Quadrature and widths
Let B(X) be the set of bounded real functions on X, let \(A\subset B(X)\), and let \(d_ n(A)\) denote the n-width of A in B(X) as defined by Kolmogorov. For \(\mu\in B'(X)\) and \(M_ n=\{\mu_ n\in B'(X)| \mu_ n(f)=\sum^{n}_{i=1}a_ if(x_ i)\}\) we define the error bounds \(e_ n(A,\mu)=\inf_{\mu_ n\in M_ n}\sup_{f\in A}| \mu (f)-\mu_ n(f)|\) and prove \(e_ n(A,\mu)\leq 2\cdot \| \mu \| \cdot d_ n(A)\). Analogously we compare stochastic error bounds for Monte Carlo methods with the n-widths of A in \(L_ 2(\mu)\). We also give lower bounds for the stochastic error and applications of our results to special function classes.
scientific article; zbMATH DE number 3951225
scientific article