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Wavelet approximation methods for pseudodifferential equations: I Stability and convergence

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References

  1. Agranovich, M.S.: On elliptic pseudodifferential operators on a closed curve. Trans. Mosc. Math. Soc.47, 23–74 (1985)

    Google Scholar 

  2. Arnold, D.N., Wendland, W.L.: The convergence of spline collocation for strongly elliptic equations on curves. Numer. Math.47, 317–341 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Wavelet analysis and the numerical solutions of partial differential equations. Progress Report. Cambridge, MA. Aware Inc.: June 1990

  4. Ben-Artzi, A., Ron, A.: On the integer translates of a compactly supported function: dual bases and linear projectors. SIAM J. Math. Anal.21, 1550–1562 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math.XLIV, 141–183 (1991)

    MathSciNet  Google Scholar 

  6. de Boor, C., Höllig, K., Riemenschneider, S.: Box-splines. Berlin Heidelberg New York: Springer 1992

    Google Scholar 

  7. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision. Mem. Am. Math. Math. Soc.93, no. 453 (1991)

    Google Scholar 

  8. Chui, C.K., Stöckler, J., Ward, J.D.: Compactly supported box spline wavelets. Approx. Theory Appl.8, 77–100 (1992)

    MATH  MathSciNet  Google Scholar 

  9. Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math.45, 485–560 (1992)

    MATH  MathSciNet  Google Scholar 

  10. Coifman, R., Meyer, Y.: Au-delà des Opérateurs Pseudo-Différentiels. (Astérisque, no. 57) Paris: Soc. Math. Fr. 1978

    MATH  Google Scholar 

  11. Costabel, M.: Principles of boundary element methods. Comp. Phys. Rep.6, 243–274 (1987)

    Article  MathSciNet  Google Scholar 

  12. Costabel, M., McLean, W.: Spline collocation for strongly elliptic equations on the torus. Numer. Math.62, 511–538 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Costabel, M., Wendland, W.: Strong ellipticity of boundary integral operators. J. Reine Angew. Math.372, 39–63 (1986)

    MathSciNet  Google Scholar 

  14. Dahmen, W., Kunoth, A.: Multilevel preconditioning. Numer. Math.63, 315–344 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dahmen, W., Micchelli, C.A.: On the local linear independence of the translates of a box spline. Studia Math.82, 243–263 (1985)

    MATH  MathSciNet  Google Scholar 

  16. Dahmen, W., Micchelli, C.A.: Biorthogonal wavelet expansion. (in preparation)

  17. Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations. II. Matrix compression and fast solution. Adv. Comput. Math.1, 259–335 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math.41, 909–996 (1988)

    MATH  MathSciNet  Google Scholar 

  19. DeVore, R., Jawerth, B., Popov, V.: Compression of wavelet decompositions. Am. J. Math.114, 737–785 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. DeVore, R., Popov, V.: Interpolation of Besov spaces. Trans. Am. Math. Soc.305, 397–414 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Glowinski, R.R., Lawton, W.M., Ravachol, M., Tenenbaum, E.: Wavelet solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Cambridge, MA: Aware Inc. (Preprint 1989)

  22. Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math.54, 463–491 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols. 1–4. (Grundlehren Math. Wiss.) Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  24. Jaffard, S.: Wavelet methods for fast resolution of elliptic problems. (Preprint 1990)

  25. Jia, R.Q., Micchelli, C.A.: Using the refinement equation for the construction of prewavelets II: Powers of two. In: Laurent, P., Le Méhanté, A., Schumaker, L.L. (eds.) pp. 209–246. Curves and Surfaces. New York: Academic Press 1991

    Google Scholar 

  26. Johnen, H., Scherer, K.: On thek-functional and moduli of continuity and some applications. In: Schempp, W., Zeller, K. (eds.) Constructive Theory of Functions of Several Variables. (Lect. Notes Math., vol. 571, pp. 119–140). Berlin Heidelberg New York: Springer 1977

    Chapter  Google Scholar 

  27. Kohn, J., Nirenberg, L.: On the algebra of pseudo-differential operators. Commun. Pure Appl. Math.,18, 269–305 (1965)

    MATH  MathSciNet  Google Scholar 

  28. Kumano-go, H.: Pseudodifferential Operators. Boston: MIT Press 1981

    Google Scholar 

  29. Mallat, S.: Multiresolution approximation and wavelet orthonormal bases ofL 2. Trans. Am. Math. Soc.315, 69–88 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  30. McLean, W.: Local and global descriptions of periodic pseudodifferential operators. Math. Nachr.150, 151–161 (1991)

    MATH  MathSciNet  Google Scholar 

  31. McLean, W.: Periodic pseudodifferential operators and periodic function spaces. Technical Report. Australia: Univ. of New South Wales 1989

    Google Scholar 

  32. Meyer, Y.: Ondelettes et Opèrateurs 1: Ondelettes. Paris: Hermann 1990

    Google Scholar 

  33. Mikhlin, S., Prössdorf, S.: Singular Integral Operators, Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  34. Prössdorf, S.: A localization principle in the theory of finite element methods. In: Friedrich, V. et al. (eds.) Probleme und Methoden der Mathematischen Physik. 8. Tagung in Karl-Marx-Stadt, 1983. (Teubner-Texte Math., Bd. 63, pp. 169–1777) Leipzig: Teubner 1984

    Google Scholar 

  35. Prössdorf, S.: Ein Lokalisierungsprinzip in der Theorie der Spline-Approximationen und einige Anwendungen. Math. Nachr.119, 239–255 (1984)

    MATH  MathSciNet  Google Scholar 

  36. Prössdorf, S., Rathsfeld, A.: A spline collocation method for singular integral equations with piecewise continuous coefficients. Integral Equations Oper. Theory7, 536–560 (1984)

    Article  MATH  Google Scholar 

  37. Prössdorf, S., Rathsfeld, A.: Mellin techniques in the numerical analysis for one-dimensional singular integral equations Report R-MATH-06/88. Berlin: Karl-Weierstrass-Institut 1988

    Google Scholar 

  38. Prössdorf, S., Schmidt, G.: A finite element collocation method for singular integral equations. Math. Nachr.100, 33–60 (1981)

    MATH  MathSciNet  Google Scholar 

  39. Prössdorf, S., Schneider, R.: A spline collocation method for multi-dimensional strongly elliptic pseudodifferential operators of order zero. Integral Equations Oper. Theory4, 399–435 (1991)

    Article  Google Scholar 

  40. Prössdorf, S., Schneider, R.: Spline approximation methods for multidimensional periodic pseudodifferential equations. Integral Equations Oper. Theory15, 626–672 (1992)

    Article  MATH  Google Scholar 

  41. Prössdorf, S., Schneider, R.: Pseudodifference operators—A symbolic calculus for approximation methods for periodic pseudodifferential equations. Preprint TH-Fb Math. Darmstadt 1991

  42. Prössdorf, S., Silbermann, B.: Numerical Analysis for Integral and Related Operator Equations. Berlin: Akademie Verlag, 1991, and Boston Basel Stuttgart: Birkhäuser 1991

    Google Scholar 

  43. Riemenschneider, S., Shen, Z.: Wavelets and pre-wavelets in low dimensions. J. Approx. Theory71, 18–38 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  44. Schmidt, G.: On ∈-collocation for pseudodifferential equations on closed curves. Math. Nachr.126, 183–196 (1986)

    MATH  MathSciNet  Google Scholar 

  45. Schmidt, G.: Splines und die näherungsweise Lösung von Pseudodifferentialgleichungen auf geschlossenen Kurven. Report R-MATH-09/86. Berlin: Karl-Weierstraß-Institut 1986

    MATH  Google Scholar 

  46. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  47. Silbermann, B.: Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren. Math. Nachr.104, 137–146 (1981)

    MATH  MathSciNet  Google Scholar 

  48. Silbermann, B.: Lokale Theorie des Reduktionsverfahrens für singuläre Integraloperatoren. Z. Anal. Anwend.1, 45–56 (1982)

    MATH  MathSciNet  Google Scholar 

  49. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970

    MATH  Google Scholar 

  50. Taylor, M.: Pseudodifferential operators, Princeton, NJ: Princeton University Press 1981

    MATH  Google Scholar 

  51. Triebel, H.: Interpolation Theory, Function Spaces and Differential Operators. Amsterdam: North-Holland 1978

    Google Scholar 

  52. Wendland, W.L.: On some mathematical aspects of boundary element methods for elliptic problems. In: Whiteman, J. (ed.) Mathematics of Finite Elements and Applications V, pp. 193–227. London: Academic Press 1985

    Google Scholar 

  53. Wendland, W.L.: Strongly elliptic boundary integral equations. In: Iserles, A., Powell, M. (eds.) The State of the Art in Numerical Analysis, pp. 511–562, Oxford: Clarendon 1987

    Google Scholar 

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Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52056, Aachen, Germany

    W. Dahmen

  2. Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 5-7, D-10117, Berlin, Germany

    S. Prössdorf

  3. Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstrasse 7, D-64289, Darmstadt, Germany

    R. Schneider

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  1. W. Dahmen
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  2. S. Prössdorf
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  3. R. Schneider
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The third author has been supported by a grant of Deutsche Forschungsgemeinschaft undergrant number Ko 634/32-1

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Dahmen, W., Prössdorf, S. & Schneider, R. Wavelet approximation methods for pseudodifferential equations: I Stability and convergence. Math Z 215, 583–620 (1994). https://doi.org/10.1007/BF02571732

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