Limits of indeterminacy of trigonometric series outside sets of the first category (Q1324921)

For two real functions \(g,h\) with the Baire property defined on \([- \pi, \pi]\) and fulfilling the inequality \(g\leq h\) on \([- \pi, \pi]\) there exists a trigonometric series \[ \sum^ \infty_{n= 0} a_ n\cos(nx+ \theta_ n), \] for which \(g(x)= \underline S(x)\), and \(h(x)= \overline S(x)\) on \([-\pi,\pi]\) except a set of the first category. Here \(\underline S(x)= \liminf_{n\to \infty} S_ n(x)\), \(\overline S(x)= \limsup_{n\to \infty} S_ n(x)\), \(S_ n(x)= \sum^ n_{k= 0} a_ k\cos(kx+ \theta_ k)\).