Error Estimates in Generalized Trigonometric Hölder-Zygmund Norms

Abstract

We consider Hölder-Zygmund spaces of $2\pi$-periodic functions $f:\mathbb{R}\to \mathbb{C}$, where the $k$-th difference with step-size $h$ of the $r$-th derivative in the $L^p$- or $C$-norm is bounded by a modulus-type function $\omega (h)$. For the Fourier sum and related approximation processes we investigate error estimates in corresponding Hölder-Zygmund norms if the smoothness of $f$ is given by other Hölder-Zygmund conditions. The convergence order for the general case can be formulated in a simple manner. This allows us to state also Jackson-type theorems for such Banach spaces. Moreover, we give explicit values for the constants appearing in these estimates.