Skip to main content
Springer Nature Link
Log in

Linear k-Monotonicity Preserving Algorithms and Their Approximation Properties

  • Conference paper
  • First Online:
Mathematical Aspects of Computer and Information Sciences (MACIS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9582))

Abstract

This paper examines the problem of finding the linear algorithm (operator) of finite rank n (i.e. with a n-dimensional range) which gives the minimal error of approximation of identity operator on some set over all finite rank n linear operators preserving the cone of k-monotonicity functions. We introduce the notion of linear relative (shape-preserving) n-width and find asymptotic estimates of linear relative n-widths for linear operators preserving k-monotonicity in the space \(C^k[0,1]\). The estimates show that if linear operator with finite rank n preserves k-monotonicity, the degree of simultaneous approximation of derivative of order \(0\le i\le k\) of continuous functions by derivatives of this operator cannot be better than \(n^{-2}\) even on the set of algebraic polynomials of degree \(k+2\) (as well as on bounded subsets of Sobolev space \(W^{(k+2)}_\infty [0,1]\)).

The results were obtained within the framework of the state task of Russian Ministry of Education and Science (project 1.1520.2014K).

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Barnabas, B., Coroianu, L., Gal, S.G.: Approximation and shape preserving properties of the Bernstein operator of max-product kind. Int. J. Math. Math. Sci. 2009, 26 (2009). Article ID 590589

    MathSciNet  MATH  Google Scholar 

  2. Boytsov, D.I., Sidorov, S.P.: Linear approximation method preserving \(k\)-monotonicity. Siberian Electron. Math. Rep. 12, 21–27 (2015)

    MATH  Google Scholar 

  3. Cai, Y., Judd, K.L.: Shape-preserving dynamic programming. Math. Meth. Oper. Res. 77, 407–421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cárdenas-Morales, D., Garrancho, P., Raşa, I.: Bernstein-type operators which preserve polynomials. Comput. Math. Appl. 62, 158–163 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cárdenas-Morales, D., Muñoz-Delgado, F.J.: Improving certain Bernstein-type approximation processes. Math. Comput. Simul. 77, 170–178 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cárdenas-Morales, D., Muñoz-Delgado, F.J., Garrancho, P.: Shape preserving approximation by Bernstein-type operators which fix polynomials. Appl. Math. Comput. 182, 1615–1622 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Floater, M.S.: On the convergence of derivatives of Bernstein approximation. J. Approximation Theor. 134(1), 130–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gal, S.G.: Shape-Preserving Approximation by Real and Complex Polynomials. Springer, Boston (2008)

    Book  MATH  Google Scholar 

  9. Gilewicz, J., Konovalov, V.N., Leviatan, D.: Widths and shape-preserving widths of sobolev-type classes of s-monotone functions. J. Approx. Theor. 140(2), 101–126 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gonska, H.H.: Quantitative Korovkin type theorems on simultaneous approximation. Math. Z. 186(3), 419–433 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Konovalov, V., Leviatan, D.: Shape preserving widths of Sobolev-type classes of \(k\)-monotone functions on a finite interval. Isr. J. Math. 133, 239–268 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Konovalov, V., Leviatan, D.: Shape-preserving widths of weighted Sobolev-type classes of positive, monotone, and convex functions on a finite interval. Constructive Approximation 19, 23–58 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Konovalov, V.N.: Estimates of diameters of Kolmogorov type for classes of differentiable periodic functions. Mat. Zametki 35(3), 369–380 (1984)

    MathSciNet  MATH  Google Scholar 

  14. Kopotun, K.A., Leviatan, D., Prymak, A., Shevchuk, I.A.: Uniform and pointwise shape preserving approximation by algebraic polynomials. Surv. Approximation Theor. 6, 24–74 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Kopotun, K., Shadrin, A.: On k-monotone approximation by free knot splines. SIAM J. Math. Anal. 34, 901–924 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kvasov, B.I.: Methods of Shape Preserving Spline Approximation. World Scientific Publ. Co., Pte. Ltd., Singapore (2000)

    Book  MATH  Google Scholar 

  17. Kvasov, B.: Monotone and convex interpolation by weighted cubic splines. Comput. Math. Math. Phys. 53(10), 1428–1439 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lewicki, G., Prophet, M.P.: Minimal shape-preserving projections onto \(\pi _n\): generalizations and extensions. Numer. Funct. Anal. Optim. 27(7–8), 847–873 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Muñoz-Delgado, F.J., Ramírez-González, V., Cárdenas-Morales, D.: Qualitative Korovkin-type results on conservative approximation. J. Approx. Theor. 94, 144–159 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pinkus, A.: nWidths in Approximation Theory. Springer, Heidelberg (1985)

    Book  Google Scholar 

  21. Popoviciu, T.: About the Best Polynomial Approximation of Continuous Functions. Mathematical Monography. Sect. Mat. Univ. Cluj, (In Romanian), fasc. III (1937)

    Google Scholar 

  22. Pǎltǎnea, R.: A generalization of Kantorovich operators and a shape-preserving property of Bernstein operators. Bull. Transilvania Univ. of Braşov, Ser. III: Math. Inf. Phys. 5(54), 65–68 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Shevaldin, V.T.: Local Approximation by Splines. UrO RAN, Ekaterinburg (2014)

    MATH  Google Scholar 

  24. Shevaldin, V., Strelkova, E., Volkov, Y.: Local approximation by splines with displacement of nodes. Siberian Adv. Math. 23(1), 69–75 (2013)

    Article  MATH  Google Scholar 

  25. Sidorov, S.P.: On the order of approximation by linear shape-preserving operators of finite rank. East J. Approximations 7(1), 1–8 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Sidorov, S.P.: Basic properties of linear shape-preserving operators. Int. J. Math. Anal. 5(37–40), 1841–1849 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Sidorov, S.P.: Linear relative \(n\)-widths for linear operators preserving an intersection of cones. Int. J. Math. Math. Sci. 2014, 7 (2014). Article ID 409219

    Article  MathSciNet  MATH  Google Scholar 

  28. Sidorov, S.: On the saturation effect for linear shape-preserving approximation in Sobolev spaces. Miskolc Mathematical Notes 16 (2015)

    Google Scholar 

  29. Vasiliev, R.K., Guendouz, F.: On the order of approximation of continuous functions by positive linear operators of finite rank. J. Approx. Theor. 69(2), 133–140 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vidensky, V.S.: On the exact inequality for linear positive operators of finite rank. Dokl. Akad. Nauk Tadzhik. SSR 24, 715–717 (1981)

    MathSciNet  Google Scholar 

  31. Volkov, Y.S., Shevaldin, V.T.: Shape preserving conditions for quadratic spline interpolation in the sense of subbotin and marsden. Trudy Inst. Mat. i Mekh. UrO RAN 18(4), 145–152 (2012)

    Google Scholar 

  32. Volkov, Y., Bogdanov, V., Miroshnichenko, V., Shevaldin, V.: Shape-preserving interpolation by cubic splines. Math. Notes 88(5–6), 798–805 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Volkov, Y., Galkin, V.: On the choice of approximations in direct problems of nozzle design. Comput. Math. Math. Phys. 47(5), 882–894 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, Saratov State University, Saratov, Russian Federation

    S. P. Sidorov

Authors
  1. S. P. Sidorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. P. Sidorov .

Editor information

Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Hamburg University of Technology, Hamburg, Germany

    Siegfried M. Rump

  3. New York University, New York, New York, USA

    Chee K. Yap

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Sidorov, S.P. (2016). Linear k-Monotonicity Preserving Algorithms and Their Approximation Properties. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-32859-1_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32858-4

  • Online ISBN: 978-3-319-32859-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics