Polynomial least deviating from zero

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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 [1] proved that, among all polynomials of the form


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


The polynomial


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


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 [2], [5]).

The integral


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

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


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


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


[a1] I.P. Natanson, "Constructive function theory" , 2 , F. Ungar (1964–1965) pp. Chapt. 6 (Translated from Russian)
[a2] T.J. Rivlin, "The Chebyshev polynomials" , Wiley (1974)
[a3] M.J.D. Powell, "Approximation theory and methods" , Cambridge Univ. Press (1981)
How to Cite This Entry:
Polynomial least deviating from zero. Encyclopedia of Mathematics. URL:
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