2013/03/17 by J. M. Almira, Almira, J. M.
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.1303.4089
openalex publication_date 2013/03/17 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We study the finite dimensional spaces V which are invariant under the action of the finite differences operator Δhm. Concretely, we prove that if V is such an space, there exists a finite dimensional translation invariant space W such that V⊆ W. In particular, all elements of V are exponential polynomials. Furthermore, V admits a decomposition V=P⊕ E with P a space of polynomials and E a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by P. Montel which states that, if f:ℝ→ ℂ is a continuous function satisfying Δh1mf(t) = Δh2mf(t)=0 for all t∈ℝ and certain h1,h2∈ℝ∖\0\ such that h1/h2\not∈ℚ, then f(t)=a0+a1t+⋯+am-1tm-1 for all t∈ℝ and certain complex numbers a0,a1,⋯,am-1. We demonstrate, with quite different arguments, the same result not only for ordinary functions f(t) but also for complex valued distributions. Finally, we also consider in this paper the subspaces V which are Δh1h2⋯ hm-invariant for all h1,⋯,hm∈ℝ.