Exponential Sampling Series: Convergence in Mellin–Lebesgue Spaces

Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the convergence in variation for the generalized sampling series. Ann. Acad. Sci. Fenn. Math. 43(2), 755–767 (2018)

Angeloni, L., Vinti, G.: A unified approach to approximation results with applications to nonlinear sampling theory. Int. J. Math. Sci. 3(1), 93–128 (2004)

Angeloni, L., Vinti, G.: Rate of approximation for nonlinear integral operators with application to signal processing. Differ. Integral Equ. 18(8), 855–890 (2005)

Bardaro, C., Butzer, P.L., Mantellini, I.: The exponential sampling theorem of signal analysis and the reproduction kernel formula in the Mellin transform setting. Sampl. Theory Signal Image Process. 13(1), 35–66 (2014)

Bardaro, C., Butzer, P.L., Mantellini, I.: The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Integral Transforms Spec. Funct. 27(1), 17–29 (2016)

Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: On the Paley–Wiener theorem in the Mellin transform setting. J. Approx. Theory 207, 60–75 (2016)

Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Mellin analysis and its basic associated metric-applications to sampling theory. Anal. Math. 42(4), 297–321 (2016)

Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: A fresh approach to the Paley–Wiener theorem for Mellin transforms and the Mellin–Hardy spaces. Math. Nachr. 290, 2759–2774 (2017)

Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 316, 269–306 (2006)

Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Trans. Inf. Theory 56(1), 614–633 (2010)

Bardaro, C., Faina, L., Mantellini, I.: A generalization of the exponential sampling series and its approximation properties. Math. Slovaca 67(6), 1481–1496 (2017)

Bardaro, C., Mantellini, I.: Approximation properties in abstract modular spaces for a class of general sampling-type operators. Appl. Anal. 85(4), 383–413 (2006)

Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling type operators. Z. Anal. Anwend. 32(3), 279–298 (2013)

Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications, vol. 9. Walter De Gruyter, Berlin (2003)

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

Butzer, P.L., Dodson, M., Ferreira, P.J.S.G., Higgins, R.J., Schmeisser, G., Stens, R.L.: Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections. Bull. Math. Sci. 4(3), 481–525 (2014)

Butzer, P.L., Ferreira, P.J.S.G., Higgins, R.J., Schmeisser, G., Stens, R.L.: The sampling theorem, Poisson’s summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals–their interconnections. Appl. Anal. 90(3–4), 431–461 (2011)

Butzer, P.L., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997)

Butzer, P.L., Jansche, S.: A self-contained approach to Mellin transform analysis for square integrable functions; applications. Integral Transforms Spec. Funct. 8(3–4), 175–198 (1999)

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

Butzer, P.L., Schmeisser, G., Stens, R.L.: Shannon’s sampling theorem for bandlimited signals and their Hilbert transform, Boas-type formulae for higher order derivatives—the aliasing error involved by their extensions from bandlimited to non-bandlimited signals. Entropy 14, 2192–2226 (2012)

Butzer, P.L., Schmeisser, G., Stens, R.L.: Basic relations valid for the Bernstein space \(B^p_\sigma \) and their extensions to functions from larger spaces with error estimates in terms of their distances from \(B^p_\sigma \). J. Fourier Anal. Appl. 19, 333–375 (2013)

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

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

Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450(2), 1083–1103 (2017)

Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinb. Math. Soc. 62(1), 265–280 (2019)

Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. Interscience Publishers, Inc., New York (1958)

Haber, S., Shisha, O.: Improper integrals, simple integrals, and numerical quadrature. J. Approx. Theory 11, 1–15 (1974)

Kivinukk, A., Tamberg, G.: Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampl. Theory Signal Image Process. 8(1), 77–95 (2009)

Mantellini, I.: Generalized sampling operators in modular spaces. Comment. Math. 38, 77–92 (1998)

Mantellini, I., Vinti, G.: Approximation results for nonlinear integral operators in modular spaces and applications. Ann. Polon. Math. 81(1), 55–71 (2003)

Osgood, C.F.: Obtaining a function of bounded coarse variation by a chabge of variable. J. Approx. Theory 44(1), 14–20 (1985)

Ostrowsky, N., Sornette, D., Parker, P., Pike, E.R.: Exponential sampling method for light scattering polydispersity analysis. Opt. Acta 28, 1059–1070 (1994)

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

Schmeisser, G., Stenger, F.: Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process. 6(2), 199–221 (2007)

Vinti, G.: A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal. 79(1–2), 217–238 (2001)