The Brenner-Hochster-Koll\'ar and Whitney Problems for Vector-valued Functions and Jets

@article{Fefferman2012TheBA,
  title={The Brenner-Hochster-Koll\'ar and Whitney Problems for Vector-valued Functions and Jets},
  author={Charles Fefferman and Garving K Luli},
  journal={arXiv: Algebraic Geometry},
  year={2012}
}
In this paper, we give analytic methods for finding m (and m+\omega) times continuously differentiable solutions of a finite system of linear equations. Along the way, we also solve a generalized Whitney problem for vector-valued functions and jets. 
View PDF on arXiv
11 Citations
Highly Influential Citations
1
Background Citations
3
Methods Citations
3
Results Citations
1

11 Citations

$C^{m}$ semialgebraic sections over the plane
for polynomials P1, · · · , Pr, Q1, · · · , Qs on R . (We allow the cases r = 0 or s = 0.) A semialgebraic function φ : E → R is a function whose graph {(x, φ(x)) : x ∈ E} is a semialgebraic set. We…
Solutions to a system of equations for $C^m$ functions
Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset…
Cm solutions of semialgebraic or definable equations
Smooth selection for infinite sets
$C^2$ interpolation with range restriction
TLDR
This paper constructs a (parameter-dependent, nonlinear) C(R) extension operator that preserves the range [λ,Λ], and provides an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E).
C A ] 5 N ov 2 02 1 On the Whitney distortion extension problem for C m ( R n ) and C ∞ ( R n )
  • 2021
C A ] 1 7 Ju l 2 02 1 C 2 Interpolation with Range Restriction
TLDR
This paper constructs a (parameter-dependent, nonlinear) C(R) extension operator that preserves the range [λ,Λ], and provides an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E).
On the Shape Fields Finiteness Principle
In this paper, we improve the finiteness constant for the finiteness principles for $C^m(\mathbb{R}^n,\mathbb{R}^d)$ and $C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)$ selection proven by Fefferman, Israel,…
Interpolation of data by smooth non-negative functions
We prove a finiteness principle for interpolation of data by nonnegative Cm functions. Our result raises the hope that one can start to understand constrained interpolation problems in which e.g. the…
Finiteness Principles for Smooth Selection
In this paper we prove finiteness principles for $${C^m{({\mathbb{R}^n},{\mathbb{R}^D)}}}$$Cm(Rn,RD) and $${C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)}$$Cm-1,1(Rn,RD) selections. In particular, we provide…
...
...

References

SHOWING 1-10 OF 24 REFERENCES
Continuous solutions to algebraic forcing equations
We ask for a given system of polynomials f_1,...,f_n and f over the complex numbers when there exist continuous functions q_1,...,q_n such that q_1 f_1+...+q_n f_n = f. This condition defines the…
Continuous closure of sheaves
We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that,…
Continuous linear combinations of polynomials. In From Fourier analysis and number theory to Radon transforms and geometry, 233–282
  • Dev. Math. 28,
  • 2013
A linear extension operator for a space of smooth functions defined on a closed subset in R
  • Dokl. Akad. Nauk SSSR
  • 1985
A Generalized Sharp Whitney Theorem for Jets
Suppose that, for each point x in a given subset E I Rn, we are given an m-jet f(x) and a convex, symmetric set ?(x) of m-jets at x. We ask whether there exist a function F I Cm,w(Rn) and a finite…
Differentiable functions defined in closed sets. A problem of Whitney
Abstract.In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝn is the restriction of a function of class 𝒞p. A necessary and sufficient…
Generalizations of Whitney's extension theorem
Étude de Quelques Algèbres Tayloriennes
Differentiable Functions Defined in Closed Sets. I
Functions Differentiable on the Boundaries of Regions
...
...