Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 11, Issue 4


Volume 13 (2015)

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

Irina Georgieva
  • Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl. 8, 1113, Sofia, Bulgaria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Clemens Hofreither / Christoph Koutschan
  • Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Veronika Pillwein
  • Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Thotsaporn Thanatipanonda
  • Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-01-29 | DOI: https://doi.org/10.2478/s11533-012-0160-1


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.

MSC: 41A05; 41A63; 44A12; 33F10; 65D05

Keywords: Harmonic interpolation; Harmonic polynomials; Radon projections; Computer tomography; Symbolic computation

  • [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 Google Scholar

  • [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/sm162-2-3CrossrefGoogle Scholar

  • [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/s002110050358CrossrefGoogle Scholar

  • [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.1999.3442CrossrefGoogle Scholar

  • [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/10.1137/040616516CrossrefGoogle Scholar

  • [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/BF01161414CrossrefGoogle Scholar

  • [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 Google Scholar

  • [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.3160340105CrossrefGoogle Scholar

  • [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 Google Scholar

  • [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 Google Scholar

  • [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.2007.04.002Web of ScienceCrossrefGoogle Scholar

  • [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_77CrossrefGoogle Scholar

  • [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_89CrossrefGoogle Scholar

  • [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)90019-3CrossrefGoogle Scholar

  • [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)90214-7CrossrefGoogle Scholar

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

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

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

  • [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/S0012-7094-75-04256-8CrossrefGoogle Scholar

  • [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-XCrossrefGoogle Scholar

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

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

  • [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)90038-6CrossrefGoogle Scholar

  • [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 Google Scholar

  • [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-XCrossrefGoogle Scholar

  • [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)80044-2CrossrefGoogle Scholar

About the article

Published Online: 2013-01-29

Published in Print: 2013-04-01

Citation Information: Open Mathematics, Volume 11, Issue 4, Pages 609–620, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-012-0160-1.

Export Citation

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Irina Georgieva and Clemens Hofreither
Numerische Mathematik, 2014, Volume 127, Number 3, Page 423

Comments (0)

Please log in or register to comment.
Log in