Polynomial least deviating from zero
polynomial deviating least from zero
An algebraic polynomial of degree , with leading coefficient 1, having minimal norm in the space
or
.
P.L. Chebyshev [1] proved that, among all polynomials of the form
![]() | (1) |
there is exactly one, viz.
![]() |
of minimal norm in , and that norm is
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The polynomial
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is the unique polynomial deviating least from zero in (among all polynomials (1)), and its norm is
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In ,
, there also exists a unique polynomial deviating least from zero; various properties of this polynomial are known (see [2], [5]).
The integral
![]() | (2) |
considered for all polynomials (1), is minimal if and only if , with respect to the weight function
, is orthogonal on
to all polynomials of degree
. If
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where , then the integral (2) is minimized by the Jacobi polynomial (cf. Jacobi polynomials) (if
by the Legendre polynomial; cf. Legendre polynomials) of degree
with leading coefficient 1.
Among all trigonometric polynomials of the form
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where and
are fixed, the polynomial of minimal norm in any of the spaces
and
(for an arbitrary
) is
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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 and
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) |
Polynomial least deviating from zero. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polynomial_least_deviating_from_zero&oldid=12323