Characterization of finite dimensional subspaces of complex functions that are invariant under linear differential operators

TL;DR

This paper proves that the only finite-dimensional complex function spaces invariant under differential operators are those spanned by polynomial-exponential products, clarifying the scope of solutions for certain differential equations.

Contribution

It establishes a unique characterization of invariant spaces under differential operators using Jordan's canonical form, highlighting their role in solving linear differential equations.

Findings

Invariant spaces are spanned by polynomial-exponential products.

Non-homogeneous linear differential equations are solvable when right-hand sides are in these spaces.

The result is proved using Jordan's canonical decomposition.

Abstract

The method to solve inhomogeneous linear differential equations that is usually taught at school relies on the fact that the right hand side function is the product of a polynomial and an exponential and that the linear spaces of those functions are invariant under differential operators (finite or ordinary). This short note uses Jordan's canonical decomposition to prove that the linear spaces spanned by products of polynomial and exponentials are the only linear complex spaces that are invariant under differential operators, therefore non-homogeneous linear finite difference or ordinary differential equations can only be generically solved when the right hand side belongs to those spaces.