On Fourier series with non-negative coefficients and two problems of R. P. Boas (Q1071986)

Let \(a_ k\geq 0\), \(b_ k\geq 0\) and \(g(x):=\sum^{\infty}_{k=1}b_ k\) sin(kx), \(f(x):=\sum^{\infty}_{k=1}a_ k\) cos(kx). Necessary and sufficient conditions on g resp. f are given such that \(\sum k^{2j+1}b_ k\) resp. \(\sum k^{2j}a_ k\) do converge.