On an estimate of the accuracy of approximation of an integral in a Hilbert space by means of multiple integrals. (Russian) Zbl 0577.60003

Gerasimenko, T. S.

Teor. Veroyatn. Mat. Stat. 29, 27-30 (1983).

Let H be a separable Hilbert space and \(\mu\) a normed measure on H. For an n-dimensional subspace \(H_ n\) of H let \(\mu_ n\) be the restriction of \(\mu\) to \(H_ n\), and \(P_ n\) the orthoprojection on \(H_ n\). The expression \(D_{n,f}:=| \int_{H}f d\mu -\int_{H_ n}f d\mu_ n|\) is considered. Two upper estimates for sup \(D_{n,f}\) are given, where the supremum is taken over all functions fulfilling the Lipschitz condition with a certain constant and exponent 1 and absolute values of which are less than some \(C>0\). The estimates are in terms of a special kernel operator and \(P_ n\) respectively in terms of the characteristic functional of \(\mu\). Furthermore, sup \(\inf_{\{H_ n\}} D_{n,f}\) is estimated from above by a partial sum of eigenvalues of the covariance estimator of \(\mu\).

Reviewer: C.Baldauf

MSC:

60B11	Probability theory on linear topological spaces
46G10	Vector-valued measures and integration
46C99	Inner product spaces and their generalizations, Hilbert spaces

Keywords:

Lipschitz condition; characteristic functional; eigenvalues of the covariance