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 [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 [2], [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) (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

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

References

[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)

Comments

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

References

[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)