Analytical functions approximations by polynomials in the given region

A function, analytical in some finite simply connected region is considered in the paper aiming at the investigation of the growth behaviour of one and many complex variable function in dependence on its best polynomial approximation. The introduction of more subtle notion characteristics of conjugate types for functions of many complex variables and the obtaining of formulae, connected them with Teilor coefficients, are also the aim of the paper. As a result the growth behaviour of a function, analytical in some finite simply connected region with Jordan bound in dependence on its best polynomial approximation has been investigated. Notions of conjugate orders and conjugate types for a function of many complex variables, analytical in multiple circular regions, have been introduced. Formulae, connected these values with Teilor coefficients, have been obtained. Notions of order and type of a function, analytical in long bar z long bar <1 circle have been extended to functions, analytical in some finite simply connected region. Formulae for the calculation of order and type of a function in dependence on coefficients of its decompostion on the given basises have been obtained. Results may find their field of application while solving of approximation theory problems, finding of orders and types of analytical functions of one or many complex variablesAvailable from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio