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Journal of Mathematical Sciences

, Volume 217, Issue 1, pp 3–22 | Cite as

A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators

  • O. L. Vinogradov
  • A. V. Gladkaya
Article
Let σ > 0, m, r ∈ ℕ, mr, let S σ,m be the space of splines of order m and minimal defect with nodes \( \frac{j\pi }{\sigma } \) (j ∈ ℤ), and let A σ,m (f) p be the best approximation of a function f by the set S σ,m in the space L p (ℝ). It is known that for p = 1,+∞,
$$ \begin{array}{l} \sup \hfill \\ {}f\in {W}_p^{(r)}\left(\mathbb{R}\right)\hfill \end{array}\frac{A_{\sigma, m}{(f)}_p}{{\left\Vert {f}^{(r)}\right\Vert}_p}=\frac{K_r}{\sigma^r}, $$
where K r are the Favard constants. In this paper, linear operators X σ,r,m with values in S σ,m such that for all p ∈ [1,+∞] and f ∈ W p (r) (),
$$ {\left\Vert f-{X}_{\sigma, r,m}(f)\right\Vert}_p\le \frac{K_r}{\sigma^r}{\left\Vert {f}^{(r)}\right\Vert}_p $$
are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.

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References

  1. 1.
    J. Favard, “Sur les méilleurs procédés d’approximation de certaines classes des fonctions par des polynomes trigonométriques,” Bull. Sci. Math., 61, 209–224, 243–256 (1937).MATHGoogle Scholar
  2. 2.
    N. I. Akhiezer and M. G. Krein, “Best approximation of differentiable periodic functions by trigonometric sums,”Dokl. AN SSSR, 15, No. 3, 107–112 (1937).Google Scholar
  3. 3.
    S. M. Nikolsky, “Approximation of functions in the mean by trigonometric polynomials,” Izv. AN SSSR, Ser. Mat., 10, 207-256 (1946).Google Scholar
  4. 4.
    M. G. Krein, “On the best approximation of continuous differentiable functions on the entire real line,” Dokl. AN SSSR, 18, No. 9, 619–623 (1938).Google Scholar
  5. 5.
    B. Nagy, “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. II. Nichtperiodischer Fall,” Ber. Verh. sӓchs. Akad. Wiss. Leipzig, 91, 3–24 (1939).MATHGoogle Scholar
  6. 6.
    N. I. Akhiezer, Lectures in the Theory of Approximation [in Russian], Moscow (1965).Google Scholar
  7. 7.
    I. J. Schoenberg, Cardinal Spline Interpolation, Second ed., SIAM (1993).Google Scholar
  8. 8.
    V. M. Tikhomirov, “Best methods of approximation and interpolation of differentiable functions in the space C[1, 1],” Mat. Sb., 80, No. 2, 290–304 (1969).Google Scholar
  9. 9.
    A. A. Ligun, “Inequalities for upper bounds of functionals,” Anal. Math., 2, No. 1, 11–40 (1976).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    N. P. Korneičuk, “Exact error bound of approximation by interpolating splines on L-metric on the classes W pr(1 ≤ p < ∞) of periodic functions,” Anal. Math., 3, No. 2, 109–117 (1977).Google Scholar
  11. 11.
    N. P. Korneichuk, Splines in Approximation Theory [in Russian], Moscow (1984).Google Scholar
  12. 12.
    N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).Google Scholar
  13. 13.
    A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
  14. 14.
    O. L. Vinogradov, “An analog of the Akhiezer–Krein–Favard sums for periodic splines of minimal defect,” Probl. Mat. Anal., Iss. 25, 29–56 (2003).Google Scholar
  15. 15.
    Sun Yongsheng and Li Chun, “Best approximation of certain classes of smooth functions on the real axis by splines of a higher order,” Mat. Zametki, 48, No. 4, 148–159 (1990).Google Scholar
  16. 16.
    G. G. Magaril-Il’yaev, “On the best approximation by splines of classes of functions on a straight line,” Trudy Mat. Inst. Steklov., 194, 148–159 (1992).MathSciNetMATHGoogle Scholar
  17. 17.
    G. G. Magaril-Il’yaev, “Mean dimension, widths, and optimal recovery of Sobolev classes of functions on the line,” Mat. Sb., 182, No. 11, 1635–1656 (1991).Google Scholar
  18. 18.
    I. J. Schoenberg, “On the remainders and the convergence of cardinal spline interpolation for almost periodic functions,” in: Studies in Spline Functions and Approximation Theory, Academic Press, New York (1976), pp. 277–303.Google Scholar
  19. 19.
    C. de Boor and I. J. Schoenberg, “Cardinal interpolation and spline functions VIII. The Budan–Fourier theorem for splines and applications,” Lect. Notes Math., 501, 1–79 (1976).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    K. Jetter, S. D. Riemenschneider, and N. Sivakumar, “Schoenberg’s exponential Euler spline curves,” Proc. Roy. Soc. Edinburgh, 118A, 21–33 (1991).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    B. M. Makarov and A. N. Podkorytov, Lectures in Real Analysis [in Russian], St.Petersburg (2011).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

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