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Some Notes on Two-Scale Difference Equations

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Functional Equations and Inequalities

Part of the book series: Mathematics and Its Applications ((MAIA,volume 518))

Abstract

In this paper, we continue our considerations in [2] on two-scale difference equations, mainly with respect to continuous solutions. Moreover, we study refinable step functions and piecewise polynomials. Also, solutions with noncompact support are considered. New algorithms for the approximative computation of continuous solutions are derived.

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References

  1. K. Baron, A. Simon and P. Volkman, Solutions d’une equation fonctionelle dans l’espace des distributions temperees, C.R. Acad. Sci. Paris 319 Série I (1994), 1249–1252.

    Google Scholar 

  2. L. Berg and G. Plonka, Compactly supported solutions of two-scale difference equations, Linear Algebra Appl. 275-276 (1998), 49–75.

    Article  MathSciNet  Google Scholar 

  3. A. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary Subdivision, Mem. Amer. Math. Soc. 453 (1991), 1–186.

    MathSciNet  Google Scholar 

  4. D. Colella and C. Heil, Characterizations of scaling functions, I. Continuous solutions, J. Math. Anal. Appl., 15 (1994), 496–518.

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

    Book  MATH  Google Scholar 

  6. I. Daubechies and J. Lagarias, Two-scale difference equations: I. Global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388–1410.

    Article  MathSciNet  MATH  Google Scholar 

  7. I. Daubechies and J. Lagarias, Two-scale difference equations: II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), 1031–1079.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescating, J. Phys. A, Math. Gen. 29, No.15 (1996), 4537–4547.

    Article  MathSciNet  MATH  Google Scholar 

  9. Förg-Rob, On a problem of R. Schilling, part 1, Math. Pannonica 5 (1995), 29–65.

    Google Scholar 

  10. R. Girgensohn, Nowhere differentiable solutions of a system of functional equations, Aeq. Math. 47 (1994), 89–59.

    MathSciNet  MATH  Google Scholar 

  11. D. H. Hyers, G. Isac, and Th. M. Rassias (eds.), Stability of functional equations in several variables, Birkhäuser Verlag, 1998.

    Google Scholar 

  12. R. Q. Jia, Refinable shift-invariant spaces: From splines to wavelets, in: Approximation Theory VIII (C. K. Chui and L. L. Schumaker, eds.), World Scientific Publishing Co., 1995, 1–30.

    Google Scholar 

  13. R. Q. Jia and J. Z. Wang, Stability and linear independence associated with wavelet decompositions, Proc. Amer. Math. Soc. 117 (1993), 1115–1124.

    Article  MathSciNet  MATH  Google Scholar 

  14. K. S. Lau and J. Z. Wang, Characterization of L p-solutions for the two-scale dilation equations, SIAM J. Math. Anal. 26 (1995), 1018–1046.

    Article  MathSciNet  MATH  Google Scholar 

  15. W. Lawton, S. L. Lee, and Z. Shen,Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995), 137–146.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841–870.

    Article  MathSciNet  Google Scholar 

  17. J. Morawiec, On continuous solution of a problem of R. Schilling, Results in Mathematics 27 (1995), 381–386.

    MathSciNet  MATH  Google Scholar 

  18. S. Paganoni Marzegalli, One parameter system of functional equations, Aeq. Math. 47 (1994), 50–59.

    Article  MATH  Google Scholar 

  19. Th. M. Rassias, ed., Topics in Mathematical Analysis. A volume dedicated to the memory of A. L. Cauchy, World Scientific Publishing Company, 1989.

    Google Scholar 

  20. Th. M. Rassias and J. Šimša, Finite Sums Decompositions in Mathematical Analysis, J. Wiley & Sons, Chichester, 1995.

    Google Scholar 

  21. G. Strang, Eigenvalues of Toeplitz matrices with 1 x 2 blocks, Z. Angew. Math. Mech. 76 (1996) Suppl. 2, 37–39.

    MathSciNet  MATH  Google Scholar 

  22. W. Volk, Properties of subspaces generated by an infinitely often differentiate function and its translates, Z. Angew. Math. Mech. 76 (1996) Suppl. 1, 575–576.

    MATH  Google Scholar 

  23. I. M. Winogradow, Elemente der Zahlentheorie, Berlin 1955.

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Rostock, D-18051, Rostock, Germany

    Lothar Berg & Gerlind Plonka

Authors
  1. Lothar Berg
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  2. Gerlind Plonka
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© 2000 Springer Science+Business Media Dordrecht

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Berg, L., Plonka, G. (2000). Some Notes on Two-Scale Difference Equations. In: Functional Equations and Inequalities. Mathematics and Its Applications, vol 518. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4341-7_2

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  • DOI: https://doi.org/10.1007/978-94-011-4341-7_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5869-8

  • Online ISBN: 978-94-011-4341-7

  • eBook Packages: Springer Book Archive

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