# Polynomial least deviating from zero

polynomial deviating least from zero

An algebraic polynomial of degree $n$, with leading coefficient 1, having minimal norm in the space $C[a,b]$ or $L_p[a,b]$.

P.L. Chebyshev  proved that, among all polynomials of the form

$$Q_n(x)=x^n+a_1x^{n-1}+\dotsb+a_n,\label{1}\tag{1}$$

there is exactly one, viz.

$$T_n(x)=2\left(\frac{b-a}{4}\right)^n\cos n\arccos\left(\frac{2x-a-b}{b-a}\right),$$

of minimal norm in $C[a,b]$, and that norm is

$$\|T_n\|_{C[a,b]}=2\left(\frac{b-a}{4}\right)^n.$$

The polynomial

$$U_n(x)=2\left(\frac{b-a}{4}\right)^{n+1}\frac{\sin((n+1)\arccos(2x-a-b)/(b-a))}{\sqrt{(b-x)(x-a)}}$$

is the unique polynomial deviating least from zero in $L_1[a,b]$ (among all polynomials \eqref{1}), and its norm is

$$\|U_n\|_{L_1[a,b]}=4\left(\frac{b-a}{4}\right)^{n+1}.$$

In $L_p[a,b]$, $1<p<\infty$, there also exists a unique polynomial deviating least from zero; various properties of this polynomial are known (see , ).

The integral

$$\int\limits_a^bQ_n^2(x)\rho(x)dx,\quad\rho(x)>0,\label{2}\tag{2}$$

considered for all polynomials \eqref{1}, is minimal if and only if $Q_n(x)$, with respect to the weight function $\rho(x)$, is orthogonal on $(a,b)$ to all polynomials of degree $n-1$. If

$$a=-1,\quad b=1,\quad\rho(x)=(1-x)^\alpha(1+x)^\beta,$$

where $\alpha,\beta>-1$, then the integral \eqref{2} is minimized by the Jacobi polynomial (cf. Jacobi polynomials) (if $\alpha=\beta=0$ by the Legendre polynomial; cf. Legendre polynomials) of degree $n$ with leading coefficient 1.

Among all trigonometric polynomials of the form

$$a\cos nx+b\sin nx+\sum_{k=0}^{n-1}(a_k\cos kx+b_k\sin kx),$$

where $a$ and $b$ are fixed, the polynomial of minimal norm in any of the spaces $C[0,2\pi]$ and $L_p[0,2\pi]$ (for an arbitrary $p\geq1$) is

$$a\cos nx+b\sin nx.$$

How to Cite This Entry:
Polynomial least deviating from zero. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polynomial_least_deviating_from_zero&oldid=44676
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article