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Birkhoff interpolation; Pell equation
Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular ``shape'' often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of ``shapes''. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., {\it Multivariate Birkhoff Interpolation\/}, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg, 1992, is not satisfied, and, at the same time, we will describe the complete solution.
[1] Barbeau E.J.: Pell's Equation. Springer, New York, 2003. MR 1949691 | Zbl 1030.11008
[2] Lorentz R.A.: Multivariate Birkhoff Interpolation. Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg, 1992. MR 1222648 | Zbl 0760.41002
[3] Gasca M., Maeztu J.I.: On Lagrange and Hermite interpolation in $\mathbb R^n$. Numer. Math. 39 (1982), 1--14. DOI 10.1007/BF01399308 | MR 0664533
[4] Stillwell J.: Elements of number theory. Undergraduate Texts in Mathematics, Springer, New York, 2003. MR 1944957 | Zbl 1112.11002
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