Skip to main content
SpringerLink
Log in
Book cover

Linear Spaces and Approximation / Lineare Räume und Approximation pp 433–448Cite as

Convergence of Spline Expansions

  • Chapter

Abstract

The aim of this lecture is to give a survey on spline expansions and their applications to function spaces. It concerns the questions for spline systems of being bases, unconditional bases, equivalent bases, bases with shift property, interpolating bases and the a.e. convergence of the spline expansions. These properties of the spline systems are discussed in various classical function spaces on the unit interval, one-dimensional torus, disc, cube, multi-dimensional torus and polydisc. The lecture covers mainly the works related to the author’s own investigations.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Billard, P., Sur la convergence presque partout des séries de Fourier-Walsh de fonctions de l’espace L2 (0,1). Studia Math. 28(1967), 363–388.

    Google Scholar 

  2. Bockariev, S.V., Existence of basis in the space of analytic functions in the disc and some properties of the Franklin system. Mat. Sbornik 95(137), (1974), 3–18 (in Russian).

    Google Scholar 

  3. Bockariev, S.V., Some inequalities for the Franklin series. Analysis Mathematica 1 (1975), 249–257.

    Article  Google Scholar 

  4. Bockariev, S.V., Divergent on a set of positive measure Fourier series for arbitrary bounded orthonormal set. Mat. Sbornik 98(140), (1975), 436–449 (in Russian).

    Google Scholar 

  5. De Boor, C., Odd-degree spline interpolation at a biinfinite knot sequence. Approximation Theory. Lecture Notes in Mathematics 556, Springer-Verlag, Berlin 1976, 30–53.

    Google Scholar 

  6. Ciesielski, Z., Properties of the orthonormal Franklin system. Studia Math. 23(1963), 141–157.

    Google Scholar 

  7. Ciesielski, Z., Properties of the orthonormal Franklin system, II. Studia Math. 27(1966), 289–323.

    Google Scholar 

  8. Ciesielski, Z., A bounded orthonormal system of polygonals. Studia Math. 31(1968), 339–346.

    Google Scholar 

  9. Ciesielski, Z., A construction of basis in C(1)(I2). Studia Math. 33(1969), 243–247.

    Google Scholar 

  10. Ciesielski, Z., Constructive function theory and spline systems. Studia Math. 53(1975), 277–302.

    Google Scholar 

  11. Ciesielski, Z., Bases and approximation by splines. Proc. Intern. Congr. of Mathematicians, Vancouver, 1974, 47–51.

    Google Scholar 

  12. Ciesielski, Z., Equivalence, unconditionality and convergence a.e. of spline bases in Lp spaces. Approximation Theory, Banach Center Publications vol. 4 (to appear).

    Google Scholar 

  13. Ciesielski, Z. and Domsta, J., Construction of an orthonormal basis in Cm (Id) and W mp (Id). Studia Math. 41(1972), 211–224.

    Google Scholar 

  14. Ciesielski, Z., Simon, P. and Sjölin, P., Equivalence of Haar and Franklin bases in L spaces. Studia Math. 60(1977), 195–210.

    Google Scholar 

  15. Ciesielski, Z. and Kwapiéfi, S., Some properties of the Haar, Walsh-Paley, Franklin and the bounded polygonal orthonormal bases in Lp spaces. Commen-tationes Mathematicae (to appear).

    Google Scholar 

  16. Demko, S., Inverses of band matrices and local convergence of spline projections. SIAM J. Numer. Anal. (to appear).

    Google Scholar 

  17. Domsta, J., A Theorem on B-splines. Studia Math. 41(1972), 291–314.

    Google Scholar 

  18. Ciesielski, Z., Approximation by spline interpolating bases. Studia Math. 58(1976), 223–237.

    Google Scholar 

  19. Ciesielski, Z., A Theorem on B-splines.II. The periodic case. Bull. Acad. Polon. Sci., Série math. astr. 24(1976), 1077–1084.

    Google Scholar 

  20. Faber, G., Über die Orthogonalfunktionen des Herrn Haar. Jahresber. Deutsch. Math. Verein. 19(1910), 104–112.

    Google Scholar 

  21. Franklin, Ph., A set of continuous orthogonal functions. Math. Ann. 100(1928), 522–529.

    Article  Google Scholar 

  22. Haar, A., Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69(1910), 331–371.

    Article  Google Scholar 

  23. Marcinkiewicz, J., Quelques théorèmes sur les séries orthogonales. Ann. Soc. Polon. Math. 16(1937), 107–115.

    Google Scholar 

  24. Paley, R.E.A.C., A remarkable series of orthogonal functions. Proc. London Math. Soc. 34(1932), 241–279.

    Article  Google Scholar 

  25. Ropela, S., Spline bases in Besov spaces. Bull. Acad. Polon. Sci., Serie math. astr. 24.(1976), 319–325.

    Google Scholar 

  26. Ropela, S., Decomposition lemma and unconditional spline bases. Bull. Acad. Polon. Sci., Serie math. astr. 24.(1976), 467–470.

    Google Scholar 

  27. Ropela, S., Properties of bounded orthonormal spline bases. Approximation Theory. Banach Center Publications, vol. 4 (to appear).

    Google Scholar 

  28. Schauder, J., Zur Theorie stetiger Abbildungen in Funktionalräumen. Math. Z. 26(1927), 47–65.

    Article  Google Scholar 

  29. Schauder, J., Eine Eigenschaft des Haarschen Orthogonalsystems. Math. Z. 28(1928), 317–320.

    Article  Google Scholar 

  30. Schipp, F., On a.e. convergence of expansions with respect to a bounded orthonormal system of polygonals. Studia Math. 58(1976), 287–290.

    Google Scholar 

  31. Schonefeld, S., Schauder bases in spaces of differentiable functions. Bull. Amer. Math. Soc. 75(1969), 586–590.

    Article  Google Scholar 

  32. Schonefeld, S., Schauder bases in the Banach spaces Ck(Tq). Trans. Amer. Math. Soc. 165(1971), 309–318.

    Google Scholar 

  33. Sjölin, P., An inequality of Paley and convergence a.e. of Walsh-Fourier series. Arkiv Math. 7(1968), 551–570.

    Article  Google Scholar 

  34. Sjölin, P., The Haar and Franklin systems are not equivalent in L. Bull. Acad. Polon. Sci., Serie math. astr. (to appear).

    Google Scholar 

  35. Subbotin, Yu. N., Spline approximation and smooth bases in C(0,2π). Mat. Zametki 12(1972), 43–51 (in Russian).

    Google Scholar 

  36. Subbotin, Yu. N., Applications of splines in approximation theory. In: Linear Operators and Approximation. ISNM, vol.20, Birkhäuser Verlag, Basel 1972, 405–418 (in Russian).

    Google Scholar 

  37. Subbotin, Yu. N., Approximation properties of splines. Approximation Theory. Lecture Notes in Math. 556, Springer-Verlag, Berlin 1976, 416–427.

    Google Scholar 

  38. Walsh, J.L., A closed set of normal orthogonal functions. Amer. J. Math. 45(1923), 5–24.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Polish Academy of Sciences, Sopot, Poland

    Zbigniew Ciesielski

Authors
  1. Zbigniew Ciesielski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, 5100, Aachen, Western Germany

    P. L. Butzer

  2. József Attila Tudományegyetem, Aradi vértanúk tere 1, Szeged, Hungary

    B. Szökefalvi-Nagy

Rights and permissions

Reprints and permissions

Copyright information

© 1978 Birkhäuser Verlag Basel

About this chapter

Cite this chapter

Ciesielski, Z. (1978). Convergence of Spline Expansions. In: Butzer, P.L., Szökefalvi-Nagy, B. (eds) Linear Spaces and Approximation / Lineare Räume und Approximation. International Series of Numerical Mathematics / Intermationale Schriftenreihe zur Numberischen Mathematik / Sùrie Internationale D’analyse Numùruque, vol 40. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7180-8_38

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7180-8_38

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-7643-0979-4

  • Online ISBN: 978-3-0348-7180-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation