Abstract
We propose an algorithm for the efficient numerical computation of the periodic Hilbert transform. The function to be transformed is represented in a basis of spline wavelets in Sobolev spaces. The underlying grids have a hierarchical structure which is locally refined during computation according to the behavior of the involved functions. Under appropriate assumptions, we prove that the algorithm can deliver a result with prescribed accuracy. Several test examples illustrate how the method works in practice.
Mathematics Subject Classification (2010). 65T60, 65R10, 44A15.
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References
K. Bittner. Fast algorithms for periodic spline wavelets on sparse grids. SIAM J. Sci. Comput., 20(4):1192–1213, 1999
S. Börm and W. Hackbusch. Hierarchical quadrature for singular integrals. Computing, 74(2):75–100, 2005
C.K. Chui. An introduction to wavelets. Wavelet Analysis and Its Applications. 1. Boston, MA: Academic Press, 1992
A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math., 45(5):485–560, 1992
W. Dahmen and A. Kunoth. Multilevel preconditioning. Numer. Math., 63(3):315– 344, 1992
W. Dahmen, S. Prössdorf, and R. Schneider. Wavelet approximation methods for pseudodifferential equations. II: Matrix compression and fast solution. Adv. Comput. Math., 1(3-4):259–335, 1993
W. Dahmen, S. Prössdorf, and R. Schneider. Wavelet approximation methods for pseudodifferential equations. I: Stability and convergence. Math. Z., 215(4):583–620, 1994
I. Daubechies. Ten lectures on wavelets. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992
A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput., 14(6):1368–1393, 1993
A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data. II. Appl. Comput. Harmon. Anal., 2(1):85–100, 1995
D. Gaier. Konstruktive Methoden der konformen Abbildung. Berlin – Göttingen – Heidelberg: Springer-Verlag, 1964
M. Golomb. Approximation by periodic spline interpolants on uniform meshes. J. Approximation Theory, 1:26–65, 1968
M.H. Gutknecht. Fast algorithms for the conjugate periodic function. Computing, 22:79–91, 1979
S.L. Hahn. Hilbert transforms in signal processing. The Artech House Signal Processing Library. Boston, MA: Artech House, 1996
P. Henrici. Applied and computational complex analysis. Volume III: Discrete Fourier analysis, Cauchy integrals, construction of conformal maps, univalent functions. Reprint. Wiley Classics Library. New York, NY: Wiley, 1993
F.W. King. Hilbert transforms. Volume 1. Encyclopedia of Mathematics and its Applications 124. Cambridge: Cambridge University Press, 2009
S. Kunis and D. Potts. Time and memory requirements of the nonequispaced FFT. Sampl. Theory Signal Image Process., 7(1):77–100, 2008
A.K. Louis, P. Maaß and A. Rieder. Wavelets. Theorie und Anwendungen. Teubner Studienb¨ucher: Mathematik. Stuttgart: Teubner, 1994
F. Martin. Analytische und numerische Verfahren zur Berechnung der Hilbert Transformation und zur Lösung funktionentheoretischer Randwertaufgaben. Ph.D. thesis, TU Bergakademie Freiberg, 2010
F. Martin and E.Wegert. Computing the Hilbert transform using biorthogonal spline wavelets, J. Math. Sci., 189(1):150–163, 2013.
S. Prössdorf and B. Silbermann. Numerical analysis for integral and related operator equations. Berlin: Akademie-Verlag, 1991
A. Rathsfeld. On the stability of piecewise linear wavelet collocation and the solution of the double layer equation over polygonal curves. In: Golberg, Michael (ed.), Boundary integral methods: numerical and mathematical aspects. Southampton: WIT Press/ Computational Mechanics Publications. Comput Eng. 1, 177–215, 1999
E. Wegert. An iterative method for solving nonlinear Riemann–Hilbert problems. J. Comput. Appl. Math., 29(3):311–327, 1990
E. Wegert. Nonlinear boundary value problems for holomorphic functions and singular integral equations. Mathematical Research. 65. Berlin: Akademie Verlag, 1992
R. Wegmann. Ein Iterationsverfahren zur konformen Abbildung. Numer. Math., 30:453–466, 1978
R. Wegmann. Methods for numerical conformal mapping. In: Kühnau, R. (ed.),
Handbook of complex analysis: geometric function theory. Volume 2. Amsterdam: Elsevier/North–Holland. 351–477, 2005
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Martin, F., Wegert, E. (2014). Computing the Hilbert Transform in Wavelet Bases on Adaptive Grids. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_21
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DOI: https://doi.org/10.1007/978-3-0348-0648-0_21
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