Unified Approach To Uniqueness, Expansion And Approximation Problems, A

This text introduces the concept of Polya property for holomorphic functions of two and several variables. The functions which have the Polya property are used to study the uniqueness problem for entire functions. This gives interesting results for a general class of generating functions. The functions with Polya property are used as generating kernels in formal generating relations to study the expansion problem for entire functions. A necessary and sufficient condition is obtained for an entire function to have a unique expansion in a series of various known polynomials generated from the formal generating relations. In this book it is shown that the uniqueness problem for entire functions of exponential type is equivalent to the approximation problem for analytic functions. Since functions with Polya property have very useful applications, the study of a class of linear operators which preserve the Polya property is embarked upon and two sufficient conditions for an operator to be Polya-property-preserving are given.