Modified Gauss-Legendre, Lobatto and Radau cubature formulas for the numerical evaluation of 2-D singular integrals.
Multivariate asymptotics for products of large powers with applications to Lagrange inversion.
Multivariate Birkhoff-Lagrange interpolation schemes and Cartesian sets of nodes.
Multivariate interpolation II of Langrange and Hermite type
Multivariate smooth interpolation that employs polyharmonic functions
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...
Multivariate spline functions, B-spline bases and polynomial interpolations II
Normal bivariate Birkhoff interpolation schemes and 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., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...
Numerical Curvature Formulae with Preassigned Knots.
Numerical curves and their applications to algebraic curves
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 , 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...
Numerical Integration by Means of Adapted Euler Summation Formulas.
Numerical methods of computation of singular and hypersingular integrals.
On approximation by Chebyshevian box splines
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 are of the form , where the functions are periodic functions of one variable. Then we consider the problem of approximation of continuous functions by Chebyshevian box splines.
On bivariate Hermite interpolation and the limit of certain bivariate Lagrange projectors
We give a new poised bivariate Hermite scheme and a formula for the interpolation polynomial. We show that the Hermite interpolation polynomial is the limit of bivariate Lagrange interpolation polynomials at Bos configurations on circles.
On Constrained Multivariate Splines and Their Approximations.
On counterexamples in multivariate approximation
On Cubature Formulae with a Minimal Number of Knots.
On error formulas for approximation by sums of univariate functions.
On exponential approximation
On global smoothness preservation in complex approximation
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.
On Hermite interpolation in .