A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

L. Angeloni and G. Vinti, Convergence and rate of approximation in B⁢Vφ⁢(ℝ+N)BV^{\varphi}(\mathbb{R}^{N}_{+}) for a class of Mellin integral operators, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25 (2014), no. 3, 217–232.

L. Angeloni and G. Vinti, A characterization of absolute continuity by means of Mellin integral operators, Z. Anal. Anwend. 34 (2015), no. 3, 343–356.

C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Inform. Theory 56 (2010), no. 1, 614–633.

C. Bardaro and I. Mantellini, A modular convergence theorem for general nonlinear integral operators, Comment. Math. (Prace Mat.) 36 (1996), 27–37.

C. Bardaro and I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type series, Jaen J. Approx. 6 (2014), no. 2, 143–165.

C. Bardaro, I. Mantellini, R. Stens, J. Vautz and G. Vinti, Generalized sampling approximation for multivariate discontinuous signals and applications to image processing, New Perspectives on Approximation and Sampling Theory, Appl. Numer. Harmon. Anal., Birkhäuser, Cham (2014), 87–114.

C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Ser. Nonlinear Anal. Appl. 9, De Gruyter, Berlin, 2003.

C. Bardaro, G. Vinti, P. L. Butzer and R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6 (2007), no. 1, 29–52.

C. Bennett and K. Rudnick, On Lorentz–Zygmund spaces, Dissertationes Math. (Rozprawy Mat.) 175 (1980), 1–67.

L. Bezuglaya and V. Katsnel’son, The sampling theorem for functions with limited multi-band spectrum, Z. Anal. Anwend. 12 (1993), no. 3, 511–534.

P. L. Butzer, A survey of the Whittaker–Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1983), no. 1, 185–212.

P. L. Butzer and G. Hinsen, Reconstruction of bounded signals from pseudo-periodic, irregularly spaced samples, Signal Process. 17 (1989), no. 1, 1–17.

P. L. Butzer and J. Lei, Errors in truncated sampling series with measured sampled values for not-necessarily bandlimited functions. Dedicated to Julian Musielak, Funct. Approx. Comment. Math. 26 (1998), 25–39.

P. L. Butzer and J. Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal. 4 (2000), no. 2, 245–255.

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation. Volume 1: One-Dimensional Theory, Pure Appl. Math. 40, Academic Press, New York-London, 1971.

P. L. Butzer, G. Schmeisser and R. L. Stens, An introduction to sampling analysis, Nonuniform Sampling, Inf. Technol. Transm. Process. Storage, Kluwer, New York (2001), 17–121.

P. L. Butzer, W. Splettstösser and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Dtsch. Math.-Ver. 90 (1988), no. 1, 1–70.

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Engrg., Springer, New York (1993), 157–183.

F. Cluni, D. Costarelli, A. M. Minotti and G. Vinti, Applications of sampling Kantorovich operators to thermographic images for seismic engineering, J. Comput. Anal. Appl. 19 (2015), no. 4, 602–617.

F. Cluni, D. Costarelli, A. M. Minotti and G. Vinti, Enhancement of thermographic images as tool for structural analysis in earthquake engineering, NDT & E International 70 (2015), 60–72.

D. Costarelli and G. Vinti, Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces, Boll. Unione Mat. Ital. (9) 4 (2011), no. 3, 445–468.

D. Costarelli and G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim. 34 (2013), no. 8, 819–844.

D. Costarelli and G. Vinti, Order of approximation for sampling Kantorovich operators, J. Integral Equations Appl. 26 (2014), no. 3, 345–368.

D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integral Equations Appl. 26 (2014), no. 4, 455–481.

D. Costarelli and G. Vinti, Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces, Numer. Funct. Anal. Optim. 36 (2015), no. 8, 964–990.

D. Costarelli and G. Vinti, Approximation by max-product neural network operators of Kantorovich type, Results Math. 69 (2016), no. 3–4, 505–519.

D. Costarelli and G. Vinti, Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting, Math. Slovaca, to appear.

M. M. Dodson, Groups and the sampling theorem, Sampl. Theory Signal Image Process. 6 (2007), no. 1, 1–27.

M. M. Dodson and A. M. Silva, Fourier analysis and the sampling theorem, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 1, 81–108.

D. E. Edmunds and M. Krbec, Two limiting cases of Sobolev imbeddings, Houston J. Math. 21 (1995), no. 1, 119–128.

K. Gröchenig, Reconstruction algorithms in irregular sampling, Math. Comp. 59 (1992), no. 199, 181–194.

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

S. Hencl, A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal. 204 (2003), no. 1, 196–227.

J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 45–89.

J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford, 1996.

J. R. Higgins and R. L. Stens, Sampling Theory in Fourier and Signal Analysis: Advanced Topics, Oxford University Press, Oxford, 1999.

A. J. Jerri, The Shannon sampling – its various extensions and applications: A tutorial review, Proc. IEEE 65 (1977), 1565–1596.

L. V. Kantorovich, Sur certains développements suivant les polynomes de la forme de S. Bernstein I, II (in Russian), C. R. Acad. Sc. URSS 1930 (1930), 563–568, 595–600.

V. A. Kotel'nikov, On the carrying capacity of "ether" and wire in telecommunications (in Russian), Material for the First All-Union Conference on Questions of Communications, Izd. Red. Upr. Svyazi RKKA, Moscow (1933), 55-64

translation in Modern Sampling Theory, Appl. Numer. Harmon. Anal., Birkhäuser, Boston (2001), 27-45.

M. A. Krasnosel’skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen, 1961.

H. Mascart, Misura di Haar su gruppi topologici, Editrice Universitaria Levrotto e Bella, Torino, 1983.

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983.

L. S. Pontryagin, Topological Groups, Gordon & Breach Science, New York, 1966.

M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 146, Marcel Dekker, New York, 1991.

M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 250, Marcel Dekker, New York, 2002.

C. E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10–21.

E. M. Stein, Note on the class L⁢log⁡LL\log L, Studia Math. 32 (1969), 305–310.

R. L. Stens, Sampling with generalized kernels, Sampling Theory and Signal Analysis: Advanced Topics, Oxford University Press, Oxford (1999), 130–157.

F. Ventriglia and G. Vinti, A unified approach for the convergence of nonlinear Kantorovich type operators, Comm. Appl. Nonlinear Anal. 21 (2014), no. 2, 45–74.

C. Vinti, A survey on recent results of the mathematical seminar in Perugia, inspired by the work of Professor P. L. Butzer, Results Math. 34 (1998), no. 1–2, 32–55.

G. Vinti, Approximation in Orlicz spaces for linear integral operators and applications, Rend. Circ. Mat. Palermo (2) (2005), no. 76, 103–127.

G. Vinti and L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, J. Approx. Theory 161 (2009), no. 2, 511–528.

G. Vinti and L. Zampogni, A unifying approach to convergence of linear sampling type operators in Orlicz spaces, Adv. Differential Equations 16 (2011), no. 5–6, 573–600.

G. Vinti and L. Zampogni, Approximation results for a general class of Kantorovich type operators, Adv. Nonlinear Stud. 14 (2014), no. 4, 991–1011.

E. T. Whittaker, On the functions which are represented by the expansion of the interpolation theory, Proc. Roy. Soc. Edinburgh 35 (1915), 181–194.