MULTIVARIATE INTERPOLATION USING POLYHARMONIC SPLINES

Authors

  • Karel Segeth Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic

DOI:

https://doi.org/10.14311/AP.2021.61.0148

Keywords:

Data interpolation, smooth interpolation, polyharmonic spline, Fourier transform

Abstract

Data measuring and further processing is the fundamental activity in all branches of science and technology. Data interpolation has been an important part of computational mathematics for a long time. In the paper, we are concerned with the interpolation by polyharmonic splines in an arbitrary dimension. We show the connection of this interpolation with the interpolation by radial basis functions and the smooth interpolation by generating functions, which provide means for minimizing the L2 norm of chosen derivatives of the interpolant. This can be useful in 2D and 3D, e.g., in the construction of geographic information systems or computer aided geometric design. We prove the properties of the piecewise polyharmonic spline interpolant and present a simple 1D example to illustrate
them.

References

A. Talmi, G. Gilat. Method for smooth approximation of data. J Comput Phys 23:93–123, 1977. doi:10.1016/0021-9991(77)90115-2.

K. Segeth. Polyharmonic splines generated by multivariate smooth interpolation. Comput Math Appl 78:3067–3076, 2019. doi:10.1016/j.camwa.2019.04.018.

K. Segeth. Some splines produced by smooth interpolation. Appl Math Comput 319:387–394, 2018. doi:10.1016/j.amc.2017.04.022.

L. Mitáš, H. Mitášová. General variational approach to the interpolation problem. Comput Math Appl 16:983–992, 1988. doi:10.1016/0898-1221(88)90255-6.

K. Segeth. A periodic basis system of the smooth approximation space. Appl Math Comput 267:436–444, 2015. doi:10.1016/j.amc.2015.01.120.

S. G. Kre˘ın (ed.). Functional analysis (Russian). 1st edition. Nauka, Moskva, 1964.

Downloads

Published

2021-02-10

Issue

Section

Refereed Articles