In this paper we introduce the Blackman- and Rogosinski-type approximation processes in an abstract Banach space setting. Historical roots of these processes go back to W. W. Rogosinski in 1926. The new definitions given use the concept of cosine operator functions. We proved that in the presented setting the Blackman- and Rogosinski-type operators possess the order of approximation, which coincides with results known in trigonometric approximation. Applications for the Fourier–Chebyshev approximation are given as well.
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