Strong summability of Fourier series of (\(\psi\) ,\(\beta\))-differentiable functions (Q912341)

Uniform estimates are given for strong means of type \(\sum^{\infty}_{k=n}\lambda_ k\phi (\delta_ k| S_ k(f;x)- f(x)|),\) where \(S_ k(f;x)\) denotes the k-th partial sums of the Fourier series of f, for functions f in the class \(C^{\psi}_{\beta}C\) introduced by Stepanec. These involve the error in best approximation of a certain (\(\psi\),\(\beta\))-derivative of f by trigonometric polynomials of a given degree. \(\phi\) must satisfy the conditions \(\phi (u)\leq e^{au}\), \(u\equiv [1,\infty)\), \(\phi\) (2u)\(\leq a\phi (u)\), \(u\in [0,1]\). The exact formulation of the result is too complicated to be quoted here.