Bivariate Generalized Exponential Sampling Series and Applications to Seismic Waves

Year 2019, Volume: 2 Issue: 4, 153 - 167, 01.12.2019

Carlo Bardaro , Giada Bevıgnanı Ilaria Mantellını Marco Seracını

https://doi.org/10.33205/cma.594066

Cited By: 7

Abstract

In this paper we introduce the generalized exponential sampling series of bivariate functions and establish some pointwise and uniform convergence results, also in a quantitative form. Moreover, we study the pointwise asymptotic behaviour of the series. One of the basic tools is the Mellin--Taylor formula for bivariate functions, here introduced. A practical application to seismic waves is also outlined.

Keywords

Bivariate generalized exponential sampling series, Mellin-Taylor formulae, moments, magnitude

Supporting Institution

Università di Perugia, Fondazione Cassa di Risparmio di Perugia (CRP)

Project Number

Università di Perugia Project "Ricerca di Base 2017"; Project Fondazione Cassa di Risparmio di Perugia, cod. nr. 2018.0419.021

Thanks

Gruppo Nazionale per l'Analisi Matematica e Applicazioni (GNAMPA) INDAM

References

• [1] L. Angeloni, D. Costarelli, G. Vinti, A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43(2), (2018), 755–767.
• [2] L. Angeloni, G. Vinti, Discrete operators of sampling type and approximation in '-variation, Math. Nachr., 291(4), (2018), 546–555.
• [3] C. Bardaro, P.L. Butzer, I. Mantellini, The exponential sampling theorem of signal analysis and the reproducing kernel formula in the Mellin transform setting, Sampling Theory in Signal and Image Process., 13(1), (2014), 35–66.
• [4] C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, On the Paley–Wiener theorem in the Mellin transform setting, J. Approx. Theory, 207, (2016), 60–75.
• [5] C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A fresh approach to the Paley–Wiener theorem for Mellin transforms and the Mellin–Hardy spaces, Math. Nachr., 290, (2017), 2759–2774.
• [6] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Information Theory, 56(1), (2010), 614–633.
• [7] C. Bardaro, L. Faina, I. Mantellini, A generalization of the exponential sampling series and its approximation properties, Math. Slovaca, 67(6), (2017), 1481–1496.
• [8] C. Bardaro, I. Mantellini, A note on the Voronovskaja theorem for Mellin–Fejer convolution operators, Appl. Math. Lett., 24, (2011), 2064–2067.
• [9] C. Bardaro, I. Mantellini, Asymptotic behaviour of Mellin–Fejer convolution operators, East J. Approx., 17(2), (2011), 181–201.
• [10] C. Bardaro, I. Mantellini, G. Schmeisser, Exponential sampling series: convergence in Mellin–Lebesgue spaces, Results Math., 74, (2019), Art. 119.
• [11] M. Bertero, E.R. Pike, Exponential sampling method for Laplace and other dilationally invariant transforms I. Singularsystem analysis. II. Examples in photon correction spectroscopy and Frauenhofer diffraction, Inverse Problems, 7, (1991), 1–20; 21–41. Bivariate generalized exponential sampling series and applications to seismic waves 167
• [12] A. Bobbio, M. Vassallo, G. Festa, Local Magnitude estimation for the Irpinia Seismic Network, Bull. Seismol. Soc. Am., 99, (2009), 2461–2470.
• [13] P.L. Butzer, S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3, (1997), 325–375.
• [14] P.L. Butzer, S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena, Suppl. Vol. 46, (special issue dedicated to Professor Calogero Vint) (1998), 99–122.
• [15] P.L. Butzer, R.L. Stens, Prediction of non-bandlimited signals in terms of splines of low degree, Math. Nachr., 132, (1987), 115–130.
• [16] P.L. Butzer, R.L. Stens, Linear prediction by samples from the past. In: R.J. Marks II (ed.) Advanced Topics in Shannon Sampling and Interpolation Theory,Springer, New York, (1993), 157–183.
• [17] D. Casasent (Ed), Optical data processing, Springer, Berlin, (1978), 241–282.
• [18] F. Gori, Sampling in optics In: R.J. Marks II (ed.), Advances Topics in Shannon Sampling and Interpolation Theory, Springer, New York, (1993), 37–83.
• [19] B. Gutenberg, C. F. Richter, Discussion: Magnitude and energy of earthquakes, Science, 83, 2147, (1936), 183–185.
• [20] J.R. Higgins, Sampling theory in Fourier and signal analysis, Foundations, Oxford Univ. Press., Oxford, (1996).
• [21] A. Kivinukk, G. Tamberg, On window methods in generalized Shannon sampling operators. In: G. Schmeisser and A. Zayed (Eds) New perspectives on approximation and sampling theory, 63–85, Appl. Numer. Harmon. Anal., Birkhaeuser Springer, Cham, (2014).
• [22] R.G. Mamedov, The Mellin transform and approximation theory (in Russian), "Elm", Baku, (1991).
• [23] N. Ostrowsky, D. Sornette, P. Parker, E.R. Pike, Exponential sampling method for light scattering polydispersity analysis, Opt. Acta, 28, (1994), 1059–1070.
• [24] C. F. Richte, An instrumental earthquake magnitude scale, Bull. Seismol. Soc. Am., 25, (1935), 1–32.
• [25] L.L. Schumaker, Spline functions: basic theory, John Wiley and Sons, New York, (1981).
• [26] E.Weber, V. Convertito, G. Iannaccone, A. Zollo, A. Bobbio, L. Cantore, M. Corciulo, M. Di Crosta, L. Elia, C. Martino, A. Romeo, C. Satriano, An advanced seismic network in the southern Apennines (Italy) for seismicity investigations and experimentation with earthquake early warning Seism. Res. Lett., 78, (2007), 622–534.
• [27] A.I. Zayed, Advances in Shannon’s Sampling Theory, CRC Press, Boca Raton (1993).