On a nonlinear moments problem (Q1260890)

The author investigates the problem to determine a function \(u\in C[a,b]\) if the value of the integral \[ I(g;u):=\int^ b_ ag(u(x))dx \] is known for all \(g\in C(\mathbb{R})\) or only those \(g\) belonging to some countable subset of \(C(\mathbb{R})\). Actually, the existence of a bounded-variation function \(\mu(x;u)\) with \[ I(g;u)={1\over 2\sqrt\pi}\int^ \pi_{- \pi}g(x)d\mu(x;u), \] for all \(2\pi\)-periodic \(g\), is stated. Then, the existence of the resolvent functional \(\mu\) is shown for the case that only a countable set of values \(\{I(g_ n;u),n\in\mathbb{N}\}\) is known. The special cases of \(\{g_ n\}\) being power or trigonometrical functions or \(u\) being smooth are considered at the end.