Approximation of $C^\infty$-functions without changing their zero-set

Manifold Approximation of Set-valued Functions

Sequences are denoted by x — although C functions are involved in the sequel, their derivative are not explicitly used so this notation for sequences will not interfere with the usual one for derivatives. A function f : R −→ P(R) is a subset of R. Notations and theorems related to manifolds are taken from Hirsch[1]. Points belonging to manifolds will be denoted with tildes. The closed n-dimensi...

APPROXIMATION OF 3D-PARAMETRIC FUNCTIONS BY BICUBIC B-SPLINE FUNCTIONS

In this paper we propose a method to approximate a parametric 3 D-function by bicubic B-spline functions

Zero Set of Sobolev Functions with Negative Power of Integrability

is of Hausdorff dimension at most n− p, by a theorem of Federer-Ziemer [3]. And for functions of bounded variation, we have Hn−1 (Σ∗) = 0, see section 5.9 of [1] for more information on fine properties of BV functions. Hence our result concerning Σ, the Lebesgue set of u of the value zero makes sense because s > n− p. In order to show Theorem 1, we will need a Poincaré type inequality which wil...

Zero-set Property of O-minimal Indefinitely Peano Differentiable Functions

Given an o-minimal expansion M of a real closed field R which is not polynomially bounded. Let P∞ denote the definable indefinitely Peano differentiable functions. If we further assume that M admits P∞ cell decomposition, each definable closed set A ⊂ Rn is the zero-set of a P∞ function f : Rn → R. This implies P∞ approximation of definable continuous functions and gluing of P∞ functions define...

On Padé Approximation of Some Practical Functions