Yu. N. Subbotin

Exact values of relative widths of classes of differentialbe functions

AbstractWe study relative widths in the spacesC andL of classes of periodic differentiable functionsWr,r=1,2,…, when in contrast to the Kolmogorov widths it is additionally required that the approximating functions belong to the classMWr with a given majorantM of the norm of the derivative of orderr. It is proved that ifM satisfies the estimate

Transformation That Changes the Geometric Structure of a Vector Field

M \geqslant \frac{4}{{\pi ^2 }}{\text{ log min (}}n{\text{,}}r{\text{) + }}O{\text{(1)}}

Sharpening of the estimates for relative widths of classes of differentiable functions

which is uniform inn andr, then the above-mentionedn-dimensional relative widths of classesWr coincide with the corresponding Kolmogorov widths. Simultaneously, we obtain a uniform (in all the parameters) estimate of the Lebesgue constants of the Zygmund normal means of Fourier series, defined by the factors 1−(k/n)r,k≤n.

On the equality of Kolmogorov and relative widths of classes of differentiable functions

We find a general solution to the problem on the motion in an incompressible continuous medium occupying at any time a whole domain D ⊂ R3 under the conditions that D is an axially symmetric cylinder and the motion is described by the Euler equation together with the continuity equation for an incompressible medium and belongs to the class of helical flows (according to I.S. Gromeka’s terminology), in which sreamlines coincide with vortex lines. This class is constructed by the method of transformation of the geometric structure of a vector field. The solution is characterized in Theorem 2 in the end of the paper.

Approximations by polynomial and trigonometric splines of third order preserving some properties of approximated functions

We propose a method of constructing vector fields with certain vortex properties by means of transformations that change the value of the field vector at every point, the form of the field lines, and their mutual position. We discuss and give concrete examples of the prospects of using the method in applications involving solution of partial differential equations, including nonlinear ones.

Norms on L of Periodic Interpolation Splines with Equidistant Nodes

A review of results on the convergence of the interpolation process for polynomial splines and derivatives in the last 50 years is given.

Harmonic wavelets in boundary value problems for harmonic and biharmonic functions

We improve the earlier obtained upper estimates for the least value of the coefficient M for which the Kolmogorov widths dn(WCr, C) of the function class WCr are equal to the relative widths Kn(WCr, MWCj, C) of the class WCr with respect to the class MWCj, j < r.

Form-preserving exponential approximation

We obtain sharper estimates of the remainders in the expression for the least value of the multiplier M for which the Kolmogorov widths dn(WCr, C) and the relative widths Kn (WCr,MWCj,C) of the class WCr with respect to the class MWCj, j < r, where r − j is odd, are equal.