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Interpolation of harmonic functions based on Radon projections

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Abstract

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr’s formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.

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Acknowledgments

The authors acknowledge the support by the Bulgarian National Science Fund through Grant DMU 03/17. The research of the first author was also supported by Bulgarian National Science Fund Grant DDVU 0230/11. The second author was supported by the project AComIn “Advanced Computing for Innovation”, Grant 316087, funded by the FP7 Capacity Programme “Research Potential of Convergence Regions”, and by the Austrian Science Fund (FWF): W1214-N15, project DK04.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Bl. 8, 1113 , Sofia, Bulgaria

    Irina Georgieva

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Bl. 25A, 1113 , Sofia, Bulgaria

    Clemens Hofreither

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  1. Irina Georgieva
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  2. Clemens Hofreither
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Correspondence to Clemens Hofreither.

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Georgieva, I., Hofreither, C. Interpolation of harmonic functions based on Radon projections. Numer. Math. 127, 423–445 (2014). https://doi.org/10.1007/s00211-013-0592-y

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