August  2022, 5(3): 197-218. doi: 10.3934/mfc.2021030

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 Published  August 2022 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, 2022, 5 (3) : 197-218. 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).

[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. Higgins, Sampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996. 
[14] J. R. Higgins, Completeness 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. Akilov, Functional 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. 

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).

[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. Higgins, Sampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996. 
[14] J. R. Higgins, Completeness 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. Akilov, Functional 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. 
[1]

Purshottam Narain Agrawal, Şule Yüksel Güngör, Abhishek Kumar. Better degree of approximation by modified Bernstein-Durrmeyer type operators. Mathematical Foundations of Computing, 2022, 5 (2) : 75-92. doi: 10.3934/mfc.2021024

[2]

H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181

[3]

Dinh Nguyen Duy Hai. Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1715-1734. doi: 10.3934/cpaa.2022043

[4]

Nikolai Dokuchaev. On strong causal binomial approximation for stochastic processes. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1549-1562. doi: 10.3934/dcdsb.2014.19.1549

[5]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235

[6]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003

[7]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3043-3054. doi: 10.3934/dcdss.2020463

[8]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[9]

Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029

[10]

Wilfried Grecksch, Hannelore Lisei. Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3095-3114. doi: 10.3934/dcdsb.2016089

[11]

Yu A. Kutoyants. On approximation of BSDE and multi-step MLE-processes. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 4-. doi: 10.1186/s41546-016-0005-0

[12]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198

[13]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

[14]

Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293

[15]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control and Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007

[16]

Marvin S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4317-4339. doi: 10.3934/dcdsb.2019121

[17]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028

[18]

Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238

[19]

Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007

[20]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977

 Impact Factor: 

Metrics

  • PDF downloads (394)
  • HTML views (301)
  • Cited by (0)

Other articles
by authors

[Back to Top]