Triangular mean value theorems and Fréchet's equation (Q1912655)

The author offers the following two main results (the first is somewhat more general in the paper than stated here). 1). Theorem 1. If the \(m\)-th iterate of the difference of a complex valued continuous function \(f\) of \(n (\geq 2)\) real variables is zero for all spans of length 1 then it is zero for all spans of any length in the \(n\)-dimensional real space and thus \(f\) is a polynomial of degree at most \(m-1\). 2). A real valued function of two real variables has the triangular mean value property for radius \(r\) if its value at the center of every equilateral (``equivalent'' in Theorem 4 is a misprint) triangle of radius \(r\) equals the arithmetic mean of its values at the vertices. Theorem 4. If a real valued function of two real variables is continuous and has the triangular mean value property for a fixed radius then it is a harmonic polynomial of at most second degree.