Local versions of the Wiener–Lévy theorem

Let $h$ be a real-analytic function on the neighborhood of some compact set $K$ on the plane, and let $f(y)$ be the Fourier--Stieltjes transform of a complex measure of a finite total variation without singular components on the Euclidean space. Then there exists another measure of a finite total variation with the Fourier--Stieltjes transform $g(y)$ such that $g(y)=h(f(y))$ whenever the value $f(y)$ belongs to $K$.