Skip to main content
SpringerLink
Log in

Shift-invariant spaces and linear operator equations

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we investigate the structure of finitely generated shift-invariant spaces and solvability of linear operator equations. Fourier trans-forms and semi-convolutions are used to characterize shift-invariant spaces. Criteria are provided for solvability of linear operator equations, including linear partial difference equations and discrete convolution equations. The results are then applied to the study of local shift-invariant spaces. Moreover, the approximation order of a local shift-invariant space is characterized under some mild conditions on the generators.

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. C. de Boor,Topics in multivariate approximation theory, inTopics in Numerical Analysis (P. Turner, ed.), Lecture Notes in Mathematics965, Springer-Verlag, Berlin, 1982, pp. 39–78.

    Chapter  Google Scholar 

  2. C. de Boor,The polynomials in the linear span of integer translates of a compactly supported function, Constructive Approximation3 (1987), 199–208.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. de Boor, R. DeVore and A. Ron,Approximation from shift-invariant subspaces of L 2 (ℝd), Transactions of the American Mathematical Society341 (1994), 787–806.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. de Boor, R. DeVore and A. Ron,The structure of finitely generated shift-invariant subspaces in L 2 (ℝd), Journal of Functional Analysis119 (1994), 37–78.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Dahmen and C. A. Micchelli,Translates of multivariate splines, Linear Algebra and its Applications52 (1983), 217–234.

    MathSciNet  Google Scholar 

  6. L. Ehrenpreis,Fourier Analysis in Several Complex Variables, Tracts in Mathematics17, Wiley-Interscience, New York, 1970.

    MATH  Google Scholar 

  7. G. B. Folland,Real Analysis, Wiley, New York, 1984.

    MATH  Google Scholar 

  8. H. Helson,Lectures on Invariant Subspaces, Academic Press, New York, 1964.

    MATH  Google Scholar 

  9. R. Q. Jia,The Toeplitz theorem and its applications to Approximation Theory and linear PDE’s, Transactions of the American Mathematical Society347 (1995), 2585–2594.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Q. Jia,Refinable shift-invariant spaces: from splines to wavelets, inApproximation Theory VIII, Vol. 2 (C. K. Chui and L. L. Schumaker, eds.), Word Scientific Publishing Co., Inc., 1995, pp. 179–208.

  11. R. Q. Jia,Shift-invariant spaces on the real line, Proceedings of the American Mathematical Society125 (1997), 785–793.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Q. Jia and J. J. Lei,Approximation by multiinteger translates of functions having global support, Journal of Approximation Theory72 (1993), 2–23.

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Q. Jia and C. A. Micchelli,On linear independence of integer translates of a finite number of functions, Proceedings of the Edinburgh Mathematical Society36 (1992), 69–85.

    Article  MathSciNet  Google Scholar 

  14. R. Q. Jia and C. A. Micchelli,Using the refinement equation for the construction of pre-wavelets II: Powers of two, inCurves and Surfaces (P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 209–246.

    Google Scholar 

  15. R. Q. Jia, S. Riemenschneider and Z. W. Shen,Solvability of systems of linear operator equations, Proceedings of the American Mathematical Society120 (1994), 815–824.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. J. Lei, R. Q. Jia and E. W. Cheney,Approximation from shift-invariant spaces by integral operators, SIAM Journal of Mathematical Analysis28 (1997), 481–498.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Ron,A characterization of the approximation order of multivariate spline spaces, Studia Mathematica98 (1991), 73–90.

    MATH  MathSciNet  Google Scholar 

  18. M. Schechter,Principles of Functional Analysis, Academic Press, New York, 1971.

    MATH  Google Scholar 

  19. I. J. Schoenberg,Contributions to the problem of approximation of equidistant data by analytic functions, Quarterly of Applied Mathematics4 (1946), 45–99, 112–141.

    MathSciNet  Google Scholar 

  20. G. Strang and G. Fix,A Fourier analysis of the finite-element variational method, inConstructive Aspects of Functional Analysis (G. Geymonat, ed.), C.I.M.E., 1973, pp. 793–840.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Canada

    Rong-Qing Jia

Authors
  1. Rong-Qing Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong-Qing Jia.

Additional information

Supported in part by NSERC Canada under Grant OGP 121336.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jia, RQ. Shift-invariant spaces and linear operator equations. Isr. J. Math. 103, 259–288 (1998). https://doi.org/10.1007/BF02762276

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02762276

Keywords

Search

Navigation