On measurable orthogonally exponential functions
Let \(X\) be a real linear topological space with \(\dim X\geq 2\) and let \(\perp\) be a relation in \(X\) such that (1) \(0\perp x\), \(x\perp 0\), \(\forall x\in X\); (2) \(x,y\in X\setminus\{0\}\), \(x\perp y\Rightarrow x,y\) linearly indepedent; (3) \(x,y\in X\), \(x\perp y\Rightarrow ax\perp\) by, \(\forall a,b\in\mathbb{R}\); (4) \(P\subset X\) 2-dimensional subspace, \(x\in P\), \(b\in\mathbb{R}\), \(b>0\Rightarrow\exists y\in P\) such that \(x\perp y\) and \(x+y\perp bx-y\). The author studies the exponential functional equation \[ f(x+y) =f(x)f(y)\text{ for }f:X\to\mathbb{C},\text{ whenever }x\perp y. \] He shows that if \(f\) is universally, Christensen, or Baire measurable (under suitable assumptions on \(X)\), then \[ f(x)=\exp\bigl(A_1(x)+iA_2(x)+cL(x,x)\bigr) \] for \(x\in X\) with some continuous linear functionals \(A_1\), \(A_2:X\to\mathbb{R}\), some bilinear functional \(L:X^2\to\mathbb{R}\) and \(c\in\mathbb{C}\).
measurable orthogonally exponential functions
scientific article; zbMATH DE number 1295497