On approximation processes defined by a cosine operator function; pp 214–224Full article in PDF format
In this paper we introduce the Blackman- and Rogosinski-type approximation processes in an abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We prove that in the presented setting the Blackman- and Rogosinski-type operators possess the order of approximation that coincides with results known in trigonometric approximation. Also applications for different types of approximations are given. An application for the Fourier series of symmetric functions with respect to η is emphasized.
1. Blackman, R. B. and Tukey, J. W. The Measurement of Power Spectra. Wiley-VCH, New York, 1958.
2. 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. Contemporary Mathematics, 1995, 190, 67–94.
3. Butzer, P. L. and Nessel, R. J. Fourier Analysis and Approximation. Vol. 1. One-Dimensional Theory. Birkhäuser Verlag, Basel, 1971.
4. 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.
5. Fejér, L. Untersuchungen über Fouriersche Reihen. Math. Ann., 1904, 58, 501–569.
6. Higgins, J. R. Sampling Theory in Fourier and Signal Analysis. Clarendon Press, Oxford, 1996.
7. Kantorovich, L. V. and Akilov, G. P. Functional Analysis. Second Edition. Elsevier, 1982.
8. Kivinukk, A. and Saksa, A. On approximation by Blackman- and Rogosinski-type operators in Banach space. Proc. Estonian Acad. Sci., 2016, 65, 205–219.
9. Kivinukk, A. and Tamberg, G. On Blackman–HarrisWindows for Shannon Sampling Series. Sampling Theory in Signal and Image Processing, 2007, 6, 87–108.
10. Korneichuk, N. Exact Constants in Approximation Theory. Cambridge University Press, 1991.
11. Labanova, O. Fejér Means of Fourier Series of p-Symmetric Functions. Master’s Thesis. Tallinn University, 2004.
12. Lasser, R. and Obermaier, J. Characterization of Blackman kernels as approximate identities. Analysis, 2002, 22, 13–19.
13. Rogosinski, W. W. Reihensummierung durch Abschnittskoppelungen. Math. Z., 1926, 25, 132–149.
14. Stepanets, A. I. Uniform Approximations by Trigonometric Polynomials. Naukova Dumka, Kiev, 1981 (in Russian).
15. Stechkin, S. B. Summation methods of S. N. Bernstein and W. Rogosinski. In Hardy, G. H. Divergent Series. Izd. Inostr. Lit., Moscow (Russian translation), 1951, 479–492.
16. Zhuk, V. V. and Natanson, G. I. Trigonometric Fourier Series and Elements of Approximation Theory. Leningrad Univ., 1983 (in Russian).
17. Zwillinger, D. and Moll, V. (eds.) Gradshteyn and Ryzhik’s Table of Integrals, Series, and Products
. Eighth edition. Academic Press, 2014Back to Issue