Exact bounds for the uniform approximation of continuous periodic functions by r-th order splines

We solve the problem of determining exact bounds for the uniform approximation of continuous periodic functions by r-th order interpolation splines in a space C and on a class H specified by the convex modulus of continuity(t).Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 217–228, February, 1972.In conclusion the author wishes to express his deep gratitude to N. P. Korneichuka for constant attention and observations which were useful to him in preparing the paper.     

On the best approximation of periodic functions of two variables by polynomial splines

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.     

Approximation of functions of two variables by harmonic splines

We construct two-dimensional splines and give two versions of an estimate of the deviation of splines from approximated functions. We compare approximations by a planar broken line and by a harmonic spline. We also substantiate the advisability of introduction of the notion of harmonic splines in mathematics.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1190–1196, September, 1995.     

On lower bounds for the approximation of functions by local splines with nonfixed nodes

For functions integrable to the power , we obtain asymptotically exact lower bounds for the approximation by local splines of degree r and defect k< r/2 in the metric of L _p      

On lower bounds for the approximation of individual functions by local splines with nonfixed nodes

For functions with the integrable βth power, where β = (r + 1 + 1/p)⁻¹, we obtain asymptotically exact lower bounds for the approximation by local splines of degreer and defectk ≥r/2 in the metric ofL _p. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1628–1637, December, 1999.     

Error bounds for the approximation of Green's kernels by splines

Summary In this paper we give error bounds for the approximation by tensor-product splines of surfaces which are defined on a square and which are smooth except along the diagonal.Supported in part by AFOSR Grant 77-3150     

Exact bounds for the uniform approximation by spline interpolants

Summary. This paper completes a result of Reimer (1984) concerning -th-degree cardinal and -periodic interpolation. The method of proof is not restricted to the case of and being odd and seems to be more elementary. Received February 1, 1993 / Revised version received September 14, 1993     

An approximation of analytical functions by local splines

Local splines are presented for the approximation of functions of one and many variables, which are analytic in the domains , where U_i(z_i) is a unit disk in the complex plane C_i,i=1,2,…,l, l=1,2, …. Results are given for functions whose r-order derivatives belong to the Hardy's class H_p,1≤p≤∞. It is shown that the approximation converge to the function at the rate for functions of one variable and An^{−(r−1/p)/(l−1)} for functions of l variables, where n is the number of points of local splines and A and C are positive constants. This work was supported by Russian Foundation of Fundumental Inverstigations     

Piece-wise linear approximation of functions of two variables

The goal of increasing computational efficiency is one of the fundamental challenges of both theoretical and applied research in mathematical modeling. The pursuit of this goal has lead to wide diversity of efforts to transform a specific mathematical problem into one that can be solved efficiently. Recent years have seen the emergence of highly efficient methods and software for solving Mixed Integer Programming Problems, such as those embodied in the packages CPLEX, MINTO, XPRESS-MP. The paper presents a method to develop a piece-wise linear approximation of an any desired accuracy to an arbitrary continuous function of two variables. The approximation generalizes the widely known model for approximating single variable functions, and significantly expands the set of nonlinear problems that can be efficiently solved by reducing them to Mixed Integer Programming Problems. By our development, any nonlinear programming problem, including non-convex ones, with an objective function (and/or constraints) that can be expressed as sums of component nonlinear functions of no more than two variables, can be efficiently approximated by a corresponding Mixed Integer Programming Problem.     

On best approximation of functions of two variables

$\mathop {\lim \sup }\limits_{r \to \infty } \frac{{E_{n_i ,m_i } (f)_L }}{{[E_{n_i ,\infty } (f)_L + E_{\infty ,m_i } (f)_L ]ln\{ 2 + min(n_i ,m_i )\} }}\underset{\raise0.3em\hbox{$\mathop {\lim \sup }\limits_{r \to \infty } \frac{{E_{n_i ,m_i } (f)_L }}{{[E_{n_i ,\infty } (f)_L + E_{\infty ,m_i } (f)_L ]ln\{ 2 + min(n_i ,m_i )\} }}\underset{\raise0.3em\hbox{