Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory

Mathematical Methods in the Applied Sciences

Volumn 21, Issue 2, 1998, Pages 129-163

  Freden, Willi ^a   Schneider, Frank ^a  

^a France   (Germany)

Author keywords

Band limited harmonic wavelets; Dirichlet's and Neumann's problem; Exact fully discrete wavelet transform; Integral formulas; Runge Walsh approximation; Scale discrete wavelets; Scaling function; Wavelets on closed surfaces

Indexed keywords

BOUNDARY VALUE PROBLEMS; CALCULATIONS; HARMONIC ANALYSIS; INTEGRATION; SURFACES; WAVELET TRANSFORMS;

BAND LIMITED HARMONIC WAVELESS; CLOSED SURFACES; EXACT FULLY DISCRETE WAVELET TRANSFORM; INTEGRAL FORMULAS; SCALE DISCRETE WAVELETS; SCALING FUNCTION; WAVELETS;

APPROXIMATION THEORY;

EID: 0031696778     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/(sici)1099-1476(19980125)21:2<129::aid-mma942>3.0.co;2-7     Document Type: Article

References (30)

1. Bäcker, M., 'Sphärische Wavelets und ihre numerische Realisation', Diploma Thesis, University of Kaiserslautern, Geomathematics Group (1995).
2. Bayer, M., 'Spherical wavelets and their applications to linear inverse satellite problems', Diploma Thesis, University of Kaiserslautern, Geomathematics Group (1996).
3. Dahlke, S. and Maass, P., 'A continuous wavelet transform on tangent bundles of spheres, J. Four. Anal. Appl., 2, 379-396 (1994).
4. 3., Numer. Funct. Anal. Optim., 16, 19-41 (1995).
5. Driscoll, J. R., Healy, D. M., 'Computing Fourier transforms and convolutions on the 2-sphere', Adv. Appl. Math., 15, 202-250 (1996).
6. Freeden, W., 'Über eine Klasse von Integralformeln der Mathematischen Geodäsie', Veröff. Geod. Inst. RWTH Aachen, No. 27, 1979.
7. Freeden, W., 'On the approximation of external gravitational potential with closed systems of (trial) functions', Bull. Géol., 54, 1-20 (1980).
8. Freeden, W., 'Least square approximation by linear combination of (multi-) poles', Report No. 344, Department of Geodetic Science, Ohio State University, Columbus, 1983.
9. Freeden, W., 'A spline interpolation method for solving boundary-value problems of potential theory from discretely given data', Numer. Meth. Part. Diff. Equations, 3, 375-398 (1987).
10. Freeden, W. and Kersten, H., 'A constructive approximation theorem for oblique derivative problem of potential theory', Math. Meth. in the Appl. Sci., 3, 104-114 (1981).
11. Freeden, W. and Reuter, R., 'Remainder terms in numerical integration formulas of the sphere', ISNM, 61, 151-170 (1982).
12. Freeden, W. and Schreiner, M., 'Orthogonal and non-orthogonal multiresolution analysis: scale discrete and exact fully discrete expansions on the sphere', Constructive Approximation (accepted for publication).
13. Freeden, W. and Windheuser, U., 'Spherical wavelet transform and its discretization', Adv. Comput. Math., 5, 51-94 (1996).
14. Freeden, W. and Windheuser, U., Combined spaherical harmonic and wavelet expansion', Appl. Comp. Harm Anal., 4, 1-37 (1997).
15. Göttelmann, J., 'Locally supported wavelets on the sphere', Department of Math, University of Mainz, Preprint No. 14 (1996).
16. Kellogg, O. D., 'Foundations of Potential Theory, Frederick Ungar Publishing Company, New York, 1928.
17. Müller, C., Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, berlin, 1966.
18. Müller, C., Foundation of the Mathematical Theory of Electromagnetic Waves, Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 115, Springer, Berlin, 1969.
19. Narcowich, F. J. and Ward, J. D., 'Nonsatationary wavelets on the m-spahere for scattered data', Appl. Comp. Harm. Anal., 3, 324-336.
20. Potts, D. and Tasche, M., 'Interpolatory wavelets on the sphere', in: Approximation Theory VIH (C. K. Chui, L. L. Schumaker, eds.) 1995, pp. 335-342 (1996).
21. Rapp, R. H., Wang, M. and Pavlis, N., The Ohio state 1991 geopotential and sea surface topography harmonic coefficient model, Report No. 410, Department of Geodetic Science, Ohio University Columbus, 1991.
22. Runge, C., 'zur Theorie der eindeutigen analytischen Funktionen', Acta Math., 6, 229-244 (1885).
23. Schneider, F., 'Inverse problems in satellite geodesy and thir approximate solution by splines and wavelets', Dissertation., University of Kaiserslautern, Geomathematics Group (1997).
24. Schreiner, M., 'A pyramid scheme for spherical wavelets', AGTM-Report 170, University of Kaiserslautern, Geomathematics Group (1997) to appear.
25. Schröder, P. and Sweldens, W., 'Spherical wavelets: efficiently representing functions on the sphere, in: Computer Graphics (SIGGRAPH '95 Proc.,) 1995.
26. Torresani, B., 'Phase space decomposition: local Fourier analysis on spheres'. Preprint (1993).
27. Walsh, J. L., 'The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions', Am. Math. Soc., 35, 499-544 (1929).
28. Walsh, J. L., 'Approximation by polynomials: uniform convergence as implid by mean convergense', Proc. Nat. Acad. Sci. U.S.A., 55, 20-25 (1996).
29. Windheuser, U., 'Sphärische Wavelets: Theorie and Anwendung in der Physikalischen Geodäsie', Dissertation, University of Kaiserslautern, Geomathematics Group (1995).
30. Ymabe, H., 'On an extension of Helly's theorem', Osaka Math. J., 2, 15-22 (1959).