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On the existence and uniqueness of a solution of the problem of uniformSK-spline-interpolation

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Abstract

We establish necessary and sufficient conditions for the existence and uniqueness of generalized interpolatingSK-splines with a uniform distribution of interpolation points.

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References

  1. A. K. Kushpel',Extreme Properties of Splines and Widths of the Classes of Periodic Functions in the Space C [in Russian], Preprint No. 84.25, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984).

    Google Scholar 

  2. A. K. Kushpel',SK-Splines and Exact Estimates of the Widths of Functional Classes in the Space C [in Russian], Preprint No. 85.51, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985).

    Google Scholar 

  3. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, “Best approximation and convergence properties of higher-order spline approximations,”J. Math. Mech.,14, No. 2, 231–243 (1965).

    MATH  MathSciNet  Google Scholar 

  4. Yu. N. Subbotin, “On the connection between finite differences and corresponding derivatives,”Tr. Mat. Inst. Akad. Nauk SSSR,78, 24–42 (1965).

    MATH  MathSciNet  Google Scholar 

  5. Yu. N. Subbotin, “Interpolating splines,” in:Proc. Conf. “Approximation Theory” (Poznan, Aug. 22–26, 1972), Warszawa, PWN (1975), pp. 221–234.

    Google Scholar 

  6. P. V. Galkin, “On the solvability of the problem of periodic spline interpolation,”Mat. Zametki,8, No. 5, 563–573 (1970).

    MathSciNet  Google Scholar 

  7. A. A. Zhensykbaev,Some Problems of the Approximation by Splines in Functional Spaces [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Dnepropetrovsk (1973).

  8. N. P. Korneichuk,Splines in the Theory of Approximation [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  9. V. G. Shevaldin, “Widths of convolution classes, with a Poisson kernel,”Mat. Zametki,51, No. 6, 126–136 (1992).

    MathSciNet  Google Scholar 

  10. V. G. Shevaldin, “Lower estimates of the widths of the classes of periodic functions with bounded fractional derivative,”Mat. Zametki,53, No. 2, 145–151 (1993).

    MATH  MathSciNet  Google Scholar 

  11. V. G. Shevaldin,Source-Like Splines and Widths of the Classes of Periodic Functions [in Russian], Author's Abstract of Doctoral-Degree Thesis (Physics and Mathematics), Ekaterinburg (1996).

  12. A. K. Kushpel',Problems of Optimal Approximation of Functional Classes [in Russian], Author's Abstract of Doctoral-Degree Thesis (Physics and Mathematics), Kiev (1989).

  13. A. I. Stepanets and A. S. Serdyuk, “On the existence of interpolatingSK-splines,”Ukr. Mat. Zh.,46, No. 11, 1546–1553 (1994).

    Article  MathSciNet  Google Scholar 

  14. V. M. Tikhomirov, “The best methods of approximation and interpolation of differentiable functions in the spaceC[−1,1]”Mat. Sb.,80, No. 2, 290–304 (1969).

    MathSciNet  Google Scholar 

  15. V. K. Dzyadyk, “On the best approximation on the classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels,”Mat. Zametki,16, No. 5, 691–701 (1974).

    MATH  MathSciNet  Google Scholar 

  16. V. K. Dzyadyk, “On the problem of the best approximation of absolutely monotonous functions and some other functions in the metric ofL by trigonometric polynomials,”Izv. Akad. Nauk SSSR, Ser. Mat.,25, 173–238 (1961).

    MathSciNet  Google Scholar 

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Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 486–492, April, 1999.

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Serdyuk, A.S. On the existence and uniqueness of a solution of the problem of uniformSK-spline-interpolation. Ukr Math J 51, 538–545 (1999). https://doi.org/10.1007/BF02591758

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  • DOI: https://doi.org/10.1007/BF02591758

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