Unified Approach To Uniqueness, Expansion And Approximation Problems, A

This book introduces the concept of Pólya property for holomorphic functions of two and several variables. The functions which have the Pólya 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 Pólya property are also 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. This result, in combination with other results, produces many interesting theorems on the approximation of analytic functions. Since functions with Pólya property have very useful applications, the study of a class of linear operators which preserve the Pólya property is embarked upon and two sufficient conditions for an operator to be Pólya-property-preserving are given.