Computing Recomposition of Maps with a New Sampling Asymptotic Formula

ABSTRACT

The aim of the present paper is to state an asymptotic property Ρ of Shannon’s sampling theorem type, based on normalized cardinal sines, and keeping constant the sampling frequency of a not necessarilly band- limited signal. It generalizes in the limit the results stated by Marvasti et al. [7] and Agud et al. [1]. We show that Ρ is fulfilled for any constant signal working for every given sampling frequency. Moreover, we conjecture that Gaussian maps of the form e^{-Λt2 } ,Λ∈R^{+}, hold Ρ. We support this conjecture by proving the equality given by for the three first coefficients of the power series representation of e^{-Λt2 } .

The aim of the present paper is to state an asymptotic property Ρ of Shannon’s sampling theorem type, based on normalized cardinal sines, and keeping constant the sampling frequency of a not necessarilly band- limited signal. It generalizes in the limit the results stated by Marvasti et al. [7] and Agud et al. [1]. We show that Ρ is fulfilled for any constant signal working for every given sampling frequency. Moreover, we conjecture that Gaussian maps of the form e

Cite this paper

nullA. Antuña, J. Guirao and M. López, "Computing Recomposition of Maps with a New Sampling Asymptotic Formula,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 2, 2011, pp. 43-49. doi: 10.4236/ojdm.2011.12006.

nullA. Antuña, J. Guirao and M. López, "Computing Recomposition of Maps with a New Sampling Asymptotic Formula,"

References

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[2] P. L. Butzer, S. Ries and R. L. Stens, “Approximation of Continuous and Discontinuous Functions by Generalized Sampling Series,” Journal of Approximation Theory, Vol. 50, No. 1, 1987, pp. 25-39. doi:10.1016/0021-9045(87)90063-3

[3] P. L. Butzer and R. L. Stens, “Sampling Theory for not Necessarily Band-Limited Functions: A Historical Overview,” SIAM Review, Vol. 34, No. 1, 1992, pp. 40-53. doi:10.1137/1034002

[4] J. A. Gubner, “A New Series for Approximating Voight Functions,” Journal of Physics A: Mathematical and General, Vol. 27, No. 19, 1994, pp. L745-L749. doi:10.1016/0021-9045(87)90063-3

[5] J. R. Higgings, “Five Short Stories about the Cardinal Series,” Bulletin of the American Mathematical Society, Vol. 12, 1985, pp.45-89. doi:10.1090/S0273-0979-1985-15293-0

[6] H. J. Landau and H. O Pollak, “Prolate Spheroidal Wave Functions, Fourier analysis and uncertainly,” Bell System Technical Journal, Vol. 40, No. 1, 1961, pp. 65-84.

[7] F. Marvasti and A. K. Jain, “Zero Crossing Bandwidth Compression, and Restoration of Nonlinearly Distorted Band-Limited Signals,” Journal of the Optical Society of America, Vol. 3, No. 5, 1986, pp. 651-654. doi:10.1364/JOSAA.3.000651

[8] D. Middleton, “An Introduction to Statistical Communication Theory,” McGraw-Hill, New York, 1960.

[9] C.E. Shannon, “Communication in the Presence of Noise,” Proceedings of the Institute of Radio Engineers, Vol. 137, 1949, pp. 10-21.

[10] E. T. Whittaker, “On the Functions which are Represented by the Expansions of the Interpolation Theory,” Proceedings of the Royal Society, Vol. 35, 1915, pp. 181-194.

[11] A. I. Zayed, “Advances in Shannon's Sampling Theory,” Ed. CRC Press, Florida, 1993.

[1] L. Agud and R. G. Cataln, “New Shannon’s Sampling Recomposition,” Revista de la Academia de Ciencias Zaragoza, Vol. 56, 2001, pp. 45-48.

[2] P. L. Butzer, S. Ries and R. L. Stens, “Approximation of Continuous and Discontinuous Functions by Generalized Sampling Series,” Journal of Approximation Theory, Vol. 50, No. 1, 1987, pp. 25-39. doi:10.1016/0021-9045(87)90063-3

[3] P. L. Butzer and R. L. Stens, “Sampling Theory for not Necessarily Band-Limited Functions: A Historical Overview,” SIAM Review, Vol. 34, No. 1, 1992, pp. 40-53. doi:10.1137/1034002

[4] J. A. Gubner, “A New Series for Approximating Voight Functions,” Journal of Physics A: Mathematical and General, Vol. 27, No. 19, 1994, pp. L745-L749. doi:10.1016/0021-9045(87)90063-3

[5] J. R. Higgings, “Five Short Stories about the Cardinal Series,” Bulletin of the American Mathematical Society, Vol. 12, 1985, pp.45-89. doi:10.1090/S0273-0979-1985-15293-0

[6] H. J. Landau and H. O Pollak, “Prolate Spheroidal Wave Functions, Fourier analysis and uncertainly,” Bell System Technical Journal, Vol. 40, No. 1, 1961, pp. 65-84.

[7] F. Marvasti and A. K. Jain, “Zero Crossing Bandwidth Compression, and Restoration of Nonlinearly Distorted Band-Limited Signals,” Journal of the Optical Society of America, Vol. 3, No. 5, 1986, pp. 651-654. doi:10.1364/JOSAA.3.000651

[8] D. Middleton, “An Introduction to Statistical Communication Theory,” McGraw-Hill, New York, 1960.

[9] C.E. Shannon, “Communication in the Presence of Noise,” Proceedings of the Institute of Radio Engineers, Vol. 137, 1949, pp. 10-21.

[10] E. T. Whittaker, “On the Functions which are Represented by the Expansions of the Interpolation Theory,” Proceedings of the Royal Society, Vol. 35, 1915, pp. 181-194.

[11] A. I. Zayed, “Advances in Shannon's Sampling Theory,” Ed. CRC Press, Florida, 1993.