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Limit theorems of the theory of discrete periodic splines

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It is known from the discrete harmonic analysis that the interpolation problem with equidistant interpolation points has a unique solution. If the right-hand sides in the interpolation problem are fixed, the spline depends on two parameters: the spline order and the number of points located between neighboring interpolation points. We find explicit expressions for the limits of interpolation spllines with respect to each parameter separately and show that both repeated limits exist. We also prove that these repeated limits are equal and their value is an interpolation trigonometric polynomial. Bibliography: 10 titles. Illustrations: 2 figures.

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References

  1. V. N. Malozemov and A. B. Pevnyi, “Discrete periodic splines and their numerical applications” [in Russian], Zh. Vychisl. Mat. Mat. Fiz. 38, No. 8, 1235–1246 (1998); English transl.: Comput. Math. Math. Phys. 38, No. 8, 1181–1192 (1998).

    MathSciNet  Google Scholar 

  2. M. G. Ber and V. N. Malozemov, “Interpolation of discrete periodic data” [in Russian] Probl. Peredachi Inf. 28, No. 4, 60-68 (1992); English transl.: Probl. Inf. Transm. 28, No. 4, 351-359 (1992).

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  3. M. von Golitschek, “On the convergence of interpolating periodic spline functions of high degree,” Numer. Math. 19, no. 2, 146–154 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  4. N. V. Chashnikov, “Limit curves for discrete periodic splines” [in Russian], DHA & CAGD, http://dha.spb.ru/reps09.shtml#0627 (June 27, 2009)

  5. V. N. Malozemov, “Limit theorems of the theory of diescrete periodic splines” [in Russian], DHA & CAGD, http://dha.spb.ru/reps10.shtml#0130 (January 30, 2010)

  6. N. V. Chashnikov, “Limit of interpolation periodic splines of a real variable” [in Russian] DHA & CAGD, http://dha.spb.ru/reps10.shtml#0313. (March 13, 2010)

  7. V. N. Malozemov and A. B. Pevnyi, “Discrete periodic B-splines” [in Russian], Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. No. 4, 13–18 (1997); English transl.: Vestn. St. Petersbg. Univ., Math. 30, No. 4, 10–14 (1997).

  8. V. N. Malozemov and S. M. Masharskii, Fundamentals of Discrete Harmonic Analysis Part 3 [in Russian], St.-Petersburg (2003).

  9. I. P. Natanoson, Theory of Functions of a Real Variable [in Russian], 3rd. Ed. Lan’, St. Petersebugr (1999); English transl.: Frederick Ungar Publishing Co., New York (1955).

  10. N. K. Bari, Trigonometric Series [in Russian], Nauka, Moscsow (1961); English transl.: Holt, Rinehart, and Winston, New York (1967).

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Authors and Affiliations

  1. St. Petersburg State University, 28, Universitetskii pr., Petrodvorets, St. Petersburg, 198504, Russia

    V. N. Malozemov

  2. IntelliJ Labs, 2A, Kantemirovskaya ul., St.-Petersburg, 197342, Russia

    N. V. Chashnikov

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  1. V. N. Malozemov
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  2. N. V. Chashnikov
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Correspondence to V. N. Malozemov.

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Translated from Problems in Mathematical Analysis 48, July 2010, pp. 53–74

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Malozemov, V.N., Chashnikov, N.V. Limit theorems of the theory of discrete periodic splines. J Math Sci 169, 188–211 (2010). https://doi.org/10.1007/s10958-010-0046-3

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  • DOI: https://doi.org/10.1007/s10958-010-0046-3

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