Approximation by Fourier sums of functions with slowly decreasing Fourier coefficients (Q1091557)

Suppose that \(f\in L[0,\pi)\) and that \(a_ k=a_ k(f)\), \(b_ k=b_ k(f)\) \(k=0,1,..\). are its Fourier coefficients. Suppose also that \(\psi\) (k) is a function of natural k and \(\beta\in R\). A functional class \(L^{\psi}_{\beta}\) is introduced by the expression \[ \sum^{\infty}_{k=1}\frac{1}{\psi (k)}(a_ k \cos (kx+\beta \pi /2)+b_ k \sin (kx+\beta \pi /2). \] Approximate properties of the classes of \(L^{\psi}_{\beta}\) type are studied. Some asymptotic estimates are also established.