Uniform approximation of sgn x by polynomials and entire functions

By Alexandre Eremenko, P. Yuditskii. 2006

55 citations recorded by the source corpus.

In 1877, E. I. Zolotarev [19, 2] found an explicit expression, in terms ofelliptic functions, of the rational function of given degree mwhich is uni-formly closest to sgn(x) on the union of two intervals [−1,−a] ∪[a,1]. Thisresult was subject to many generalizations, and it has applications in electricengineering.Surprisingly, to the best of our knowledge, the similar problem for poly-nomials was not solved yet, so we investigate it in this paper.For comparison, we mention here the results on the uniform approxima-tion of |x|

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d1e3985c4ea299916730220353532775d16a275f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Sanjeev Arora
2. Simon Shaolei Du
3. Yi Zhang

In Numerical Analysis

1. Guang Hao Low
2. Isaac L. Chuang
3. Sanjeev Arora
4. Simon Shaolei Du
5. Nathan Wiebe
6. Daniel Litinski
7. András Gilyén
8. Danial Motlagh

In Statistics and Probability

1. Sanjeev Arora
2. Simon Shaolei Du
3. Isaac L. Chuang
4. Guang Hao Low
5. Yi Zhang

Citations in the corpus

• Methodology of Resonant Equiangular Composite Quantum Gates (2016, background): “Remarkably, the Chebyshev error of this approximation problem is known [52]:”
• Over-parameterized Adversarial Training: An Analysis Overcoming the Curse of Dimensionality (2020, methodology): “Interestingly, a nearly matching lower bound on the degree had been shown by [12] much prior to these constructions.”