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Multivariate smooth interpolation that employs polyharmonic functions

Segeth, Karel (2019)

Programs and Algorithms of Numerical Mathematics

We study the problem of construction of the smooth interpolation formula presented as the minimizer of suitable functionals subject to interpolation constraints. We present a procedure for determining the interpolation formula that in a natural way leads to a linear combination of polyharmonic splines complemented with lower order polynomial terms. In general, such formulae can be very useful e.g. in geographic information systems or computer aided geometric design. A simple computational example...

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,...

Numerical curves and their applications to algebraic curves

H. Gevorgian, H. Hakopian, A. Sahakian (1996)

Studia Mathematica

Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme N = n 1 , . . . , n s ; n , i.e., the sequence of multiplicities of nodes associated with the degree of interpolating polynomials, is considered. We continue the investigation of canonical decomposition of schemes and define so called maximal schemes. Some numerical results concerning the factorization...

On approximation by Chebyshevian box splines

Zygmunt Wronicz (2002)

Annales Polonici Mathematici

Chebyshevian box splines were introduced in [5]. The purpose of this paper is to show some new properties of them in the case when the weight functions w j are of the form w j ( x ) = W j ( v n + j · x ) , where the functions W j are periodic functions of one variable. Then we consider the problem of approximation of continuous functions by Chebyshevian box splines.

On global smoothness preservation in complex approximation

George A. Anastassiou, Sorin G. Gal (2002)

Annales Polonici Mathematici

By using the properties of convergence and global smoothness preservation of multivariate Weierstrass singular integrals, we establish multivariate complex Carleman type approximation results with rates. Here the approximants fulfill the global smoothness preservation property. Furthermore Mergelyan's theorem for the unit disc is strengthened by proving the global smoothness preservation property.

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