A cardinal number connected to the solvability of systems of difference equations in a given function class

Abstract

Let ℝ^ℝ denote the set of real valued functions defined on the real line. A map D: ℝ^ℝ → ℝ^ℝ is said to be a difference operator if there are real numbers a _i, b _i (i = 1, …, n) such that (Dƒ)(x) = ∑ ⁿ _i=1 a _i ƒ(x + b _i) for every ƒ ∈ ℝ^ℝand x ∈ ℝ. By a system of difference equations we mean a set of equations S = {D _i ƒ = g _i: i ∈ I}, where I is an arbitrary set of indices, D _i is a difference operator and g _i is a given function for every i ∈ I, and ƒ is the unknown function. One can prove that a system S is solvable if and only if every finite subsystem of S is solvable. However, if we look for solutions belonging to a given class of functions then the analogous statement is no longer true. For example, there exists a system S such that every finite subsystem of S has a solution which is a trigonometric polynomial, but S has no such solution; moreover, S has no measurable solutions.

This phenomenon motivates the following definition. Let $\mathcal{F}$ be a class of functions. The solvability cardinal sc($\mathcal{F}$) of $\mathcal{F}$ is the smallest cardinal number κ such that whenever S is a system of difference equations and each subsystem of S of cardinality less than κ has a solution in $\mathcal{F}$, then S itself has a solution in $\mathcal{F}$. In this paper we determine the solvability cardinals of most function classes that occur in analysis. As it turns out, the behaviour of sc($\mathcal{F}$) is rather erratic. For example, sc(polynomials) = 3 but sc(trigonometric polynomials) = ω ₁, sc({ƒ: ƒ is continuous}) = ω ₁ but sc({f : f is Darboux}) = (2^ω)⁺, and sc(ℝ^ℝ) = ω. We consistently determine the solvability cardinals of the classes of Borel, Lebesgue and Baire measurable functions, and give some partial answers for the Baire class 1 and Baire class α functions.