QUADRATIC FUNCTIONAL EQUATIONS ASSOCIATED WITH BOREL FUNCTIONS AND MODULE ACTIONS
QUADRATIC FUNCTIONAL EQUATIONS ASSOCIATED WITH BOREL FUNCTIONS AND MODULE ACTIONS
For a Borel function <TEX>${\psi}:\mathbb{R}{\times}\mathbb{R}{\rightarrow}\mathbb{R}$</TEX> satisfying the functional equation <TEX>$\psi$</TEX> (s + t, u + v) + <TEX>$\psi$</TEX>(s - t, u - v) = <TEX>$2\psi$</TEX>(s, u) + <TEX>$2\psi$</TEX>(t, v), we show that it satisfies the functional equation <TEX>$$\psi$$</TEX>(s, t) = s(s - t)<TEX>$$\psi$$</TEX>(1, 0) + <TEX>$$st\psi$$</TEX>(1, 1) + t(t - …