Bulletin of the
Korean Mathematical Society BKMS

For a Borel function $\psi:\mathbb R\times\mathbb R\to\mathbb R$ satisfying the functional equation $\psi(s+t,u+v)+\psi(s-t,u-v)=2\psi(s,u)+2\psi(t,v)$, we show that it satisfies the functional equation $$\psi(s,t)=s(s-t)\psi(1,0)+st\psi(1,1)+t(t-s)\psi(0,1).$$ Using this, we prove the stability of the functional equation $$f(ax+ay,bz+bw)+f(ax-ay,bz-bw)=2abf(x,z)+2abf(y,w)$$ in Banach modules over a unital $C^\star$-algebra.

Keywords: stability, functional equation, quadratic form, Borel function

MSC numbers: Primary 39B22, 39B82