# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021030
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## On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function

 1 Tallinn University, Narva Str. 25, 10120 Tallinn, Estonia 2 Estonian Maritime Academy, Tallinn Univ. of Technology, Kopli 101, 11712 Tallinn, Estonia

* Corresponding author: A. Kivinukk

Received  June 2021 Revised  September 2021 Early access November 2021

Fund Project: 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

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.

Citation: Andi Kivinukk, Anna Saksa. On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021030
##### References:
 [1] N. I. Akhiezer, Lectures in the Theory of Approximation, Second revised and enlarged edition, Izdat. "Nauka", Moscow, 1965 (in Russian).  Google Scholar [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.   Google Scholar [3] C. Bardaro, P. 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.  Google Scholar [4] R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, Dover Publications, Inc., New York 1959.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] P. L. Butzer and S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 99-122.   Google Scholar [8] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag, Basel-Stuttgart, 1971. Google Scholar [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.  Google Scholar [10] P. L. Butzer, W. Splettstösser and R. L. Stens, The sampling theorems and linear prediction in signal analysis, Jahresber. Deutsch. Math-Verein, 90 (1988), 1-70.   Google Scholar [11] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303. Springer-Verlag, Berlin, 1993.  Google Scholar [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. Google Scholar [13] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996.   Google Scholar [14] J. R. Higgins, Completeness and Basis Properties of Sets of Special Functions, Cambridge: Cambridge University Press, 1977.  doi: 10.1017/CBO9780511566189.008.  Google Scholar [15] L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2$^{nd}$ edition, Pergamon Press, Oxford-Elmsford, N.Y., 1982.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] A. Kivinukk, A. 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.  Google Scholar [19] A. Kivinukk and A. Šeletski, On the steklov averages in operator cosine function framework, Filomat, 00 (2021), 00–00 (accepted for publication). Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. Kivinukk and G. Tamberg, Blackman-type windows for sampling series, J. Comput. Anal. Appl., 7 (2005), 361-371.   Google Scholar [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.  Google Scholar [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.  Google Scholar [25] D. Popa and I. Raça, Steklov averages as positive linear operators, Filomat, 30 (2016), 1195-1201.  doi: 10.2298/FIL1605195P.  Google Scholar [26] W. W. Rogosinski, Reihensummierung durch Abschnittskoppelungen, Math. Z., 25 (1926), 132-149.  doi: 10.1007/BF01283830.  Google Scholar [27] M. Sova, Cosine operator functions, Rozprawy Mat., 49 (1966), 47pp.  Google Scholar [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. Google Scholar [29] A. I. Stepanets, Uniform Approximations by Trigonometric Polynomials, De Gruyter, 2001 (Russian original: Naukova Dumka, Kiev, 1981).  Google Scholar [30] A. F. Timan, Theory of Approximation of Functions of a Real Variable, MacMillan, New York, 1963.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] V. V. Zhuk and G. I. Natanson, Trigonometric Fourier Series and Elements of Approximation Theory, Leningrad. Univ., Leningrad, 1983 [in Russian].  Google Scholar [34] D. Zwillinger and V. Moll (eds.), Grandshteyn and Ryzhik's Table of Integrals, Series and Products, Eighth edition. Academic Press, 2014.   Google Scholar

show all references

##### References:
 [1] N. I. Akhiezer, Lectures in the Theory of Approximation, Second revised and enlarged edition, Izdat. "Nauka", Moscow, 1965 (in Russian).  Google Scholar [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.   Google Scholar [3] C. Bardaro, P. 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.  Google Scholar [4] R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, Dover Publications, Inc., New York 1959.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] P. L. Butzer and S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 99-122.   Google Scholar [8] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag, Basel-Stuttgart, 1971. Google Scholar [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.  Google Scholar [10] P. L. Butzer, W. Splettstösser and R. L. Stens, The sampling theorems and linear prediction in signal analysis, Jahresber. Deutsch. Math-Verein, 90 (1988), 1-70.   Google Scholar [11] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303. Springer-Verlag, Berlin, 1993.  Google Scholar [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. Google Scholar [13] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996.   Google Scholar [14] J. R. Higgins, Completeness and Basis Properties of Sets of Special Functions, Cambridge: Cambridge University Press, 1977.  doi: 10.1017/CBO9780511566189.008.  Google Scholar [15] L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2$^{nd}$ edition, Pergamon Press, Oxford-Elmsford, N.Y., 1982.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] A. Kivinukk, A. 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.  Google Scholar [19] A. Kivinukk and A. Šeletski, On the steklov averages in operator cosine function framework, Filomat, 00 (2021), 00–00 (accepted for publication). Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. Kivinukk and G. Tamberg, Blackman-type windows for sampling series, J. Comput. Anal. Appl., 7 (2005), 361-371.   Google Scholar [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.  Google Scholar [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.  Google Scholar [25] D. Popa and I. Raça, Steklov averages as positive linear operators, Filomat, 30 (2016), 1195-1201.  doi: 10.2298/FIL1605195P.  Google Scholar [26] W. W. Rogosinski, Reihensummierung durch Abschnittskoppelungen, Math. Z., 25 (1926), 132-149.  doi: 10.1007/BF01283830.  Google Scholar [27] M. Sova, Cosine operator functions, Rozprawy Mat., 49 (1966), 47pp.  Google Scholar [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. Google Scholar [29] A. I. Stepanets, Uniform Approximations by Trigonometric Polynomials, De Gruyter, 2001 (Russian original: Naukova Dumka, Kiev, 1981).  Google Scholar [30] A. F. Timan, Theory of Approximation of Functions of a Real Variable, MacMillan, New York, 1963.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] V. V. Zhuk and G. I. Natanson, Trigonometric Fourier Series and Elements of Approximation Theory, Leningrad. Univ., Leningrad, 1983 [in Russian].  Google Scholar [34] D. Zwillinger and V. Moll (eds.), Grandshteyn and Ryzhik's Table of Integrals, Series and Products, Eighth edition. Academic Press, 2014.   Google Scholar
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