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.
1. Butzer, P. L. and Gessinger, A. Ergodic theorems for semigroups and cosine operator functions at zero and infinity with rates; applications to partial differential equations. A survey. In Mathematical Analysis, Wavelets, and Signal Processing (Ismail, M. E., Nashed, M. Z., Zayed, A. L., and Ghaleb, A. F., eds), Contemp. Math., 1995, 190, 67–94.
2. Butzer, P. L. and Nessel, R. J. Fourier Analysis and Approximation. Vol. 1, Birkhäuser Verlag, Basel-Stuttgart, 1971.
http://dx.doi.org/10.1007/978-3-0348-7448-9
3. Butzer, P. L. and Stens, R. L. Chebyshev transform methods in the theory of best algebraic approximation. Abh. Math. Sem. Univ. Hamburg, 1976, 45, 165–190.
4. Lasser, R. and Obermaier, J. Characterization of Blackman kernels as approximate identities. Analysis, 2002, 22, 13–19.
http://dx.doi.org/10.1524/anly.2002.22.1.13
5. Rogosinski, W. W. Reihensummierung durch Abschnittskoppelungen. Math. Z., 1926, 25, 132–149.
http://dx.doi.org/10.1007/BF01283830
6. Stechkin, S. B. Summation methods of S. N. Berstein and W. Rogosinski. In Hardy, G. H. Divergent Series, Moscow (Russian Edition), 1951, 479–492.
7. Stepanets, A. I. Uniform Approximations by Trigonometric Polynomials. Naukova Dumka, Kiev, 1981 (in Russian).
8. Zhuk, V. V. and Natanson, G. I. Trigonometric Fourier Series and Elements of Approximation Theory. Leningrad University, Leningrad, 1983 (in Russian).
