# 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}
}
• Published 11 September 2012
• Mathematics
• arXiv: Algebraic Geometry
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.
$C^{m}$ semialgebraic sections over the plane
• Mathematics
Journal of the Mathematical Society of Japan
• 2022
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
• Mathematics
Revista Matemática Iberoamericana
• 2020
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 $E \subset Cm solutions of semialgebraic or definable equations • Mathematics Advances in Mathematics • 2021$C^2$interpolation with range restriction • Mathematics, Computer Science Revista Matemática Iberoamericana • 2022 This paper constructs a (parameter-dependent, nonlinear) C(R) extension operator that preserves the range [λ,Λ], and provides an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E). C A ] 5 N ov 2 02 1 On the Whitney distortion extension problem for C m ( R n ) and C ∞ ( R n ) • 2021 Smooth Selection for Infinite Sets • Mathematics • 2021 Whitney’s extension problem asks the following: Given a compact set E ⊂ R and a function f : E → R, how can we tell whether there exists F ∈ C(R) such that F |E = f? A 2006 theorem of Charles C A ] 1 7 Ju l 2 02 1 C 2 Interpolation with Range Restriction • Computer Science, Mathematics • 2021 This paper constructs a (parameter-dependent, nonlinear) C(R) extension operator that preserves the range [λ,Λ], and provides an efficient algorithm to compute such an extension using O(N logN) operations, where N = #(E). On the Shape Fields Finiteness Principle • Mathematics • 2020 In this paper, we improve the finiteness constant for the finiteness principles for$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
• Mathematics, Computer Science
• 2016
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
• Mathematics
• 2015
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