Form preservation under approximation by local exponential splines of an arbitrary order

We continue the study of the properties of local L-splines with uniform knots (such splines were constructed in the authors’ earlier papers) corresponding to a linear differential operator L of order r with constant coefficients and real pairwise different roots of the characteristic polynomial. Sufficient conditions (which are also necessary) are established under which an L-spline locally inherits the property of the generalized k-monotonicity (k ≤ r ? 1) of the input data, which are the values of the approximated function at the nodes of a uniform grid shifted with respect to the grid of knots of the L-spline. The parameters of an L-spline that is exact on the kernel of the operator L are written explicitly.     

Local approximation by splines with displacement of nodes

We consider the problem of approximating a function defined on a uniform mesh by the method of local polynomial spline-approximation where the mesh of the nodes of the spline is chosen displaced relative to the mesh of the initial data. Conditions are established for the local form preservation by the spline of the initial data.We study the approximative properties of the method for the case of the simplest local approximation formula and find the optimal values of the displacement parameters.     

Local Approximation by Parabolic Splines in the Mean for Large Averaging Intervals

Mathematical Notes - Local parabolic splines on the axis $$\mathbb R$$ with equidistant nodes realizing the simplest local approximation scheme are considered. But, instead of the values of...     

On Lebesgue constants of local parabolic splines

Lebesgue constants (the norms of linear operators from C to C) are calculated exactly for local parabolic splines with an arbitrary arrangement of knots, which were constructed by the second author in 2005, and for N.P. Korneichuk’s local parabolic splines, which are exact on quadratic functions. Both constants are smaller than the constants for interpolating parabolic splines.     

Local exponential splines with arbitrary knots

We construct local L-splines that have an arbitrary arrangement of knots and preserve the kernel of a linear differential operator L of order r with constant coefficients and real pairwise distinct roots of the characteristic polynomial.     

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L _r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L ₃ = D(D ² ? β ²) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.     

Upper bounds for uniform Lebesgue constants of interpolation periodic sourcewise representable splines

A method for the construction of biorthogonal bases of multiwavelets from known bases of multiscaling functions is given. It is similar to the method presented in the author’s 2014 paper joint with N.I. Chernykh and is based on the same principle: the construction of multiwavelets based on k multiscaling functions employs an analog of the vector product of vectors in a 2k-dimensional space.     

Non-uniformity and representative sampling in NAA

An instrumental procedure of activation analysis has been developed to study the distribution of some minor components in synthetic granates, ferrites and glass charges in some environmental materials. The determination of the degree of non-uniformity by NAA has been used for the preparation of uniform mixtures and for representative sample estimation in the analysis of nonhomogeneous materials.     

Approximation by local trigonometric splines

For the class of 1-periodic functions, we use the linear noninterpolating method of trigonometric spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data, i.e., the values of a function from at the points of a uniform grid. The approximation error is calculated exactly for this class of functions in the uniform metric. It coincides with the Kolmogorov and Konovalov widths.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 354–363.Original Russian Text Copyright © 2005 by K. V. Kostousov, V. T. Shevaldin.This revised version was published online in April 2005 with a corrected issue number.