On the representation by bivariate ridge functions

UDC 517.5 We consider the problem of representation of a bivariate function by sums of ridge functions. It is shown that if a function of a certain smoothness class is represented by a sum of finitely many arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation.

Holomorphically projective mappings of (pseudo-) Kähler manifolds preserve the class of differentiability

In this paper we study fundamental equations of holomorphically projective mappings of (pseudo-) K?hler manifolds with respect to the smoothness class of metrics Cr, r ? 1. We show that holomorphically projective mappings preserve the smoothness class of metrics.

QUADRATIC/LINEAR RATIONAL SPLINE INTERPOLATION

We describe the construction of an interpolating quadratic/linear rational spline S of smoothness class C 2 for a strictly convex (or strictly concave) function y on [a, b]. On uniform mesh x i = a + ih, i = 0,..., n, in the case of sufficiently smooth function y the expansions of S and its derivatives are obtained. They give the superconvergence of order h 4 for the first derivative, of order h 3 for the second derivative and of order h 2for the third derivative of S in certain points. Corresponding numerical examples are given.

LINEAR/LINEAR RATIONAL SPLINE INTERPOLATION

For a strictly monotone function y on [a, b] we describe the construction of an interpolating linear/linear rational spline S of smoothness class C 1. We show that for the linear/linear rational splines we obtain ¦S(xi ) − y(xi )¦8 = O(h 4) on uniform mesh xi = a + ih, i = 0,…, n. We prove also the superconvergence of order h3 for the first derivative and of order h2 for the second derivative of S in certain points. Numerical examples support the obtained theoretical results. This work was supported by the Estonian Science Foundation grant 8313.

CONVERGENCE RATE OF RATIONAL SPLINE HISTOPOLATION

The convergence rate of histopolation on arbitrary nonuniform mesh with linear/linear rational splines of class C1 is studied. Established convergence rate depends on Lipschitz smoothness class of the function to histopolate. Corresponding numerical examples are given.

Critical values lie on a line

We prove that critical values set of a differentiable map lies on a line of certain smoothness class.

Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on Sullivan's dual Cantor set. We show the limit sets are themselves ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the ${\cal C}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then the conjugacy (with a different extension) is in fact already ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\cal C}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Smoothness classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.