Optimale Quadraturformeln mit semidefiniten Peano-Kernen. - Optimal Integration Formulas with Semidefinite Peano-Kernels.
Optimality of the range for which equivalence between certain measures of smoothness holds
Recently it was proved for 1 < p < ∞ that , a modulus of smoothness on the unit sphere, and , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence does not hold either for p = ∞ or for p = 1.
Optimal-order quadratic interpolation in vertices of unstructured triangulations
We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition that nodes of interpolation are vertices of a triangulation. For an extensive class of triangulations we prove that every inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property: For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation error...
Optimization of Rational Approximations by Continued Fractions
The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.To get guaranteed machine enclosures of a special function f(x), an upper bound ε(f) of the relative error is needed, where ε(f) itself depends on the error bounds ε(app); ε(eval) of the approximation and evaluation error respectively. The approximation function g(x) ≈ f(x) is a rational function (Remez algorithm), and with sufficiently high polynomial degrees ε(app) becomes...
Order of approximation of functions of two variables by new type gamma operators.
Orthogonal Algebraic Polynomial Schauder Bases of Optimal Degree.
Orthogonal Laurent polynomials and quadratures on the unit circle and the real half-line.
Orthogonality in normed linear spaces. A survey I: Main properties.
Orthonormal polynomial vectors and least squares approximation for a discrete inner product.
Oscillation Matrices with Spline Smoothing.
Padé and Hermite-Padé approximation and orthogonality.
Padé approximations to certain power series.
Padé-Type Approximants of Exp (-z) Whose Denominators Are (1+z/n)n.
Pál-type interpolation and quadrature formulae on Laguerre abscissas.
Parameter Tuning and Repeated Application of the IMT-type Transformation in Numerical Quadrature.
Penalized Least Squares Fitting
* Supported by the Army Research Office under grant DAAD-19-02-10059.Bounds on the error of certain penalized least squares data fitting methods are derived. In addition to general results in a fairly abstract setting, more detailed results are included for several particularly interesting special cases, including splines in both one and several variables.
Permanence de relations de récurrence dans certains développements asymptotiques. (Permanence of recurrence relations in certain asymptotic expansions).
Perturbations of an Ostrowski type inequality and applications.
Perturbations of Best Approximation Problems.
Piecewise approximation and neural networks
The paper deals with the recently proposed autotracking piecewise cubic approximation (APCA) based on the discrete projective transformation, and neural networks (NN). The suggested new approach facilitates the analysis of data with complex dependence and relatively small errors. We introduce a new representation of polynomials that can provide different local approximation models. We demonstrate how APCA can be applied to especially noisy data thanks to NN and local estimations. On the other hand,...