Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Let σ > 0, m, r ∈ ℕ, m ≥ r, 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 (r) 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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).

    MATH  Google Scholar 

  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. S. M. Nikolsky, “Approximation of functions in the mean by trigonometric polynomials,” Izv. AN SSSR, Ser. Mat., 10, 207-256 (1946).

  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. B. Nagy, “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. II. Nichtperiodischer Fall,” Ber. Verh. sӓchs. Akad. Wiss. Leipzig, 91, 3–24 (1939).

    MATH  Google Scholar 

  6. N. I. Akhiezer, Lectures in the Theory of Approximation [in Russian], Moscow (1965).

  7. I. J. Schoenberg, Cardinal Spline Interpolation, Second ed., SIAM (1993).

  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).

  9. A. A. Ligun, “Inequalities for upper bounds of functionals,” Anal. Math., 2, No. 1, 11–40 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  10. N. P. Korneičuk, “Exact error bound of approximation by interpolating splines on L-metric on the classes W r p (1 ≤ p < ∞) of periodic functions,” Anal. Math., 3, No. 2, 109–117 (1977).

  11. N. P. Korneichuk, Splines in Approximation Theory [in Russian], Moscow (1984).

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

    Google Scholar 

  13. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  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).

  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).

  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).

    MathSciNet  MATH  Google Scholar 

  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. 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.

  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).

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Jetter, S. D. Riemenschneider, and N. Sivakumar, “Schoenberg’s exponential Euler spline curves,” Proc. Roy. Soc. Edinburgh, 118A, 21–33 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  21. B. M. Makarov and A. N. Podkorytov, Lectures in Real Analysis [in Russian], St.Petersburg (2011).

Download references

Author information

Authors and Affiliations

  1. St.Petersburg State University, St.Petersburg, Russia

    O. L. Vinogradov & A. V. Gladkaya

Authors
  1. O. L. Vinogradov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Gladkaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to O. L. Vinogradov or A. V. Gladkaya.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 440, 2015, pp. 8–35.

Translated by O. L. Vinogradov and A. V. Gladkaya.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vinogradov, O.L., Gladkaya, A.V. A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators. J Math Sci 217, 3–22 (2016). https://doi.org/10.1007/s10958-016-2950-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-016-2950-7

Search

Navigation