On Generalized Derivative Sampling Series Expansion

  • Zurab A. Piranashvili
  • Tibor K. PogányEmail author


Master generalized sampling series expansion is presented for entire functions (signals) coming from a class whose members satisfy an extended exponential boundedness condition. Firstly, estimates are given for the remainder of Maclaurin series of those functions and consequent derivative sampling results are obtained and discussed.

The established results are employed in evaluating the related remainder term of signals which occur in sampling series expansion of stochastic processes and random fields (not necessarily stationary or homogeneous) whose spectral kernel satisfies the relaxed exponential boundedness. The derived truncation error upper bounds enable to obtain mean-square master generalized derivative sampling series expansion formulae either for harmonizable Piranashvili-type stochastic processes or for random fields.

Finally, being the sampling series convergence rate exponential, almost sure P sampling series expansion formulae are presented.


Whittaker–Kotel’nikov–Shannon sampling theorem Derivative sampling Exponentially bounded signals Entire functions Truncation error upper bound Harmonizable stochastic process Piranashvili process Karhunen process Loève process Weak sense stationary process Mean-square convergence Almost sure P convergence 


  1. 1.
    M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55 (National Bureau of Standards, Washington, D. C., 1964); 9th Reprinted edition, Dover Publications, New York, 1972Google Scholar
  2. 2.
    Yu.K. Belyaev, Analytical random processes. Teor. Veroyat. Primenen. III/4, 437–444 (1959). (in Russian)Google Scholar
  3. 3.
    P.L. Butzer, W. Splettstösser, R.L. Stens, The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math.-Verein. 90(1), 1–70 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 1 (McGraw–Hill, New York, 1953)zbMATHGoogle Scholar
  5. 5.
    I.I. Gikhman, A.V. Skorokhod, Introduction to the Theory of Stochastic Processes. 2nd edn. (Nauka, Moscow, 1971). (in Russian)Google Scholar
  6. 6.
    J.R. Higgins, Five short stories about the cardinal series. Bull. Am. Math. Soc. (N.S.) 12(1), 45–89 (1985)Google Scholar
  7. 7.
    J.R. Higgins, Sampling theorem and the contour integral method. Appl. Anal. 41, 155–171 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J.R. Higgins, Derivative sampling – a paradigm example of multichannel methods, in Sampling Theory in Fourier and Signal Analysis: Advanced Topics, ed. by J.R. Higgins, R.L. Stens (Oxford University Press, Oxford, 1999), pp. 67–95Google Scholar
  9. 9.
    C. Houdré, Harmonizability, V-boundedness, (2, p)-boundedness of stochastic processes. Probab. Theory Relat. Fields 84, 39–54 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    C. Houdré, On the linear prediction of multivariate (2, p)-bounded processes. Ann. Probab. 19(2), 843–867 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    C. Houdré, On the spectral SLLN and the pointwise ergodic theorem in L α. Ann. Probab. 20(4), 1731–1753 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C. Houdré, Wavelets, probability, and statistics: some bridges, in Wavelets: Mathematics and Applications, ed. by J.J. Benedetto, M.W. Frazier (CRC Press, Boca Raton, 1994), pp. 365–399zbMATHGoogle Scholar
  13. 13.
    C. Houdré, Reconstruction of band limited processes form irregular samples. Ann. Probab. 23(2), 674–696 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. Hurwitz, R. Courant, Allgemeine Funktionentheorie und elliptische Funktionen Geometrische Funktionentheorie (Springer, Berlin, 1922)zbMATHCrossRefGoogle Scholar
  15. 15.
    J.A. Jerri, The Shannon sampling theorem – its various extensions and applications: a tutorial review. Proc. IEEE 65(11), 1565–1596 (1977)zbMATHCrossRefGoogle Scholar
  16. 16.
    A. Jonquière, Note sur la série \(\sum _{n=1}^\infty \tfrac {x^n}{n^s}\). Bull. Soc. Math. France 17, 142–152 (1889)Google Scholar
  17. 17.
    Y. Kakihara, Multidimensional Second Order Stochastic Processes (World Scientific, Singapore, 1997)zbMATHCrossRefGoogle Scholar
  18. 18.
    A.J. Lee, On band limited stochastic processes. SIAM J. Appl. Math. 30, 267–277 (1976)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A.J. Lee, Characterization of bandlimited functions and processes. Inform. Control 31, 258–271 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    A.J. Lee, Approximate interpolation and the sampling theorem. SIAM J. Appl. Math. 32, 730–743 (1977)MathSciNetCrossRefGoogle Scholar
  21. 21.
    A.J. Lee, Sampling theorems for nonstationary processes. Trans Am. Math. Soc. 242, 225–241 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    D.S. Mitrinović, Analytic Inequalities. In cooperation with Vasić P. M. Die Grundlehren der mathematischen Wissenschaften, Band 165 (Springer, New York, 1970)Google Scholar
  23. 23.
    A.Ya. Olenko, T.K. Pogány, Direct Lagrange–Yen type interpolation of random fields. Theory Stoch. Process. 9(25)(3–4), 242–254 (2003)Google Scholar
  24. 24.
    A.Ya. Olenko, T.K. Pogány, A precise upper bound for the error of interpolation of stochastic processes. Theory Probab. Math. Stat. 71, 151–163 (2005)Google Scholar
  25. 25.
    A.Ya. Olenko, T.K. Pogány, Time shifted aliasing error upper bounds for truncated sampling cardinal series. J. Math. Anal. Appl. 324(1), 262–280 (2006)Google Scholar
  26. 26.
    A.Ya. Olenko, T.K. Pogány, On sharp bounds for remainders in multidimensional sampling theorem. Sampl. Theory Signal Image Process. 6(3), 249–272 (2007)Google Scholar
  27. 27.
    A.Ya. Olenko, T.K. Pogány, Universal truncation error upper bounds in sampling restoration. Georgian Math. J. 17, 765–786 (2010)Google Scholar
  28. 28.
    A.Ya. Olenko, T.K. Pogány, Universal truncation error upper bounds in irregular sampling restoration. Appl. Anal. 90(3–4), 595–608 (2010)Google Scholar
  29. 29.
    A.Ya. Olenko, T.K. Pogány, Average sampling reconstruction of harmonizable processes. Commun. Stat. Theory Methods 40(19–20), 3587–3598 (2011)Google Scholar
  30. 30.
    Z.A. Piranašvili, The problem of interpolation of random processes. Teor. Verojatnost. i Primenen. 12(4), 708–717 (1967). (in Russian)MathSciNetGoogle Scholar
  31. 31.
    Z. Piranashvili, On the extension of Kotelnikoff–Shannon formula. Bull. Georgian Acad. Sci. 154(1), 52–54 (1996)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Z. Piranashvili, Kotelnikov-Shannon’s generalized formula for the random fields. Bull. Georgian Acad. Sci. 163(3), 448–451 (2001)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Z. Piranashvili, On the generalization of one interpolation formula. Bull. Georgian Acad. Sci. 166(2), 251–254 (2002)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Z. Piranashvili, On the generalized formula of exponentially convergent sampling. Bull. Georgian Acad. Sci. 170(1), 50–53 (2004)MathSciNetGoogle Scholar
  35. 35.
    Z. Piranashvili, On estimation of the remainder term of the generalized sampling series. Bull. Georgian Natl. Acad. Sci. (N.S.) 3(2), 30–32 (2009)Google Scholar
  36. 36.
    Z.A. Piranashvili, On the generalized fast convergent sampling series. Bull. Georgian Natl. Acad. Sci. (N.S.) 5(2), 19–24 (2011)Google Scholar
  37. 37.
    Z. Piranashvili, Some remarks on certain sampling formulas. Bull. Georgian Natl. Acad. Sci. (N.S.) 7(3), 15–19 (2013)Google Scholar
  38. 38.
    Z.A. Piranashvili, T.K. Pogány, Some new generalizations of the Kotel’nikov–Shannon formula for stochastic signals, in Probability Theory and Mathematical Statistics Conference Dedicated to 100th Birth Anniversary of A. N. Kolmogorov, held September 21–27, Tbilisi, Georgia (2003). (unpublished conference talk)Google Scholar
  39. 39.
    T. Pogány, An approach to the sampling theorem for continuous time processes. Austral. J. Stat. 31(3), 427–432 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    T. Pogány, On the irregular-derivative sampling with uniformly dense sample points for bandlimited stochastic processes, in Exploring Stochastic Laws. Festschrift in Honour of the 70th Birthday of Academician Vladimir Semenovich Korolyuk, ed. by A.V. Skorokhod, Yu.A. Borovskikh (VSP Utrecht, The Netherlands, 1995), pp. 395–408Google Scholar
  41. 41.
    T. Pogány, On certain correlation properties of the sampling cardinal series expansion of stationary stochastic processes. Glasnik Mat. Ser. III 31(51), 353–362 (1996)MathSciNetzbMATHGoogle Scholar
  42. 42.
    T. Pogány, Some irregular derivative sampling theorems, in Functional Analysis, ed. by D. Butković et al., vol. IV. Proceedings of Postgraduate School and Conference. (held in Dubrovnik, Croatia, November 10–17, 1993), Matematisk Institut, Aarhus Universitet (1997), pp. 205–219Google Scholar
  43. 43.
    T. Pogány, Aliasing in the sampling restoration of non-band-limited homogeneous random fields. Math. Pannon. 8(1), 155–164 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    T. Pogány, Almost sure sampling restoration of bandlimited stochastic signals, in Sampling Theory in Fourier and Signal Analysis: Advanced Topics, ed. by J.R. Higgins, R.L. Stens (Oxford University Press, Oxford, 1999), 203–232Google Scholar
  45. 45.
    T. Pogány, Multidimensional Lagrange-Yen type interpolation via Kotel’nikov–Shannon sampling formulae. Ukrainian Math. J. 55(11), 1810–1827 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    T.K. Pogány, Whittaker-type derivative sampling reconstruction of stochastic L α( Ω)-processes. Appl. Math. Comput. 187(1), 384–394 (2007)MathSciNetzbMATHGoogle Scholar
  47. 47.
    T. Pogány, P. Peruničić, On the sampling theorem for homogeneous random fields. Theory Probab. Math. Stat. 53, 153–159 (1996)MathSciNetzbMATHGoogle Scholar
  48. 48.
    T. Pogány, P. Peruničić, On the multidimensional sampling theorem. Glas. Mat. Ser. III 36(56), 155–167 (2001)MathSciNetzbMATHGoogle Scholar
  49. 49.
    M. Priestley, Non-linear and Non-stationary Time Series (Academic Press, New York, 1988)Google Scholar
  50. 50.
    M.M. Rao, Harmonizable processes: structure theory. Enseign. Math. (2) 28(3–4), 295–351 (1982)Google Scholar
  51. 51.
    Ju.A. Rozanov, Spectral analysis of abstract functions. Theor. Probab. Appl. 4(3), 271–287 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    A.M. Yaglom, in Correlation Theory of Stationary and Related Random Functions I. Basic Results. Springer Series in Statistics (Springer, New York, 1987)Google Scholar
  53. 53.
    A.M. Yaglom, in Correlation Theory of Stationary and Related Random Functions II. Supplementary Notes and References. Springer Series in Statistics (Springer, New York, 1987)Google Scholar

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Authors and Affiliations

  1. 1.Vladimir Chavchanidze Institute of Cybernetics, Georgian Technical UniversityTbilisiGeorgia
  2. 2.Faculty of Maritime StudiesUniversity of RijekaRijekaCroatia
  3. 3.Institute of Applied MathematicsÓbuda UniversityBudapestHungary

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