Approximation and optimization on the Wiener space
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 \(\{n^{-1/2}\}\).
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