# American Institute of Mathematical Sciences

September  2007, 6(3): 569-585. doi: 10.3934/cpaa.2007.6.569

## Limits of radial basis function interpolants

 1 Justus–Liebig University, Mathematics Institut, Arndtstr. 2, 35392 Giessen, Germany 2 Jagiellonian University, Mathematics Institute, ul. Remonta 4, 30–059 Krakow, Poland

Received  February 2006 Revised  April 2006 Published  June 2007

We solve some open problems posed by Fornberg et al. in [6], [9] and [12], related to radial basis functions with parameters. They concern the limits of interpolants using these radial basis functions when the aforementioned parameters tend to zero--which makes them "increasingly flat" in a term coined by Fornberg. These aspects of radial basis function interpolation are useful because they concern the numerical problems with ill-conditioned matrices for small parameters and how to solve the interpolation problems efficiently in the face of this ill-conditioning. Finally, there are some interesting links between radial basis function interpolation and polynomial interpolation coming out of this research. While answering several such conjectures, we also develop a number of new techniques--some of them with number-theoretic arguments--for attacking similar problems.
Citation: Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
 [1] Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115 [2] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164 [3] Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439 [4] Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543 [5] Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695 [6] Charles Fefferman. Interpolation by linear programming I. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477 [7] Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453 [8] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [9] Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera, S. H. Pau. A general multipurpose interpolation procedure: the magic points. Communications on Pure & Applied Analysis, 2009, 8 (1) : 383-404. doi: 10.3934/cpaa.2009.8.383 [10] Anita Mayo. Accurate two and three dimensional interpolation for particle mesh calculations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1205-1228. doi: 10.3934/dcdsb.2012.17.1205 [11] V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223 [12] Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems & Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147 [13] Lucio Boccardo, Daniela Giachetti. A nonlinear interpolation result with application to the summability of minima of some integral functionals. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 31-42. doi: 10.3934/dcdsb.2009.11.31 [14] Gleb Beliakov. Construction of aggregation operators for automated decision making via optimal interpolation and global optimization. Journal of Industrial & Management Optimization, 2007, 3 (2) : 193-208. doi: 10.3934/jimo.2007.3.193 [15] Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675 [16] Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40 [17] Alexander Balandin. The localized basis functions for scalar and vector 3D tomography and their ray transforms. Inverse Problems & Imaging, 2016, 10 (4) : 899-914. doi: 10.3934/ipi.2016026 [18] Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451 [19] Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629 [20] Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037

2018 Impact Factor: 0.925