Related papers: Asymptotic Coefficients and Errors for Chebyshev P…
Uniform polynomial approximation, also called minimax approximation or Chebyshev approximation, consists in searching polynomial approximation that minimizes the worst case error. Optimality conditions for the uniform approximation of…
For the Hahn and Krawtchouk polynomials orthogonal on the set $\{0, \ldots,N\}$ new identities for the sum of squares are derived which generalize the trigonometric identity for the Chebyshev polynomials of the first and second kind. These…
Due to their importance in both data analysis and numerical algorithms, low rank approximations have recently been widely studied. They enable the handling of very large matrices. Tight error bounds for the computationally efficient…
The Chebyshev or $\ell_{\infty}$ estimator is an unconventional alternative to the ordinary least squares in solving linear regressions. It is defined as the minimizer of the $\ell_{\infty}$ objective function \begin{align*}…
The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge…
The low-rank matrix approximation problem is ubiquitous in computational mathematics. Traditionally, this problem is solved in spectral or Frobenius norms, where the accuracy of the approximation is related to the rate of decrease of the…
In this paper, we derive optimality conditions (Chebyshev approximation) for multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions is very elegant. The optimality conditions are based on the notion…
Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where…
We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $\Phi_n=\{\varphi_k\}_{k=1}^n$ on the interval $[0,q]_{\mathbb{Z}}=[0,q]\cap \mathbb{Z}$. In particular, we establish that the polynomial of…
Given a measure $\mu$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} \mu = w \, {\rm d} \sigma + {\rm d} \mu_s$, with respect to the rotation-invariant Lebesgue measure $\sigma$ on $\partial…
The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in…
This paper provides error analyses of the algorithms most commonly used for the evaluation of the Chebyshev polynomial of the first kind $T_N(x)$. Some of these algorithms are shown to be backward stable. This means that the computed value…
The Chernoff bound is one of the most widely used tools in theoretical computer science. It's rare to find a randomized algorithm that doesn't employ a Chernoff bound in its analysis. The standard proofs of Chernoff bounds are beautiful but…
A Chebyshev expansion is a series in the basis of Chebyshev polynomials of the first kind. When such a series solves a linear differential equation, its coefficients satisfy a linear recurrence equation. We interpret this equation as the…
We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension…
We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these…
It is well known that matrices with low Hessenberg-structured displacement rank enjoy fast algorithms for certain matrix factorizations. We show how $n\times n$ principal finite sections of the Gram matrix for the orthogonal polynomial…
We solve principal component regression (PCR), up to a multiplicative accuracy $1+\gamma$, by reducing the problem to $\tilde{O}(\gamma^{-1})$ black-box calls of ridge regression. Therefore, our algorithm does not require any explicit…
The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in $\C^d.$ We study this problem on general sets, but devote special attention to product sets…
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem…