“Analytical” functions of polynomial type

• Main Author: Todorov, Vladimir Todorov
• Format: Conference Proceeding
• Language: English
• Subjects: Algebra; Differential equations; Euclidean geometry; Euclidean space; Fields (mathematics); Functions (mathematics); Lie groups; Mathematical analysis; Metric space; Operators (mathematics); Partitions; Polynomials; Rings (mathematics); Topology; Unity
• ISSN: 0094-243X; 1551-7616
• DOI: 10.1063/5.0042081

We deal in this paper with functions of the type f(t,x)=∑k=0∞ak(t)xk where t ∈ T, x ∈ X. In what follows we shall consider different types of spaces T it supposed to be a separable metric space of different kinds and X, which suppose to be some kind of a topological algebra. Next suppose that ak(t)k∈ℕ+ is a partition of unity. More generally cα(t)α∈N+n we consider here a complex - valued (if necessary) partition of unity in ℝn with multi-indexes if n > 1. A partition of unity is locally finite hence the function f (t, x) is a polynomial concerning the variable x which should explain the name of this note. Note that x it may belong to various algebras, Banach one, operator algebras, functional algebras one etc. The aim of this note is to see how this point of view may help to solve some problems of mathematical physics in non-standard way. It may be Banach of different types, some Lie algebra etc. Here is our basic stock of examples: 1) Consider a topological space T and countable partition of unity {ak (t)}, (k ∈ N+) in it. Clearly that there is a locally finite open (or even point finite) cover U = {Uk} (k ∈ N+) for which supp (ak) ⊂ Uk. Because ordt U < ∞ we have ak ≡ 0 for almost all k and thus f(t,x)=∑k=0∞ak(t)xk is a polynomial of x ∈ X for every point t ∈ T. Note that we suppose here that X admits algebraic structure a field or a module or a ring, and it is not necessary to be loaded with some topology. 2) In addition, we can consider some structures on T or X. For example, if T = ℝn and ak(t) ∈ C∞(ℝn) then one may consider derivatives ∂tα f(t, x), so ∂αf(t,x)=∑k=0∞∂tαak(t)xk where α=(α1,α2,…,αn)∈ℕ+n is a multi-index. Moreover, if X is an n-dimensional Euclidean space ℝn one can consider the function f(t,x)=∑| α |=0∞aα(t)xα where as usual xα=x1α1x2α2⋯xnαn follows next then for an arbitrary differential operator D we have Df(t,x)=∑| α |=0∞D(aα(t)xα);(1) note that we deal here with finite sums. In the following text we discuss some properties and examples of ”analytical” functions of polynomial type.