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On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function

  • * Corresponding author: A. Kivinukk

    * Corresponding author: A. Kivinukk 
Research supported partially by the EU, European Reg. Develop. Fund ASTRA project for 2016-2022 of Estonian Doctoral School in Mathematics and Statistics; Tallinn University TLU TEE
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  • In this article, we investigate the approximation properties of general cosine-type operators, especially Rogosinski-type operators, in Banach space when there is a cosine operator function. We apply our approach to both trigonometric Rogosinski operators and Shannon sampling operators. Moreover, for some operators, we derived orders of approximation that include numerical estimates of the constants contained in it. We announced a new direction for approximation issues in the Mellin framework.

    Mathematics Subject Classification: Primary: 41A65; Secondary: 41A17.


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  • [1] N. I. Akhiezer, Lectures in the Theory of Approximation, Second revised and enlarged edition, Izdat. "Nauka", Moscow, 1965 (in Russian).
    [2] M. V. Babushkin and V. V. Zhuk, On approximation of periodical functions by generalized Rogosinski sums (in Russian), Transactions of Tula State University. Natural Sciences, 2 (2014), 5-29. 
    [3] C. BardaroP. L. Butzer and I. Mantellini, The foundations of fractional calculus in the Mellin transform setting with applications, J. Fourier Anal. Appl., 21 (2015), 961-1017.  doi: 10.1007/s00041-015-9392-3.
    [4] R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, Dover Publications, Inc., New York 1959.
    [5] P. L. Butzer and A. Gessinger, Ergodic theorems for semigroups and cosine operator functions at zero and infinity with rates; applications to partial differential equations. A survey,, Contemp. Math., 190 (1995), 67-94.  doi: 10.1090/conm/190/02293.
    [6] P. L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), 325-376.  doi: 10.1007/BF02649101.
    [7] P. L. Butzer and S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 99-122. 
    [8] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag, Basel-Stuttgart, 1971.
    [9] P. L. Butzer and R. L. Stens, Chebyshev transform methods in the theory of best algebraic approximation, Abh. Math. Sem. Univ. Hamburg, 45 (1976), 165-190.  doi: 10.1007/BF02992913.
    [10] P. L. ButzerW. Splettstösser and R. L. Stens, The sampling theorems and linear prediction in signal analysis, Jahresber. Deutsch. Math-Verein, 90 (1988), 1-70. 
    [11] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303. Springer-Verlag, Berlin, 1993.
    [12] G. Heinzel, A. Rüdiger and R. Schilling, Spectrum and Spectral Density Estimation by the Discrete Fourier Transform (DFT), Including a Comprehensive List of Window Functions and Some New Flat-Top Windows, (Technical report), Max Planck Institute (MPI) für Gravitationsphysik / Laser Interferometry and Gravitational Wave Astronomy, 2002.
    [13] J. R. HigginsSampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996. 
    [14] J. R. HigginsCompleteness and Basis Properties of Sets of Special Functions, Cambridge: Cambridge University Press, 1977.  doi: 10.1017/CBO9780511566189.008.
    [15] L. V. Kantorovich and  G. P. AkilovFunctional Analysis, 2$^{nd}$ edition, Pergamon Press, Oxford-Elmsford, N.Y., 1982. 
    [16] A. Kivinukk, On the measure of approximation for some linear means of trigonometric Fourier series, J. Approx. Theory, 88 (1997), 193-208.  doi: 10.1006/jath.1996.3022.
    [17] A. Kivinukk, Approximation of continuous functions by Rogosinski-type sampling series, In Modern Sampling Theory: Mathematics and Applications, Birkhäuser Verlag, Boston, (2001), 229–244.
    [18] A. KivinukkA. Saksa and M. Zeltser, On a cosine operator function framework of approximation processes in Banach space, Filomat, 33 (2019), 4213-4228.  doi: 10.2298/FIL1913213K.
    [19] A. Kivinukk and A. Šeletski, On the steklov averages in operator cosine function framework, Filomat, 00 (2021), 00–00 (accepted for publication).
    [20] A. Kivinukk and G. Tamberg, On sampling series based on some combinations of sinc functions, Proc. Estonian Acad. Sci. Phys. Math., 51 (2002), 203-220.  doi: 10.3176/phys.math.2002.4.01.
    [21] A. Kivinukk and G. Tamberg, On sampling operators defined by the Hann window and some of their extensions, Sampl. Theory Signal Image Process., 2 (2003), 235-257.  doi: 10.1007/BF03549397.
    [22] A. Kivinukk and G. Tamberg, Blackman-type windows for sampling series, J. Comput. Anal. Appl., 7 (2005), 361-371. 
    [23] A. Kivinukk and G. Tamberg, On Blackman-Harris windows for Shannon sampling series, Sampl. Theory Signal Image Process., 6, (2007), 87–108. doi: 10.1007/BF03549465.
    [24] D. Lutz, Strongly continuous operator cosine functions, In Functional Analysis. Proc., Dubrovnik 1981, Lecture Notes in Math., (eds. D. Butković, H. Kaljević and S. Kurepa), Lect Notes in Math. 948 (1982), 73–97.
    [25] D. Popa and I. Raça, Steklov averages as positive linear operators, Filomat, 30 (2016), 1195-1201.  doi: 10.2298/FIL1605195P.
    [26] W. W. Rogosinski, Reihensummierung durch Abschnittskoppelungen, Math. Z., 25 (1926), 132-149.  doi: 10.1007/BF01283830.
    [27] M. Sova, Cosine operator functions, Rozprawy Mat., 49 (1966), 47pp.
    [28] S. B. Stechkin, Summation methods of S. N. Bernstein and W. Rogosinski, In G. H. Hardy, Divergent Series, Moscow, (Russian Edition), (1951), 479–492.
    [29] A. I. Stepanets, Uniform Approximations by Trigonometric Polynomials, De Gruyter, 2001 (Russian original: Naukova Dumka, Kiev, 1981).
    [30] A. F. Timan, Theory of Approximation of Functions of a Real Variable, MacMillan, New York, 1963.
    [31] O. L. Vinogradov and V. V. Zhuk, Estimates for functionals with a known finite set of moments in term of moduli of continuity, and behavior of constants in the Jackson-type inequalities,, St. Petersburg Math. J., 24 (2013), 691-721.  doi: 10.1090/S1061-0022-2013-01261-1.
    [32] V. V. Zhuk, Inequalities of the type of the generalized Jackson theorem for the best approximations,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 404 (2012), 135-156.  doi: 10.1007/s10958-013-1435-1.
    [33] V. V. Zhuk and G. I. Natanson, Trigonometric Fourier Series and Elements of Approximation Theory, Leningrad. Univ., Leningrad, 1983 [in Russian].
    [34] D. Zwillinger and  V. Moll (eds.)Grandshteyn and Ryzhik's Table of Integrals, Series and Products, Eighth edition. Academic Press, 2014. 
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