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Book review

Author(s) Egorov, A.D.
Sobolevsky, P.I.
Yanovich, L.A.
Title Functional integrals: Approximate evaluation and applications
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by Vasile Postolica

This is an updated and revised translation of the original work entitled "Approximate evaluation of continual integrals", Nauka and Takhnika, Minsk, 1985, 1987, being devoted to functional integrals defined on linear topological spaces in order to present a general scheme for the issues of evaluation of the functional integrals. It contains fifteen chapters, an ample bibliography, and an index.

Chapter 1, "Backgrounds from analysis on linear topological spaces" contains an introduction to cylindric functions, functional polynomials, derivatives, functional integrals and connections with random processes.

Chapter 2, "Integrals with respect to Gaussian measures and some quasimeasures: exact fomulae, Wick polynomials, diagrams" gives results on the concepts mentioned above and related topics.

Chapter 3, "Integration in linear topological spaces of some special classes" is devoted to inductive and projective limits of linear topolofical spaces, generalized function spaces and integrals in product spaces.

Chapter 4, "Approximate interpolation-type fomulae" contains interpolation of functionals, repeated interpolation, some construction rules for divided difference operators and approximate interpolation formulae.

Chapter 5, "Formulae based on characteristic functional approximations, which preserve a given number of elements" deals with approximations of characteristic functionals and approximate formulae based on the reductions of the number of terms in approximations.

Chapter 6, "Integrals with respect to Gaussian measures" contains many formulae of evaluation for integrals with respect to Gaussian measures and convergence of functional quadrature processes.

Chapter 7, "Integrals with respect to conditional Wiener measure" deals with approximations of conditional Wiener process which preserve a given number of moments and formulae for first accuracy, third accuracy and arbitrary accuracy degree.

Chapter 8, "Integrals with respect to measures which correspond to uniform processes with independent increments" contains useful formulae for first, third, fifth and arbitrary accuracy degree, integrals with respect to measures generated by multi-dimensional processes, convergence of composite formulae and cubature formulae for multiple probabilistic integrals.

Chapter 9, "Approximations which agree with diagram approaches" deals with formulae of evaluation which are exact for polynomials of Wick powers, approximate integration of functionals of Wick exponents, exact formulae for diagrams of a given type and approximate formulae for integrals with respect to quasimeasures coupled with some extensions.

Chapter 10, "Approximations of integrals based on interpolation of measure" is concerned with formulae of approximation of integrals with respect to Ornstein-Uhlenbeck, Wiener and modular measures and some considerations on formulae based on measure interpolation for integrals fo non-differentiable functionals.

Chapter 11, "Integrals with respect to measures generated by solutions of stochastic equations. Integrals over manifolds" is devoted to approximate formulae for integrals with respect to measures generated by solutions of stochastic (differential) equations, solutions of Ito equations and approximate formulae of integrals over manifolds.

Chapter 12, "Quadrature formulae for integrals of special form" contains several quadrature formulae in special cases.

Chapter 13, "Evaluation of integrals by Monte-Carlo method" debates the estimation of integrals with respect to Wiener measure and arbitrary Gaussian measure in the space of continuous functions through the agency of Monte-Carlo method.

Chapter 14, "Approximate formulae for multiple integrals with respect to Gaussian measure" contains formulae for the third, fifth and seventh accuracy degree and cubature formulae for multiple integrals of a certain kind.

Chapter 15, "Some special problems of functional integration" considers "an application of Gaussian functional integrals to solution of differential and integral equations, computation of the ground-state energy for quantum-mechanics systems and the problem of approximation of linear functionals", giving explicit expressions for a class of functional integrals.

This textbook is a valuable contribution to the development and applications of the main methods of evaluation for functional integrals, being written very clearly. The research workers in the field of functional integration will find here important evaluation methods useful for theory and for applied problems.