Abstract
Homogeneous polynomials are vital in the study of analytic functions on Banach spaces, as they are the components of the Taylor series that represent the functions locally. As most of the classical Banach spaces are Banach lattices, it is natural to work with polynomials that are coherent with the lattice structure. Thus, we study regular homogeneous polynomials, as they have a modulus that is a positive homogeneous polynomial. We demonstrate some applications, to analysis on Banach spaces with an unconditional basis, and to the radius of analyticity in a real Banach space. We also look at an application to the computation of the radius of convergence of power series on Banach lattices. We then consider the class of orthogonally additive polynomials, particularly on spaces of continuous functions, where some interesting geometric phenomena are seen.
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