 t(s 1) ((t + 1)e) s 1 / ) π(s 1) ( t(s 1) ln(2) + (s 1) ln(t + 1) + s 1 1 ln(π(s 1))) 2(s 1) 2 ( ) 2(s 1) s e π(s 1) s 1 g(t), with g(t) := t + 1 t ln(2) + ln(t + 1) since s 1 ln(π(s 1))/2 is positive. It is easy to verify that max t 1 g(t) = 1/ ln(2). This yields ( ) 2(s 1) ( ) 2(s 1) s e s e L (s 1)π 2 (s 1) = (s 1) ln(2) (s 1)π (s 1), 2 ln(2) and completes the proof. 5. Extensions In this section, we extend some of the previous results assuming now that the functions f are from the space F = F p,s which is the completion of H s with respect to the

12 12 HICKERNELL, SLOAN, AND WASILKOWSKI following norm f p,s := f(c) p + f p L p 1/p ( ) 1/p = f p L p, where p [1, ]. Of course, for p = we have f,s = max f L. This norm differs from 1,s by using L p instead of L 1 -norms. To stress that now a different space than F 1,s is being considered, we write error(a; F p,s ) instead error(a). For simplicity, we assume through this section that (3) is satisfied and that c = 0. Note that for p > 1 and unbounded D it could happen that the approximation problem is not well-defined since the corresponding embedding operator could be unbounded. As follows from [9], the problem is well-defined iff (17) ( bk where (18) ψ k (t) = a k ψ p k (x)dx ) 1/p <, k, bk t ρ k (x)dx for t 0, t a k ρ k (x)dx otherwise, where here and elsewhere, p denotes the conjugate to p, i.e., 1 p + 1 p = 1. This is why we assume (17) throughout the rest of this paper. Of course, (17) trivially holds when D is bounded. It also holds for p = 1 since then p = and the left-hand bk side of (17) should formally be replaced by ess sup t ck ρ t k (x)dx which obviously is equal to 1/2. Theorem 4. The error of any algorithm A is bounded by ( ( (19) error(a; F p,s ) h (x,t )dx where ) p dt ) 1/p h (x,t ) = ρ (x ) M (x,t ) A(M (,t ))(x ). If A is based on piecewise constant interpolation then we have equality in (19). Proof. Since the proof of (19) is quite straightforward for p = 1, we show it only for p > 1. To simplify the notation, we write m(x,t ) to denote m(x,t ) := M(x,t ) A(M(,t ))(x ). Of course, h (x,t ) = ρ (x ) m (x,t ).,

13 PIECEWISE CONSTANT ALGORITHM 13 We begin with the proof of (19). sing (4), we have by Hölder s inequality ρ(x) f(x) A(f)(x) dx D = ρ(x) f D (t )m (x,t )dt dx ρ (x ) f (t )m (x,t ) dt dx = f (t ) h (x,t )dx dt f Lp ( ( h (x,t )dx f p,s ( ) p dt ) 1/p ( ) ) p 1/p h (x,t )dx dt. This proves (19). We now show equality when A(m (,t )) is a piecewise constant function interpolating m (,t ). To that end, define ( ) p 1 g (t ) := sign(t ) h(x,t )dx with sign(t ) := sign(t k ), k and f(y) := g (t )M (y,t )dt. Of course, from (4), f = g. Since p p p = p, it is easy to check that ( ( ) ) p 1/p f p,s = h (x,t )dx dt. Moreover, ( f A( f))ρ L1 = ρ(y) f(y) A( f)(y) dy D = ρ(y) g D (t )m(y,t )dt dy ( ) p 1 = ρ(y) h(x D,t )dx sign(t )m(y,t )dt dy ( ) p 1 = = h(x,t )dx ( h (x,t )dx ) p h(y,t )dy dt dt = f p p,s,

14 14 HICKERNELL, SLOAN, AND WASILKOWSKI with the third-last equality due to the fact that sign(t )m(y,t ) = m(y,t ) 0, y,t. Dividing ( f A( f))ρ L1 by f p 1 p/p p,s (to obtain f p,s = f p,s ) completes the proof of the equality. Let A q,s be the algorithm from Section 4 and denote δ,i (x,t ) :=,i (M (,t ))(x ) = k δ k,ik (x ik,t ik ). From Lemma 1 we have the following proposition. Proposition 2. Let and t. Then (i) For every x, δ,i (x,t ) { 1, 0, 1}. (ii) For every x there exists at most one i such that δ,i (x,t ) = 1. (iii) For every i, the ρ -probability of the set of x s with δ,i (x,t ) = 1 is at most 2 i. (iv) If δ,i (x,t ) 0 for some x and i, then t k [x k i k,1,x k i k ] for every k.,2 i k 1 As in the proof of Theorem 2, let E q,s (f) = s i q+1 k=1 k,i k (f). Recall that i (M (,t )) 0 if either i k = 1 for some k, or i k 2 for some k /. Hence E q,s (M (,t ))(x ) = δ,i (x,t ) with P(q,s,) = From Theorem 4, this yields error(a q,s, F) = where (20) b (t ) := ρ (x ) i P(q,s,) { } j N + : j 2, j q + 1 s +. b p i P(q,s,) (t )dt 1/p, δ,i (x,t )dx. Due to its definition (18), ψ k (t) converges to zero with t b k and/or t a k. Moreover, from (iv) of Proposition 2, we know that δ,i (x,t ) 0 implies ψ k (t k ) 2 i k for every k. This means that we can replace the set P(q,s,) by the even smaller set { } P(q,s,,t ) = j N + : j 2, j q + 1 s +, and 2 j k ψ k (t k ), k.

15 PIECEWISE CONSTANT ALGORITHM 15 This leads to b (t ) j P(q,s,,t ) 2 j min { 2 ψ (t ), 2 B(max{q s, 1}, ) }, where as always, ψ (t ) = k ψ k(t k ). We summarize this in the following Proposition. Proposition 3. Let (3) hold. Then for any s 2 and any q s, error(a q,s ; F p,s ) = b p (t )dt 1/p. Moreover, { b (t ) min 2 ψ (t ), } B(max{q s, 1}, ). 2 For p > 1 and unbounded D, the above error bound is quite complicated, due to the presence of integrals of b p. Suppose now that D is bounded, say D = [0, 1] s. Then the integrals of b can be replaced by B(max{q s, 1}, ) 2 leading to the following upper bound 1/p (21) error(a q,s ; F p,s ) ( 2 B(max{q s, 1}, ) ) p. Of course, if p = 1, then p = and the bound (21) takes the following form error(a q,s ; F 1,s ) max 2 B(max{q s, 1}, ), which coincides with the bound from Section 4 for the case when (3) holds. For p > 1, (21) can further be estimated from above leading to error(a q,s ; F p,s ) 2 s/p max 2 B(max{q s, 1}, ). This means that error upper bounds from Section 4 also hold for p > 1 modulo the multiplicative factor 2 s/p. In particular, we have the following consequence of Theorem 3. Theorem 5. Let p > 1 and D = [0, 1] s. Then for every s 2 and q 2(s 1), error(a q,s ; F p,s ) s 21/p (s 1)π ( ) 2(s 1) e ln(card(aq,s )) 1 (s 1) 2 1/(2p) ln(2) card(a q,s ).

16 16 HICKERNELL, SLOAN, AND WASILKOWSKI 6. Integration Problem In this section, we briefly discuss the problem of approximating weighted integrals I ρ (f) = f(x)ρ(x)dx D by algorithms (often called quadratures) Q of the form Q(f) = n i=1 f(xi )g i. The worst case error of Q (with respect to F p,s ) is defined by error(q; F p,s, Int) := sup I ρ (f) Q(f). f p,s 1 Consider now the following quadrature (22) Q q,s (f) := I ρ (A q,s (f)). It is easy to see that Q q,s = s 2 i 1 (Q k,ik Q k,ik 1), where Q k,i (f) = 2 i f(c k ) + i q k=1 j=1 x k i,j defined by (6). Clearly, error(q q,s ; F p,s, Int) error(a q,s ; F p,s ). f(x k i,j) Hence all error bounds for A q,s obtained in the previous sections also hold for Q q,s. Acknowledgments. The research has been partially supported by the Hong Kong Research Grants Council grant HKB/2020/02P, the Hong Kong Baptist niversity Faculty Research Grant FRG/00-01/II-62, the Australian Research Council, and the National Science Foundation under Grant CCR References [1] F. J. Hickernell, I. H. Sloan, and G. W. Wasilkowski, On tractability of weighted integration over bounded and unbounded regions in R s, Mathematics of Computation, to appear, [2] Y. Li and G. W. Wasilkowski, Worst case complexity of weighted approximation and integration over R d, J. Complexity 18, pp , [3] E. Novak, Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Mathematics 1349, Springer, [4] A. Pinkus, n-widths in Approximation Theory, Springer-Verlag, Berlin, [5] S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 4, pp , [6] V. N. Temlyakov, Approximation of Periodic Functions, Nova Science Publishers, Inc., New York, [7] J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-Base Complexity, Academic Press, New York, [8] G. W. Wasilkowski and H. Woźniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, J. of Complexity 11, pp. 1-56, [9] G.W. Wasilkowski and H. Woźniakowski, Complexity of weighted approximation over R 1, J. of Approximation Theory 103, pp , 2000.

17 PIECEWISE CONSTANT ALGORITHM 17 Department of Mathematics, Hong Kong Baptist niversity, Kowloon Tong, Hong Kong address: School of Mathematics, niversity of New South Wales, Sydney 2052, Australia address: Department of Computer Science, niversity of Kentucky, 773 Anderson Hall, Lexington, KY , SA address: