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Smooth approximation and its application to some 1D problems. (English) Zbl 1313.65017
Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 2–5, 2012. In honor of the 60th birthday of Michal Křížek. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-60-8/pbk). 243-252 (2012).
The paper deals with the exact interpolation of scattered data. It reflects the situation when data at finite number of nodes in one, two, or three dimensions are given and one seeks a smooth curve or surface that honors the data points. The type of the interpolation function is defined through a variational problem with constrains which allows to specify the requirements on smoothness in the form of the integral of squared magnitudes of the interpolating function and of its chosen (possibly all) derivatives. The author briefly summarizes previous results of A. Talmi and G. Gilat [J. Comput. Phys. 23, 93–123 (1977; Zbl 0354.65005)] showing that the approach has a unique solution. He then introduces several basis systems for the 1D case. Some of them are further compared to classical polynomial and rational function interpolation. The comparison is made by numerical experiments, and illustrates the flexibility of the approach.
For the entire collection see [Zbl 1277.00031].
MSC:
65D05 Numerical interpolation
41A29 Approximation with constraints
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