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''polynomial deviating least from zero''
 
''polynomial deviating least from zero''
  
An algebraic polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737201.png" />, with leading coefficient 1, having minimal norm in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737202.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737203.png" />.
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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
 
P.L. Chebyshev [[#References|[1]]] proved that, among all polynomials of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737204.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$Q_n(x)=x^n+a_1x^{n-1}+\dotsb+a_n,\label{1}\tag{1}$$
  
 
there is exactly one, viz.
 
there is exactly one, viz.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737205.png" /></td> </tr></table>
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$$T_n(x)=2\left(\frac{b-a}{4}\right)^n\cos n\arccos\left(\frac{2x-a-b}{b-a}\right),$$
  
of minimal norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737206.png" />, and that norm is
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of minimal norm in $C[a,b]$, and that norm is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737207.png" /></td> </tr></table>
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$$\|T_n\|_{C[a,b]}=2\left(\frac{b-a}{4}\right)^n.$$
  
 
The polynomial
 
The polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737208.png" /></td> </tr></table>
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$$U_n(x)=2\left(\frac{b-a}{4}\right)^{n+1}\frac{\sin((n+1)\arccos(2x-a-b)/(b-a))}{\sqrt{(b-x)(x-a)}}$$
  
is the unique polynomial deviating least from zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p0737209.png" /> (among all polynomials (1)), and its norm is
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is the unique polynomial deviating least from zero in $L_1[a,b]$ (among all polynomials \eqref{1}), and its norm is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372010.png" /></td> </tr></table>
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$$\|U_n\|_{L_1[a,b]}=4\left(\frac{b-a}{4}\right)^{n+1}.$$
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372012.png" />, there also exists a unique polynomial deviating least from zero; various properties of this polynomial are known (see [[#References|[2]]], [[#References|[5]]]).
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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
 
The integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\int\limits_a^bQ_n^2(x)\rho(x)dx,\quad\rho(x)>0,\label{2}\tag{2}$$
  
considered for all polynomials (1), is minimal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372014.png" />, with respect to the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372015.png" />, is orthogonal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372016.png" /> to all polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372017.png" />. If
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372018.png" /></td> </tr></table>
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$$a=-1,\quad b=1,\quad\rho(x)=(1-x)^\alpha(1+x)^\beta,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372019.png" />, then the integral (2) is minimized by the Jacobi polynomial (cf. [[Jacobi polynomials|Jacobi polynomials]]) (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372020.png" /> by the Legendre polynomial; cf. [[Legendre polynomials|Legendre polynomials]]) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372021.png" /> with leading coefficient 1.
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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
 
Among all trigonometric polynomials of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372022.png" /></td> </tr></table>
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$$a\cos nx+b\sin nx+\sum_{k=0}^{n-1}(a_k\cos kx+b_k\sin kx),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372024.png" /> are fixed, the polynomial of minimal norm in any of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372026.png" /> (for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372027.png" />) is
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372028.png" /></td> </tr></table>
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$$a\cos nx+b\sin nx.$$
  
 
====References====
 
====References====
Line 49: Line 50:
  
 
====Comments====
 
====Comments====
The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073720/p07372030.png" /> are called (normalized) Chebyshev polynomials of the first, respectively second, kind (cf. [[Chebyshev polynomials|Chebyshev polynomials]]).
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The polynomials $T_n$ and $U_n$ are called (normalized) Chebyshev polynomials of the first, respectively second, kind (cf. [[Chebyshev polynomials|Chebyshev polynomials]]).
  
 
====References====
 
====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>
 
<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>

Latest revision as of 15:05, 14 February 2020

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)
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=12323
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