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Zeitschrift für Analysis und ihre Anwendungen

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Volume 9, Issue 4, 1990, pp. 343–349
DOI: 10.4171/ZAA/406

Published online: 1990-08-31

Error Estimates in Generalized Trigonometric Hölder-Zygmund Norms

Jürgen Prestin[1] and Siegfried Prössdorf

(1) Universität zu Lübeck, Germany

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.

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Prestin Jürgen, Prössdorf Siegfried: Error Estimates in Generalized Trigonometric Hölder-Zygmund Norms. Z. Anal. Anwend. 9 (1990), 343-349. doi: 10.4171/ZAA/406