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2013 | 11 | 4 | 609-620

Tytuł artykułu

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

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EN

Abstrakty

EN
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.

Twórcy

  • Bulgarian Academy of Sciences
  • Johannes Kepler University
  • Johannes Kepler University
  • Johannes Kepler University
  • Johannes Kepler University

Bibliografia

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

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bwmeta1.element.doi-10_2478_s11533-012-0160-1
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