We study multivariate tensor product problems in the worst case and average case settings. They are de ned on functions of d variables. For arbitrary d, w e p r o vide explicit upper bounds on the costs of algorithms which compute an "-approximation to the solution. The cost bounds are of the form (c(d) + 2) 1 2 + 3 ln 1=" d ; 1 4 (d;1) 1 " 5 : Here c(d) is the cost of one function evaluation (or one linear functional evaluation), and i 's do not depend on d they are determined by the properties of the problem for d = 1. F or certain tensor product problems, these cost bounds do not exceed c(d) K " ;p for some numbers K and p, both independent o f d. We apply these general estimates to certain integration and approximation problems in the worst and average case settings. We also obtain an upper bound, which is independent o f d, for the number, n(" d), of points for which discrepancy (with unequal weights) is at most ", n(" d)

The inverse of the star-discrepancy depends linearly on the dimension

a b s t r a c tMany recent papers considered the problem of multivariate integration, and studied the tractability of the problem in the worst case setting as the dimensionality d increases. The typical question is: can we find an algorithm for which the error is bounded polynomially in d, or even independently of d? And the general answer is: yes, if we have a suitably weighted function space.Since there are important problems with infinitely many variables, here we take one step further: we consider the integration problem with infinitely many variables -thus liberating the dimension -and we seek algorithms with small error and minimal cost. In particular, we assume that the cost for evaluating a function depends on the number of active variables. The choice of the cost function plays a crucial role in the infinite dimensional setting. We present a number of lower and upper estimates of the minimal cost for product and finite-order weights. In some cases, the bounds are sharp.

Abstract. We present formulas that allow us to decompose a function f of d variables into a sum of 2 d terms f u indexed by subsets u of {1, . . . , d}, where each term f u depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections {P j } d j=1 , where P j (f ) does not depend on the variable x j . We present an explicit formula for f u , which is new even for the anova and anchored decompositions; both are special cases of the general decomposition. We show that the decomposition is minimal in the following sense: if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z, then, for every choice of {P j } d j=1 , the terms f u = 0 for all subsets u containing z. Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms f u to be mutually orthogonal.

We study the =-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. 4 all consists of all linear functionals while 4 std consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1Â= and d. For 4 all we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. For 4 std we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences. Academic PressArticle ID jcom.1999.0512, available online at http:ÂÂwww.idealibrary.com on 402 0885-064XÂ99 30.00

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We study multivariate approximation with the error measured in L ∞ and weighted L 2 norms. We consider the worst case setting for a general reproducing kernel Hilbert space of functions of d variables with a bounded or integrable kernel. Here d can be arbitrarily large. We analyze algorithms that use standard information consisting of n function values, and we are especially interested in the optimal order of convergence, i.e., in the maximal exponent b for which the worst case error of such an algorithm is of order n −b . We prove that b ∈ [2 p 2 /(2 p + 1), p] for weighted L 2 approximation and b ∈ [2 p( p − 1/2)/(2 p + 1), p − 1/2] for L ∞ approximation, where p is the optimal order of convergence for weighted L 2 approximation among all algorithms that may use arbitrary linear functionals, as opposed to function values only. Under a mild assumption on the reproducing kernels we have p > 1/2. It was shown in our previous paper that the optimal order for L ∞ approximation and linear information is p − 1/2. We do not know if our bounds are sharp for standard information.We also study tractability of multivariate approximation, i.e., we analyze when the worst case error bounds depend at most polynomially on d and n −1 . We present necessary and sufficient conditions on tractability and illustrate our results for the weighted Korobov spaces with arbitrary smoothness and for the * Corresponding author.weighted Sobolev spaces with the Wiener sheet kernel. Tractability conditions for these spaces are given in terms of the weights defining these spaces.

Abstract. We prove that for the space of functions with mixed first derivatives bounded in L 1 norm, the weighted integration problem over bounded or unbounded regions is equivalent to the corresponding classical integration problem over the unit cube, provided that the integration domain and weight have product forms. This correspondence yields tractability of the general weighted integration problem.