Bivariate interpolating polynomials and splines (I) |
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Authors: | Xiong Zhenziang |
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Affiliation: | (1) Department of Maths. & Phys, Beijing University of Aeronautics & Astronautics, 100083 Beijing, PRC |
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Abstract: | The multivariate splines which were first presented by de Boor as a complete theoretical system have intrigued many mathematicians
who have devoted many works in this field which is still in the process of development. The author of this paper is interested
in the area of interpolation with special emphasis on the interpolation methods and their approximation orders. But such B-splines
(both univariate and multivariate) do not interpolated directly, so I approached this problem in another way which is to extend
my interpolating spline of degree 2n-1 in univariate case (See7]) to multivariate case. I selected triangulated region which
is inspired by other mathematician’s works (e.g. 2] and 3]) and extend the interpolating polynomials from univariate to
m-variate case (See 10])In this paper some results in the case m=2 are discussed and proved in more concrete details. Based on these polynomials, the interpolating splines (it is defined by
me as piecewise polynomials in which the unknown partial derivatives are determined under certain continuous conditions) are
also discussed. The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated.
We limited our discussion on the rectangular domain which is partitioned into equal right triangles. As to the case in which
the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains, we will
discuss in the next paper. |
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