smoothness class
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2021 ◽  
Vol 73 (5) ◽  
pp. 579-588
Author(s):  
R. A. Aliev ◽  
A. A. Asgarova ◽  
V. E. Ismailov

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.


Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 3115-3122 ◽  
Author(s):  
Irena Hinterleitner

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.


2013 ◽  
Vol 18 (2) ◽  
pp. 250-259 ◽  
Author(s):  
Erge Ideon ◽  
Peeter Oja

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.


2010 ◽  
Vol 15 (4) ◽  
pp. 447-455 ◽  
Author(s):  
Erge Ideon ◽  
Peeter Oja

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.


2007 ◽  
Vol 12 (1) ◽  
pp. 29-38 ◽  
Author(s):  
Malle Fischer ◽  
Peeter Oja

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.


2004 ◽  
Vol 2004 (9) ◽  
pp. 757-776
Author(s):  
Azat Ainouline

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


1997 ◽  
Vol 17 (3) ◽  
pp. 531-564 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER

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.


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