## Approximation and optimization on the Wiener space.(English)Zbl 0718.41046

Let $$f\in C=C[0,1]$$, $$N^*_ n(f)=(f(1/n),...,f(k/n),...,f(1))= y\in {\mathbb{R}}^ n$$, and $$\phi: {\mathbb{R}}^ n\to L_ q[0,1]$$ be a measurable mapping. Choosing $$\tilde f=\phi(y)$$ as an approximation to f causes the individual errors $$\| f-\tilde f\|_ q=(\int^{1}_{0}| f(t)-\tilde f(t)|^ q dt)^{1/q}$$ if $$1\leq q<\infty$$ and $$\| f- \tilde f\|_{\infty}=\max_{0\leq t\leq 1}| f(t)-\tilde f(t)|.$$ Let $$1\leq p<\infty$$ and w be the Wiener measure on the space C. Then the p-average error of $$\phi$$ and $$N^*_ n$$ and the p-average radius of $$N^*_ n$$ are defined by $e_ p(\phi,N^*_ n,L_ q)=(\int_{C} \| f-\phi \cdot N^*_ n(f)\|^ p_ qw(df))^{1/p}$ and $$r_ p(N^*_ n,L_ q)=\inf_{\phi} e_ p(\phi,N^*_ n,L_ q)$$ respectively. Similarly, the p-average error and the p-average radius for the global optimization problem are defined by $e_ p(\phi,N^*_ n,Opt)= (\int_{C} (\max_{\beta\leq t\leq 1}f(t)-f(\phi \cdot N(f)))^ pw(df))^{1/p}$ and $$r_ p(N^*_ n,Opt)=\inf_{\phi}e_ p(\phi,N^*_ n,Opt)$$ respectively, where $$\phi: {\mathbb{R}}^ n\to [0,1]$$ is a measurable mapping. The author mainly proves the following results: (1) The sequence $$\{r_ p(N^*_ n,L_ q)\}$$ is weakly equivalent to the sequence $$\{n^{-\ell}\}$$ in the case $$1\leq q<\infty$$; (2) The sequence $$\{r_ p(N^*_ n,L_{\infty})\}$$ is strongly equivalent to the sequence $$\{(\ell_ nn/2n)^{1/2}\}$$ and (3) The sequence $$\{r_ p(N^*_ n,Opt)\}$$ is weakly equivalent to the sequence $$\{n^{-1/2}\}$$.

### MSC:

 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

### Keywords:

Wiener space; global optimization problem
Full Text:

### References:

 [1] Billingsley, P, Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201 [2] Erdélyi, A, () [3] Kac, M, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke math. J., 21, 501-509, (1954) · Zbl 0056.10201 [4] Lee, D, Approximation of linear operators on a Wiener space, Rocky mountain J. math., 16, 641-659, (1986) · Zbl 0679.41014 [5] Lee, D; Wasilkowski, G.W, Approximation of linear functionals on a Banach space with a Gaussian measure, J. complexity, 2, 12-43, (1986) · Zbl 0602.65036 [6] Mockus, J, Bayesian approach to global optimization, (1989), Kluwer Academic Dordrecht · Zbl 0693.49001 [7] Papageorgiou, A; Wasilkowski, G.W, On the average complexity of multivariate problems, J. complexity, 6, 1-23, (1990) · Zbl 0723.68050 [8] Parthasarathy, K.R, Probability measures on metric spaces, (1967), Academic Press New York · Zbl 0153.19101 [9] Speckman, P, Lp approximation of autoregressive Gaussian processes, (1979), Department of Statistics, University of Oregon, Report [10] Suldin, A.V, Wiener measure and its applications to approximation methods II, Izv. vvssh. uchebn. zaved. mat., 18, 165-179, (1960), [In Russian] [11] Törn, A; Žilinskas, A, Global optimization, () · Zbl 0752.90075 [12] Traub, J.F; Wasilkowski, G.W; Woźniakowski, H, Information-based complexity, (1988), Academic Press New York · Zbl 0674.68039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.