2020/04/19 by Miklós Laczkovich, Laczkovich, Miklos
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations #primary 39B52 #secondary 22A20
paper · pdf · doi:10.48550/arxiv.2004.08936
openalex publication_date 2020/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a topological Abelian semigroup with unit, let E be a Banach\nspace, and let C(G,E) denote the set of continuous functions f colon G\→\nE. A function f\∈ C(G,E) is a generalized polynomial, if there is an n\≥\n0 such that \Δh1 \… \Δ_hn+1 f=0 for every h1 ,\…\n, hn+1 \∈ G, where \Δh is the difference operator. We say that\nf\∈ C(G,E) is a polynomial, if it is a generalized polynomial, and the\nlinear span of its translates is of finite dimension; f is a w-polynomial, if\nu\∘ f is a polynomial for every u\∈ E^*, and f is a local polynomial,\nif it is a polynomial on every finitely generated subsemigroup. We show that\neach of the classes of polynomials, w-polynomials, generalized polynomials,\nlocal polynomials is contained in the next class. If G is an Abelian group\nand has a dense subgroup with finite torsion free rank, then these classes\ncoincide. We introduce the classes of exponential polynomials and\nw-expo -nential polynomials as well, establish their representations and\nconnection with polynomials and w-polynomials. We also investigate spectral\nsynthesis and analysis in the class C(G,E). It is known that if G is a\ncompact Abelian group and E is a Banach space, then spectral synthesis holds\nin C(G,E). On the other hand, we show that if G is an infinite and discrete\nAbelian group and E is a Banach space of infinite dimension, then even\nspectral analysis fails in C(G,E). If, however, G is discrete, has finite\ntorsion free rank and if E is a Banach space of finite dimension, then\nspectral synthesis holds in C(G,E).\n