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

Abstract

Let ${\mathbb{R}}^{\mathbb{R}}$ denote the set of real valued functions defined on the real line. A map $D:{\mathbb{R}}^{\mathbb{R}}\to {\mathbb{R}}^{\mathbb{R}}$ is a {\it difference operator}, if there are real numbers $a_i,b_i(i=1,...,n)$ such that $(Df)(x)={\sum }_{i=1}^n{a}_{i}f(x+b_i)$ for every $f\in {\mathbb{R}}^{\mathbb{R}}$ and $x\in \mathbb{R}$. A {\it system of difference equations} is a set of equations $S=D_{i}f=g_i:i\in I$, where $I$ is an arbitrary set of indices, $D_i$ is a difference operator and $g_i$ is a given function for every $i\in I$, and $f$ 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 fails. 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. This phenomenon motivates the following definition. Let $\mathcal{F}$ be a class of functions. The {\it solvability cardinal} $\solc (\iF)$ of $\mathcal{F}$ is the smallest cardinal $\kappa$ such that whenever $S$ is a system of difference equations and each subsystem of $S$ of cardinality less than $\kappa$ has a solution in $\mathcal{F}$, then $S$ itselfhas 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 $\solc ({\cal F})$ is rather erratic. For example, $\solc (\text{polynomials})=3$ but $\solc (\text{trigonometric polynomials})=ω_1$, $\solc ({f: f\ \text{is continuous}}) = ω_1$ but $\solc ({f: f\ \text{is Darboux}}) =(2^ω)^+$, and $\solc (\R^\R)=ω$. 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 $\alpha$ functions.