On estimating best approximations of functions defined by differential equations

Finite difference methods are used to estimate the error in approximations to functions defined by differential equations. The problem of minimising the maximum of this error estimate can be solved by linear programming in the linear case, and by a method due to the authors in the nonlinear case. It is shown by examples that this new approach can improve substantially on techniques which minimise the maximum residual in the differential equation, and a convergence result as the mesh spacing tends to zero is given for the linear case.     

Zeros of Bernoulli-type functions and best approximations

The zero sets of (D+a)ⁿg(t) with in the (t,a)-plane are investigated for and .The results are used to determine entire interpolations to functions , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η>0 to .     

On the number of zeros of certain rational harmonic functions

Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function , where is a rational function of degree 1$">, has no more than complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an -point gravitational lens.

     

Asymptotic behavior of the best approximations for the functions inL p

The asymptotic behavior of the ratioE _n (f)_p/(f, / n) _p is studied for individual periodic functionsf L_p Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 867–870, June, 1993.     

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L ₁-approximations of classes W ^r F, r ∈ ?, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes W ^r H ^ω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ?, k ∈ ?, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.