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Bernstein-type inequalities for splines defined on the real axis

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We obtain exact Bernstein-type inequalities for splines \( s \in {S_{m,h}}\bigcap {{L_2}\left( \mathbb{R} \right)} \), as well as the exact inequalities estimating, for splines sS m, h , h > 0; the L p -norms of the Fourier transforms of their kth derivative in terms of the L p -norms of the Fourier transforms of the splines themselves.

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References

  1. 1.

    N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).

  2. 2.

    V. M. Tikhomirov, “Widths of sets in functional spaces and the theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960).

  3. 3.

    Yu. N. Subbotin, “On piecewise-polynomial interpolation,” Mat. Zametki, 1, No. 1, 24–29 (1967).

  4. 4.

    V. F. Babenko and A. S. Pichugov, “Inequalities of Bernstein type for polynomial splines in L 2,” Ukr. Mat. Zh., 43, No. 3, 420–422 (1991); English translation: Ukr. Math. J., 43, No. 3, 385–387 (1991).

  5. 5.

    N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extreme Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).

  6. 6.

    G. G. Magaril-Il’yaev, “On the best approximation of classes of functions on the straight line by splines,” Tr. Mat. Obshch. Inst. Ros. Akad. Nauk, 194, 48–159 (1992).

  7. 7.

    V. F. Babenko and S. A. Spektor, “Bernstein-type inequalities for splines in the space L 2(R),” Visn. Dnipropetr. Univ., 16, No. 6/1, 21–27 (2008).

  8. 8.

    L. Hörmander, Linear Partial Differential Operators, Springer, Berlin (1963).

  9. 9.

    L. R. Volevich and B. P. Paneyakh, “Some spaces of distributions and imbedding theorems,” Usp. Mat. Nauk, 20, No. 1, 3–74 (1965).

  10. 10.

    V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems [in Russian], Institute of Mathematics of the Ukrainian National Academy of Sciences, Kiev (2010).

  11. 11.

    N. A. Strelkov, “Projection-grid widths and lattice packings,” Mat. Sb., 182, No. 10, 1513–1533 (1991).

  12. 12.

    N. A. Strelkov, “Universal optimal splashes,” Mat. Sb., 188, No. 1, 147–160 (1997).

  13. 13.

    C. K. Chui, An Introduction to Wavelets [Russian translation], Mir, Moscow (2001).

  14. 14.

    B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], AFTs, Moscow (1999).

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Author information

Correspondence to V. F. Babenko.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 5, pp. 603–611, May, 2011.

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Babenko, V.F., Zontov, V.A. Bernstein-type inequalities for splines defined on the real axis. Ukr Math J 63, 699–708 (2011). https://doi.org/10.1007/s11253-011-0536-6

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Keywords

  • Real Axis
  • Approximation Theory
  • Ukrainian National Academy
  • Polynomial Spline
  • Linear Partial Differential Operator