2000 character limit reached
The Fekete problem in segmental polynomial interpolation (2403.09378v1)
Published 14 Mar 2024 in math.NA and cs.NA
Abstract: In this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant.
- A. Alonso Rodríguez, L. Bruni Bruno and F. Rapetti, Towards nonuniform distributions of unisolvent weights for Whitney finite element spaces on simplices: the edge element case, Calcolo, 59(4):37 (2022).
- A. Alonso Rodríguez, L. Bruni Bruno and F. Rapetti, Whitney edge elements and the Runge phenomenon, J. Comput. Appl. Math., 427:115117 (2023).
- A. Alonso Rodríguez and F. Rapetti, On a generalization of the Lebesgue’s constant, J. Comput. Phys., 428:109964 (2021).
- B. Bojanov, Interpolation and integration based on averaged values, in Approximation and probability, Polish Acad. Sci. Inst. Math., Warsaw, 72 (2006), pp. 25–47.
- L. P. Bos, On Fekete points for a real simplex, Indag. Math. (N.S.), 34(2) (2023), pp. 274–293
- Computing multivariate Fekete and Leja points by numerical linear algebra, SIAM J. Num. Anal., 48(5) (2010), pp. 1984–1999.
- L. P. Bos and N. Levenberg, On the calculation of approximate Fekete points: the univariate case, Electron. Trans. Numer. Anal., 30 (2008), pp. 377–397.
- L. P. Bos, M. Taylor and B. Wingate, Tensor product Gauss-Lobatto points are Fekete points for the cube, Math. Comput., 70 (2001), pp. 1543–1547.
- A. Bossavit, Computational electromagnetism. Academic Press, Inc., San Diego, CA, 1998
- L. Bruni Bruno, Weights as degrees of freedoom for high order Whitney finite elements, Ph.D. thesis, University of Trento, 2022.
- L. Bruni Bruno and W. Erb, Polynomial interpolation of function averages on interval segments, submitted (2023). ArXiv: https://arxiv.org/abs/2309.00328.
- E. W. Cheney, Introduction to Approximation Theory, AMS Chelsea Publishing, Providence, RI, 1982.
- E. W. Cheney and W. Light, A Course in Approximation Theory, AMS, Providence, RI, 2000.
- P. J. Davis, Interpolation and Approximation, Dover Publications, New York, 1975.
- L. Fejér, Bestimmung derjenigen Abszissen eines Intervalles für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle [−1,1]11[-1,1][ - 1 , 1 ] ein möglichst kleines Maximum besitzt, Ann. Scula Norm. Sup. Pisa Sci. Fis. Mat. Ser. II., 1 (1932), pp. 263–276.
- L. Fejér, Lagrangesche Interpolation und die Zugehörigen Konjugierten Punkte, Math. Ann., 106 (1932), pp. 1–55.
- M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17(1) (1923), pp. 228–249
- M. Gerritsma, Edge functions for spectral element methods, in Spectral and high order methods for partial differential equations, Selected papers from the ICOSAHOM ’09 conference, J. S. Hesthaven and E. M. Rønquist, eds., Lect. Notes Comput. Sci. Eng. 76, Spinger, Heidelberg, 2011, pp. 199–207.
- J. Harrison, Continuity of the integral as a function of the domain, J. Geom. Anal., 8:5 (1998), pp. 769–795.
- J. S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal., 35 (1998), pp. 655–676.
- High order geometric methods with exact conservation properties, J. Comput. Phys. 257 (2014), part B, 1444–1471.
- B. A. Ibrahimoglu, Lebesgue functions and Lebesgue constants in polynomial interpolation J. Inequal. Appl., (2016), Paper No. 93, pp. 15.
- T. J. Rivlin, Chebyshev polynomials, Wiley, New York, 2nd edition, 1990.
- N. Robidoux, Polynomial histopolation, superconvergent degrees of freedom, and pseudospectral discrete Hodge operators, technical report (2006). Available at https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7d9598cb27819bc85cedeb149368d4e213ce9f1c
- C. Runge, Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten, Zeitschrift fur Mathematik und Physik, 46, (1901), pp. 224–243.
- Computing approximate Fekete points by QR factorizations of Vandermonde matrices, Comput. Math. Appl., 57(8) (2009), pp. 1324–1336.
- B. Sündermann, Lebesgue constants in Lagrangian interpolation at the Fekete points, Mitt. Math. Ges. Hamburg, 11(2) (1983), pp. 204–211.
- G. Szegő, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Providence, 1975.
- Two results on polynomial interpolation in equally spaced points, J. Approx. Theory, 65(3) (1991), pp. 247–260.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.