Abstract
In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.
Highlights
Introduction and PreliminariesLet X be an infinite dimensional complex Banach space
For every entire function f there exists a sequence of continuous n-homogeneous polynomials such that
The following example shows that the product of two functions of unbounded type is not necessarily a function of unbounded type because there are invertible analytic functions of unbounded type
Summary
Introduction and PreliminariesLet X be an infinite dimensional complex Banach space. A function P : X → C is an n-homogeneous polynomial if there exists a symmetric n-linear map BP defined on the Cartesian power X n to C such that P( x ) = BP ( x, . . . , x ). N =1 is an entire function of unbounded type on X. The set of entire functions of unbounded type is not a linear space. The following proposition shows that the set of entire functions of unbounded type has some kind of ideal property.
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