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Order of the best spline approximations of some classes of functions

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Abstract

The rate of decrease of the upper bounds of the best spline approximations Em,n(f)p with undetermined n nodes in the metric of the space Lp(0, 1) (1≤p≤∞) is studied in a class of functionsf(x) for which ∥f m+1 (x)∥Lq(0, 1)≤1(1≤q≤t8) or var {f(m) (x); 0, 1}≤1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n → ∞, and asymptotic formulas are obtained for p=∞ and 1≤q≤∞ in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.

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Literature cited

  1. 1.

    G. Birkhoff, “Local spline approximation by moments,” Z. Math. Mech.,13, No. 9, 987–990 (1967).

  2. 2.

    S. N. Bernshtein, Collected Works [in Russian], Vol. 1, Moscow (1952), pp. 11–104.

  3. 3.

    I. P. Natanson, Constructive Theory of Functions [in Russian], Moscow-Leningrad (1949).

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Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 31–42, January, 1970.

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Subbotin, Y.N., Chernykh, N.I. Order of the best spline approximations of some classes of functions. Mathematical Notes of the Academy of Sciences of the USSR 7, 20–26 (1970). https://doi.org/10.1007/BF01093336

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Keywords

  • Break Line
  • Asymptotic Formula
  • Spline Function
  • Simultaneous Approximation
  • Spline Approximation