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''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 [[#References|[1]]] 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 [[#References|[2]]], [[#References|[5]]]).

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|Jacobi polynomials]]) (if $\alpha=\beta=0$ by the Legendre polynomial; cf. [[Legendre polynomials|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

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,   "Complete collected works" , '''2''' , Moscow-Leningrad  (1947)  pp. 23–51  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Tikhomirov,   "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Timan,   "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.M. Nikol'skii,   "Quadrature formulas" , Hindushtan Publ. Comp. , London  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.K. Suetin,   "Classical orthogonal polynomials" , Moscow  (1976)  (In Russian)</TD></TR></table>



====Comments====
The polynomials $T_n$ and $U_n$ are called (normalized) Chebyshev polynomials of the first, respectively second, kind (cf. [[Chebyshev polynomials|Chebyshev polynomials]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''2''' , F. Ungar  (1964–1965)  pp. Chapt. 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Rivlin,   "The Chebyshev polynomials" , Wiley  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.J.D. Powell,   "Approximation theory and methods" , Cambridge Univ. Press  (1981)</TD></TR></table>