# Harmonic interpolation based on Radon projections along the sides of regular polygons

Irina Georgieva 1 , Clemens Hofreither 2 , Christoph Koutschan 3 , Veronika Pillwein 3 , and Thotsaporn Thanatipanonda 3
• 1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl. 8, 1113, Sofia, Bulgaria
• 2 Doctoral College Computational Mathematics, Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria
• 3 Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria

## Abstract

Given information about a harmonic function in two variables, consisting of a finite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique solution in the case when the chords form a regular polygon. Numerical experiments for this and more general cases are presented.

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• [1] Bojanov B., Draganova C., Surface approximation by piece-wise harmonic functions, In: Algorithms for Approximation V, University College, Chester, 2005, available at http://roar.uel.ac.uk/618

• [2] Bojanov B., Georgieva I., Interpolation by bivariate polynomials based on Radon projections, Studia Math., 2004, 162(2), 141–160 http://dx.doi.org/10.4064/sm162-2-3

• [3] Bojanov B., Petrova G., Numerical integration over a disc. A new Gaussian quadrature formula, Numer. Math., 1998, 80(1), 39–59 http://dx.doi.org/10.1007/s002110050358

• [4] Bojanov B., Petrova G., Uniqueness of the Gaussian quadrature for a ball, J. Approx. Theory, 2000, 104(1), 21–44 http://dx.doi.org/10.1006/jath.1999.3442

• [5] Bojanov B., Xu Y., Reconstruction of a polynomial from its Radon projections, SIAM J. Math. Anal., 2005, 37(1), 238–250 http://dx.doi.org/10.1137/040616516

• [6] Cavaretta A.S. Jr., Micchelli C.A., Sharma A., Multivariate interpolation and the Radon transform, Math. Z., 1980, 174(3), 263–279 http://dx.doi.org/10.1007/BF01161414

• [7] Cavaretta A.S. Jr., Micchelli C.A., Sharma A., Multivariate interpolation and the Radon transform. II. Some further examples, In: Quantitive Approximation, Bonn, August 20–24, 1979, Academic Press, New York-London, 1980, 49–62

• [8] Davison M.E., Grünbaum F.A., Tomographic reconstruction with arbitrary directions, Comm. Pure Appl. Math., 1981, 34(1), 77–119 http://dx.doi.org/10.1002/cpa.3160340105

• [9] Georgieva I., Hofreither C., Uluchev R., Interpolation of mixed type data by bivariate polynomials, In: Constructive Theory of Functions, Sozopol, June 3–10, 2010, Marin Drinov Academic Publishing House, Sofia, 2012, 93–107

• [10] Georgieva I., Ismail S., On recovering of a bivariate polynomial from its Radon projections, In: Constructive Theory of Functions, Varna, June 1–7, 2005, Marin Drinov Academic Publishing House, Sofia, 2006, 127–134

• [11] Georgieva I., Uluchev R., Smoothing of Radon projections type of data by bivariate polynomials, J. Comput. Appl. Math., 2008, 215(1), 167–181 http://dx.doi.org/10.1016/j.cam.2007.04.002

• [12] Georgieva I., Uluchev R., Surface reconstruction and Lagrange basis polynomials, In: Large-Scale Scientific Computing, Sozopol, June 5–9, 2007, Lecture Notes in Comput. Sci., 4818, Springer, Berlin, 2008, 670–678 http://dx.doi.org/10.1007/978-3-540-78827-0_77

• [13] Georgieva I., Uluchev R., On interpolation in the unit disk based on both Radon projections and function values, In: Large-Scale Scientific Computing, Sozopol, June 4–8, 2009, Lecture Notes in Comput. Sci., 5910, Springer, Berlin, 2010, 747–755 http://dx.doi.org/10.1007/978-3-642-12535-5_89

• [14] Hakopian H., Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type, J. Approx. Theory, 1982, 34(3), 286–305 http://dx.doi.org/10.1016/0021-9045(82)90019-3

• [15] Hamaker C., Solmon D.C., The angles between the null spaces of X-rays, J. Math. Anal. Appl., 1978, 62(1), 1–23 http://dx.doi.org/10.1016/0022-247X(78)90214-7

• [16] Jain A.K., Fundamentals of Digital Image Processing, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1989

• [17] Koutschan C., Advanced Applications of the Holonomic Systems Approach, PhD thesis, RISC, Johannes Kepler University, Linz, 2009

• [18] Koutschan C., HolonomicFunctions, available at http://www.risc.uni-linz.ac.at/research/combinat/software/HolonomicFunctions/

• [19] Logan B.F., Shepp L.A., Optimal reconstruction of a function from its projections, Duke Math. J., 1975, 42(4), 645–659 http://dx.doi.org/10.1215/S0012-7094-75-04256-8

• [20] Marr R.B., On the reconstruction of a function on a circular domain from a sampling of its line integrals, J. Math. Anal. Appl., 1974, 45(2), 357–374 http://dx.doi.org/10.1016/0022-247X(74)90078-X

• [21] Natterer F., The Mathematics of Computerized Tomography, Classics Appl. Math., 32, SIAM, Philadelphia, 2001

• [22] Nikolov G., Cubature formulae for the disk using Radon projections, East J. Approx., 2008, 14(4), 401–410

• [23] Petkovšek M., Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symbolic Comput., 1992, 14(2–3), 243–264 http://dx.doi.org/10.1016/0747-7171(92)90038-6

• [24] Radon J., Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Berichte über die Verhandlungen der Sächsische Akademie der Wissenschaften zu Leipzig, 1917, 69, 262–277

• [25] Zeilberger D., A holonomic systems approach to special functions identities, J. Comput. Appl. Math., 1990, 32(3), 321–368 http://dx.doi.org/10.1016/0377-0427(90)90042-X

• [26] Zeilberger D., The method of creative telescoping, J. Symbolic Comput., 1991, 11(3), 195–204 http://dx.doi.org/10.1016/S0747-7171(08)80044-2

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