Tractability of Multivariate Approximation Defined over Hilbert Spaces with Exponential Weights

Journal of Approximation Theory

et al. 2016

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We consider L 2 -approximation of elements of a Hermite space of analytic functions over R s . The Hermite space is a weighted reproducing kernel Hilbert space of real valued functions for which the Hermite coefficients decay exponentially fast. The weights are defined in terms of two sequences a = {a j } and b = {b j } of positive real numbers. We study the nth minimal worst-case error e(n, APP s ; Λ std ) of all algorithms that use n information evaluations from the class Λ std which only allows function evaluations to be used.We study (uniform) exponential convergence of the nth minimal worst-case error, which means that e(n, APP s ; Λ std ) converges to zero exponentially fast with increasing n. Furthermore, we consider how the error depends on the dimension s.To this end, we study the minimal number of information evaluations needed to compute an ε-approximation by considering several notions of tractability which are defined with respect to s and log ε −1 . We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. It turns out that the conditions on the weight sequences are almost the same as for the information class Λ all which uses all linear functionals. The results are also constructive as the considered algorithms are based on tensor products of Gauss-Hermite rules for multivariate integration. The obtained results are compared with the analogous results for integration in the same Hermite space. This allows us to give a new sufficient condition for EC-weak tractability for integration.

Section: Exponential Convergence and Tractabilitymentioning

confidence: 99%

Section: Hermite Spaces With Infinite Smoothnessmentioning

confidence: 99%

Section: Hermite Spaces With Infinite Smoothnessmentioning

confidence: 99%

Section: Proof Of Point 3 (Ec-weak Tractability)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximation in Hermite spaces of smooth functions

Irrgeher

Journal of Approximation Theory

et al. 2016

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L∞-Approximation in Korobov spaces with exponential weights

Woźniakowski

2017

Journal of Complexity

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We study multivariate L ∞ -approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences a = {a j } and b = {b j } of positive real numbers bounded away from zero. We study the minimal worstcase error e L∞−app,Λ (n, s) of all algorithms that use n information evaluations from a class Λ in the s-variate case. We consider two classes Λ in this paper: the class Λ all of all linear functionals and the class Λ std of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that e L∞−app,Λ (n, s) converges to zero exponentially fast with increasing n. Furthermore, we consider how the error depends on the dimension s. To this end, we define the notions of κ-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC stands for "exponential convergence". In particular, EC-polynomial tractability means that we need a polynomial number of information evaluations in s and 1+log ε −1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ.L 2 -approximation for functions from the same function space has been considered in [2]. It is surprising that most results for L ∞ -approximation coincide with their counterparts for L 2 -approximation. This allows us to deduce also results for L p -approximation for p ∈ [2, ∞].

Integration and approximation in cosine spaces of smooth functions

Irrgeher

Mathematics and Computers in Simulation

2018

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We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend on two sequences of real numbers and decay exponentially. As a consequence the functions are infinitely often differentiable, and therefore it is natural to expect exponential convergence of the worst-case error. We give conditions on the weight sequences under which we have exponential convergence for the integration as well as the approximation problem. Furthermore, we investigate the dependence of the errors on the dimension by considering various notions of tractability. We prove sufficient and necessary conditions to achieve these tractability notions.Keywords: numerical integration, function approximation, cosine space, worst-case error, exponential convergence, tractability 2010 MSC: 41A63, 41A25, 65C05, 65D30, 65Y20Without loss of generality, see, e.g., [15] or [12, Section 4], we approximate S s by a linear algorithm A n,s using n information evaluations which are given by linear functionals from the class Λ ∈ {Λ all , Λ std }. More precisely, we approximate S s by algorithms of the formObviously, for multivariate integration only the class Λ std makes sense. Furthermore, we remark that in this paper we consider only function spaces for which Λ std ⊂ Λ all .We measure the error of an algorithm A n,s in terms of the worst-case error, which is defined aswhere · Hs , · Gs denote the norms in H s and G s , respectively. The nth minimal (worstcase) error is given bywhere the infimum is taken over all admissible algorithms A n,s . When we want to emphasize that the nth minimal error is taken with respect to algorithms using information from the class Λ ∈ {Λ all , Λ std }, we write e(n, S s ; Λ). For n = 0, we consider algorithms that do not use information evaluations and therefore we use A 0,s ≡ 0. The error of A 0,s is called the initial (worst-case) error and is given bySince we will study a class of weighted reproducing kernel Hilbert spaces with exponentially decaying weights, which will be introduced in Section 1.2, we are concerned with spaces H s of smooth functions. We remark that reproducing kernel Hilbert spaces of a similar flavor were previously considered in [2,3,5,6,7,8,9]. In this case it is natural to expect that, by using suitable algorithms, we should be able to obtain errors that converge to zero very quickly as n increases, namely exponentially fast. By exponential convergence (EXP) for the worst-case error we mean that there exist a number q ∈ (0, 1) and functions p, C, M : N → (0, ∞) such that e(n, S s ) ≤ C(s) q (n/M (s)) p (s) for all s, n ∈ N.(1)If the function p in (1) can be taken as a constant function, i.e., p(s) = p > 0 for all s ∈ N, we say that we achieve uniform exponential convergence (UEXP) for e(n, S s ). Furthermore, we denote by p * (s) and p * the largest possible rates p(s) and p such that EXP and UEXP holds, respectively. When stu...