August  2022, 5(3): 219-229. doi: 10.3934/mfc.2021031

Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces

Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli 1, I-06123 Perugia, Italy

* Corresponding author: Carlo Bardaro

Received  August 2021 Revised  October 2021 Published  August 2022 Early access  November 2021

Fund Project: Carlo Bardaro and Ilaria Mantellini have been partially supported by the "Gruppo Nazionale per l'Analisi Matematica e Applicazioni (GNAMPA)" of the "Istituto di Alta Matematica (INDAM)" as well as by the projects "Ricerca di Base 2019 of University of Perugia (title: Misura, Integrazione, Approssimazione e loro Applicazioni)" and "Progetto Fondazione Cassa di Risparmio cod. nr. 2018.0419.021 (title: Metodi e Processi di Intelligenza artificiale per lo sviluppo di una banca di immagini mediche per fini diagnostici (B.I.M.))"

In this paper we study boundedness properties of certain semi-discrete sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed. These results pave the way to the norm-convergence of these operators.

Citation: Carlo Bardaro, Ilaria Mantellini. Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces. Mathematical Foundations of Computing, 2022, 5 (3) : 219-229. doi: 10.3934/mfc.2021031
References:
[1]

L. AngeloniD. Costarelli and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767.  doi: 10.5186/aasfm.2018.4343.

[2]

L. AngeloniD. Costarelli and G. Vinti, Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing, Ann. Acad. Sci. Fenn. Math., 45 (2020), 751-770.  doi: 10.5186/aasfm.2020.4532.

[3]

F. AsdrubaliG. BaldinelliF. BianchiD. CostarelliA. RotiliM. Seracini and G. Vinti, Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput., 317 (2018), 160-171.  doi: 10.1016/j.amc.2017.08.058.

[4]

S. Balsamo and I. Mantellini, On linear combinations of general exponential sampling series, Results Math., 74 (2019), Paper No. 180, 19 pp. doi: 10.1007/s00025-019-1104-x.

[5]

C. BardaroP. L. Butzer and I. Mantellini, The exponential sampling theorem of signal analysis and the reproduction kernel formula in the Mellin transform setting, Sampl. Theory Signal Image Process., 13 (2014), 35-66.  doi: 10.1007/BF03549572.

[6]

C. BardaroP. L. Butzer and I. Mantellini, The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics, Integral Transforms and Special Functions, 27 (2016), 17-29.  doi: 10.1080/10652469.2015.1087401.

[7]

C. BardaroP. L. ButzerR. L. Stens and G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Information Theory, 56 (2010), 614-633.  doi: 10.1109/TIT.2009.2034793.

[8]

C. BardaroL Faina and I. Mantellini, A generalization of the exponential sampling series and its approximation properties, Math. Slovaca., 67 (2017), 1481-1496.  doi: 10.1515/ms-2017-0064.

[9]

C. BardaroL. Faina and I. Mantellini, Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series, Math. Nachr., 289 (2016), 1702-1720.  doi: 10.1002/mana.201500225.

[10]

C. Bardaro and I. Mantellini, A quantitative Voronovskaja formula for generalized sampling operators, East J. Approx., 15 (2009), 459-471. 

[11]

C. Bardaro and I. Mantellini, Asymptotic formulae for linear combinations of generalized sampling type operators, Z. Anal. Anwend., 32 (2013), 279-298.  doi: 10.4171/ZAA/1485.

[12]

C. Bardaro and I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type series, Jaen Journal on Approximation, 6 (2014), 143-165. 

[13]

C. Bardaro and I. Mantellini, On a Durrmeyer type modifcation of the Exponential sampling series, Rend. Circ. Mat. Palermo (2), 70 (2021), 1289-1304.  doi: 10.1007/s12215-020-00559-6.

[14]

C. Bardaro, I. Mantellini and G. Schmeisser, Exponential sampling series: Convergence in Mellin-Lebesgue spaces, Results Math., 74 (2019), Paper No. 119, 20 pp. doi: 10.1007/s00025-019-1044-5.

[15]

C. BardaroG. VintiP. L. Butzer and R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampling Theory Signal Image Processing, 6 (2007), 29-52.  doi: 10.1007/BF03549462.

[16]

M. Bertero and E. R. Pike, Exponential sampling method for Laplace and other dilationally invariant transforms: I. Singular-system analysis, II. Examples in photon correction spectroscopy and Frauenhofer diffraction, Inverse Problems, 7 (1991), 1–20, 21–41. doi: 10.1088/0266-5611/7/1/004.

[17]

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.

[18]

P. L. Butzer and S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena, Suppl., (special issue dedicated to Professor Calogero Vinti), 46 (1998), 99–122.

[19]

P. L. Butzer and S. Jansche, A self-contained approach to Mellin transform analysis for square integrable functions; applications, Integral Transform. Spec. Funct., 8 (1999), 175-198.  doi: 10.1080/10652469908819226.

[20]

P. L. Butzer, G. Schmeisser and R. L. Stens, An introduction to sampling analysis, In: Marvasti, F. (ed. ) Nonuniform Sampling, Theory and Practice, 17–121. Kluwer Academic/Plenum Publishers, New York, (2001).

[21]

P. L. ButzerW. Splettstöẞer and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein., 90 (1988), 1-70. 

[22]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, In: Marks II, R.J. (ed. ) Advanced Topics in Shannon Sampling and Interpolation Theory, 157–183. Springer, New York, (1993).

[23]

D. Casasent, Optical signal processing, In: Casasent, D. (ed. ) Optical Data Processing, 241–282. Springer, Berlin, (1978). doi: 10.1007/BFb0057988.

[24]

D. CostarelliA. M. Minotti and G. Vinti, Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl., 450 (2017), 1083-1103.  doi: 10.1016/j.jmaa.2017.01.066.

[25]

D. Costarelli, M. Piconi and G. Vinti, On the convergence properties of Durrmeyer-Sampling type operators in Orlicz spaces, to appear, arXiv: 2007.02450v1, 2021.

[26]

D. Costarelli, M Seracini and G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comput., 347, (2020), 125046, 18 pp. doi: 10.1016/j. amc. 2020.125046.

[27]

D. CostarelliM. Seracini and G. Vinti, A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci., 43 (2020), 114-133.  doi: 10.1002/mma.5838.

[28]

D. Costarelli and G. Vinti, An inverse result of approximation by sampling Kantorovich series, Proc. Edinb. Math. Soc., 62 (2019), 265-280.  doi: 10.1017/S0013091518000342.

[29]

D. Costarelli and G. Vinti, Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels, Anal. Math. Phys., 9 (2019), 2263-2280.  doi: 10.1007/s13324-019-00334-6.

[30] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis, Foundations. Oxford Univ. Press, Oxford, 1996. 
[31]

A. Kivinukk and G. Tamberg, Interpolating generalized Shannon sampling operators, their norms and approximation properties, Sampl. Theory Signal Image Process., 8 (2009), 77-95.  doi: 10.1007/BF03549509.

[32]

A. Kivinukk and G. Tamberg, On window methods in generalized Shannon sampling operators., In New perspectives on approximation and sampling theory, 63–85, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2014).

[33]

A. S. Kumar and S. Bajpeyi, Direct and inverse results for Kantorovich type exponential sampling series, Results Math., 75 (2020), Paper No. 119, 17 pp. doi: 10.1007/s00025-020-01241-0.

[34]

A. S. Kumar and D. Ponnaian, Approximation by generalized bivariate Kantorovich sampling type series, J. Anal., 27 (2019), 429-449.  doi: 10.1007/s41478-018-0085-6.

[35]

A. S. Kumar and B. Shivam, Inverse approximation and GBS of bivariate Kantorovich type sampling series, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 82, 15 pp. doi: 10.1007/s13398-020-00805-7.

[36]

N. OstrowskyD. SornetteP. Parker and E. R. Pike, Exponential sampling method for light scattering polydispersity analysis, Opt. Acta, 28 (1981), 1059-1070.  doi: 10.1080/713820704.

[37]

S. Ries and R. L. Stens, Approximation by generalized sampling series, In: Sendov, Bl., Petrushev, P., Maalev, R., Tashev, S. (eds. )Constructive Theory of Functions, pp. 746–756. Pugl. House Bulgarian Academy of Sciences, Sofia, (1984).

[38]

G. Schmeisser, Interconnections between the multiplier methods and the window methods in generalized sampling, Sampl. Theory Signal Image Process., 9 (2010), 1-24.  doi: 10.1007/BF03549522.

[39] A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, 1993. 

show all references

References:
[1]

L. AngeloniD. Costarelli and G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767.  doi: 10.5186/aasfm.2018.4343.

[2]

L. AngeloniD. Costarelli and G. Vinti, Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing, Ann. Acad. Sci. Fenn. Math., 45 (2020), 751-770.  doi: 10.5186/aasfm.2020.4532.

[3]

F. AsdrubaliG. BaldinelliF. BianchiD. CostarelliA. RotiliM. Seracini and G. Vinti, Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput., 317 (2018), 160-171.  doi: 10.1016/j.amc.2017.08.058.

[4]

S. Balsamo and I. Mantellini, On linear combinations of general exponential sampling series, Results Math., 74 (2019), Paper No. 180, 19 pp. doi: 10.1007/s00025-019-1104-x.

[5]

C. BardaroP. L. Butzer and I. Mantellini, The exponential sampling theorem of signal analysis and the reproduction kernel formula in the Mellin transform setting, Sampl. Theory Signal Image Process., 13 (2014), 35-66.  doi: 10.1007/BF03549572.

[6]

C. BardaroP. L. Butzer and I. Mantellini, The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics, Integral Transforms and Special Functions, 27 (2016), 17-29.  doi: 10.1080/10652469.2015.1087401.

[7]

C. BardaroP. L. ButzerR. L. Stens and G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Information Theory, 56 (2010), 614-633.  doi: 10.1109/TIT.2009.2034793.

[8]

C. BardaroL Faina and I. Mantellini, A generalization of the exponential sampling series and its approximation properties, Math. Slovaca., 67 (2017), 1481-1496.  doi: 10.1515/ms-2017-0064.

[9]

C. BardaroL. Faina and I. Mantellini, Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series, Math. Nachr., 289 (2016), 1702-1720.  doi: 10.1002/mana.201500225.

[10]

C. Bardaro and I. Mantellini, A quantitative Voronovskaja formula for generalized sampling operators, East J. Approx., 15 (2009), 459-471. 

[11]

C. Bardaro and I. Mantellini, Asymptotic formulae for linear combinations of generalized sampling type operators, Z. Anal. Anwend., 32 (2013), 279-298.  doi: 10.4171/ZAA/1485.

[12]

C. Bardaro and I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type series, Jaen Journal on Approximation, 6 (2014), 143-165. 

[13]

C. Bardaro and I. Mantellini, On a Durrmeyer type modifcation of the Exponential sampling series, Rend. Circ. Mat. Palermo (2), 70 (2021), 1289-1304.  doi: 10.1007/s12215-020-00559-6.

[14]

C. Bardaro, I. Mantellini and G. Schmeisser, Exponential sampling series: Convergence in Mellin-Lebesgue spaces, Results Math., 74 (2019), Paper No. 119, 20 pp. doi: 10.1007/s00025-019-1044-5.

[15]

C. BardaroG. VintiP. L. Butzer and R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampling Theory Signal Image Processing, 6 (2007), 29-52.  doi: 10.1007/BF03549462.

[16]

M. Bertero and E. R. Pike, Exponential sampling method for Laplace and other dilationally invariant transforms: I. Singular-system analysis, II. Examples in photon correction spectroscopy and Frauenhofer diffraction, Inverse Problems, 7 (1991), 1–20, 21–41. doi: 10.1088/0266-5611/7/1/004.

[17]

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.

[18]

P. L. Butzer and S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena, Suppl., (special issue dedicated to Professor Calogero Vinti), 46 (1998), 99–122.

[19]

P. L. Butzer and S. Jansche, A self-contained approach to Mellin transform analysis for square integrable functions; applications, Integral Transform. Spec. Funct., 8 (1999), 175-198.  doi: 10.1080/10652469908819226.

[20]

P. L. Butzer, G. Schmeisser and R. L. Stens, An introduction to sampling analysis, In: Marvasti, F. (ed. ) Nonuniform Sampling, Theory and Practice, 17–121. Kluwer Academic/Plenum Publishers, New York, (2001).

[21]

P. L. ButzerW. Splettstöẞer and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein., 90 (1988), 1-70. 

[22]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, In: Marks II, R.J. (ed. ) Advanced Topics in Shannon Sampling and Interpolation Theory, 157–183. Springer, New York, (1993).

[23]

D. Casasent, Optical signal processing, In: Casasent, D. (ed. ) Optical Data Processing, 241–282. Springer, Berlin, (1978). doi: 10.1007/BFb0057988.

[24]

D. CostarelliA. M. Minotti and G. Vinti, Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl., 450 (2017), 1083-1103.  doi: 10.1016/j.jmaa.2017.01.066.

[25]

D. Costarelli, M. Piconi and G. Vinti, On the convergence properties of Durrmeyer-Sampling type operators in Orlicz spaces, to appear, arXiv: 2007.02450v1, 2021.

[26]

D. Costarelli, M Seracini and G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comput., 347, (2020), 125046, 18 pp. doi: 10.1016/j. amc. 2020.125046.

[27]

D. CostarelliM. Seracini and G. Vinti, A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci., 43 (2020), 114-133.  doi: 10.1002/mma.5838.

[28]

D. Costarelli and G. Vinti, An inverse result of approximation by sampling Kantorovich series, Proc. Edinb. Math. Soc., 62 (2019), 265-280.  doi: 10.1017/S0013091518000342.

[29]

D. Costarelli and G. Vinti, Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels, Anal. Math. Phys., 9 (2019), 2263-2280.  doi: 10.1007/s13324-019-00334-6.

[30] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis, Foundations. Oxford Univ. Press, Oxford, 1996. 
[31]

A. Kivinukk and G. Tamberg, Interpolating generalized Shannon sampling operators, their norms and approximation properties, Sampl. Theory Signal Image Process., 8 (2009), 77-95.  doi: 10.1007/BF03549509.

[32]

A. Kivinukk and G. Tamberg, On window methods in generalized Shannon sampling operators., In New perspectives on approximation and sampling theory, 63–85, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2014).

[33]

A. S. Kumar and S. Bajpeyi, Direct and inverse results for Kantorovich type exponential sampling series, Results Math., 75 (2020), Paper No. 119, 17 pp. doi: 10.1007/s00025-020-01241-0.

[34]

A. S. Kumar and D. Ponnaian, Approximation by generalized bivariate Kantorovich sampling type series, J. Anal., 27 (2019), 429-449.  doi: 10.1007/s41478-018-0085-6.

[35]

A. S. Kumar and B. Shivam, Inverse approximation and GBS of bivariate Kantorovich type sampling series, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 82, 15 pp. doi: 10.1007/s13398-020-00805-7.

[36]

N. OstrowskyD. SornetteP. Parker and E. R. Pike, Exponential sampling method for light scattering polydispersity analysis, Opt. Acta, 28 (1981), 1059-1070.  doi: 10.1080/713820704.

[37]

S. Ries and R. L. Stens, Approximation by generalized sampling series, In: Sendov, Bl., Petrushev, P., Maalev, R., Tashev, S. (eds. )Constructive Theory of Functions, pp. 746–756. Pugl. House Bulgarian Academy of Sciences, Sofia, (1984).

[38]

G. Schmeisser, Interconnections between the multiplier methods and the window methods in generalized sampling, Sampl. Theory Signal Image Process., 9 (2010), 1-24.  doi: 10.1007/BF03549522.

[39] A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, 1993. 
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