A note on invariant subspaces and the solution of some classical functional equations

2013/10/29 by J. M. Almira, Almira, J. M., Kh. F. Abu-Helaiel +1
Engineering · Mathematics · #Advanced Control Systems Optimization #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Numerical methods for differential equations #math.CA

paper · pdf · doi:10.48550/arxiv.1310.7844

7 pages, submitted to a Journal

arxiv created 2013/10/29 · openalex publication_date 2013/10/29 · arxiv updated 2013/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of C(ℂ^d,ℂ) which are invariant under affine transformations T_a,b(f)(z)=f(az+b), or closed vector subspaces of C(ℝ^d,ℝ) which are translation and dilation invariant. These spaces have been recently classified by Sternfeld and Weit, and Pinkus, respectively, so that we use this information to give a direct characterization of the continuous solutions of the corresponding functional equations.

Related

• Characterization of finite dimensional subspaces of complex functions that are invariant under linear differential operators
• The invariant subspace problem for the space of smooth functions on the\n real line
• Funktionalanalysis Teil I
• On linear operators with an invariant subspace of functions
• Applications of fixed point theorems in the theory of invariant subspaces