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Natural and smoothing quadratic spline. (An elementary approach)

Jiří Kobza, Dušan Zápalka (1991)

Applications of Mathematics

For quadratic spine interpolating local integrals (mean-values) on a given mesh the conditions of existence and uniqueness, construction under various boundary conditions and other properties are studied. The extremal property of such's spline allows us to present an elementary construction and an algorithm for computing needed parameters of such quadratic spline smoothing given mean-values. Examples are given illustrating the results.

Normal bivariate Birkhoff interpolation schemes and Pell equation

Marius Crainic, Nicolae Crainic (2009)

Commentationes Mathematicae Universitatis Carolinae

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., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...

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