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Polyharmonic splines on grids $\mathbb{Z}\times a\mathbb{Z} ^{n}$ and their limits


Authors: O. Kounchev and H. Render
Journal: Math. Comp. 74 (2005), 1831-1841
MSC (2000): Primary 41A05, 65D10; Secondary 41A15
DOI: https://doi.org/10.1090/S0025-5718-05-01753-9
Published electronically: February 14, 2005
MathSciNet review: 2164099
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Abstract: Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form $\mathbb{Z}\times a\mathbb{Z} ^{n}$ having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines $I_{a}$on such grids for the limiting process $a\rightarrow0,$ $a>0.$ For a large class of data functions defined on $\mathbb{R}\times\mathbb{R} ^{n}$ it turns out that there exists a limit function $I.$ This limit function is shown to be a polyspline of order $p$ on strips. By the theory of polysplines we know that the function $I$ is smooth up to order $2\left( p-1\right) $everywhere (in particular, they are smooth on the hyperplanes $\left\{ j\right\} \times\mathbb{R} ^{n}$, which includes existence of the normal derivatives up to order $2\left( p-1\right))$ while the RBF interpolants $I_{a}$ are smooth only up to the order $2p-n-1.$ The last fact has important consequences for the data smoothing practice.


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  • 1. Bejancu, A., Kounchev, O., Render, H., Cardinal interpolation with biharmonic polysplines on strips. Curve and surface fitting (Saint-Malo, 2002), 41-58, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2003. MR 2042434
  • 2. Bejancu, A., Kounchev, O., Render, H., The cardinal interpolation on hyperplanes with polysplines, submitted.
  • 3. Buhmann, M.D., Multivariate Cardinal Interpolation with Radial-Basis Functions, Constr. Approx. 6 (1990), 225-255. MR 1054754 (91f:41001)
  • 4. Buhmann, M., Micchelli, C., On radial basis approximation on periodic grids, Math. Proc. Camb. Phil. Soc. 112 (1992), 317-334. MR 1171168 (93h:41001)
  • 5. Hörmander, L., The Analysis of Linear Partial Differential Operators II. Pseudo-Differential Operators, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. MR 0705278 (85g:35002b)
  • 6. Jetter, K., Multivariate Approximation from the Cardinal Interpolation Point of View. Approximation Theory VII, E.W. Cheney, C.K. Chui and L.L. Schumaker (eds.), pp. 131-161. MR 1212572 (94d:41004)
  • 7. Kounchev, O.I., Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, London-San Diego, 2001. MR 1852149 (2002h:41001)
  • 8. Kounchev, O., Render, H., Multivariate cardinal splines via spherical harmonics, submitted
  • 9. Kounchev, O., Render, H.,Wavelet Analysis of cardinal L-splines and Construction of multivariate Prewavelets, In: Proceedings ``Tenth International Conference on Approximation Theory'', St. Louis, Missouri, March 26-29, 2001. MR 1924893 (2003h:42060)
  • 10. Kounchev, O., Render, H., The approximation order of polysplines, Proc. Amer. Math. Soc. 132 (2004), no. 2, 455-461. MR 2022369 (2004i:41016)
  • 11. Kounchev, O., Render, H., Rate of convergence of polyharmonic splines to polysplines. Submitted.
  • 12. Kounchev, O., Wilson, M., Application of PDE methods to visualization of heart data. In: Michael J. Wilson, Ralph R. Martin (Eds.): Mathematics of Surfaces, Lecture Notes in Computer Science 2768, Springer-Verlag, 2003; pp. 377-391.
  • 13. Liu, Y., Lu, G., Simultaneous Approximations for functions in Sobolev spaces by derivatives of polyharmonic cardinal splines, J. Approx. Theory 101 (1999), 49-62. MR 1724025 (2000j:41022)
  • 14. Madych, W.R., Nelson, S.A., Polyharmonic Cardinal Splines, J. Approx. Theory 60 (1990), 141-156. MR 1033167 (90j:41022)
  • 15. Madych, W.R., Nelson, S.A., Multivariate interpolation and conditionally positive definite functions, II; Math. Comp., 54(189) (1990), 211-230. MR 0993931 (90e:41007)
  • 16. Stein, E.M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971. MR 0304972 (46:4102)

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Additional Information

O. Kounchev
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 8, 1113 Sofia, Bulgaria
Email: kounchev@math.bas.bg; kounchev@math.uni-duisburg.de

H. Render
Affiliation: Departamento de Matemáticas y Computatión, Universidad de la Rioja, Edificio Vives, Luis de Ulloa, s/n 26004, Logroño, Spain
Email: render@math.uni-duisburg.de; herender@dmc.unirioja.es

DOI: https://doi.org/10.1090/S0025-5718-05-01753-9
Keywords: Radial basis functions, interpolation, polyharmonic splines, polysplines.
Received by editor(s): August 14, 2003
Received by editor(s) in revised form: June 25, 2004
Published electronically: February 14, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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