Chebyshev polynomials
of the first kind
Polynomials that are orthogonal on the interval with the weight function
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For the standardized Chebyshev polynomials one has the formula
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and the recurrence relation
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by which one can determine the sequence
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The orthonormalized Chebyshev polynomials are:
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The leading coefficient of , for
, is
. Hence Chebyshev polynomials with leading coefficient 1 are defined by the formula
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The zeros of , given by
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frequently occur as interpolation nodes in quadrature formulas. The polynomial is a solution of the differential equation
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The polynomials deviate as least as possible from zero on the interval
, that is, for any other polynomial
of degree
with leading coefficient 1 one has the following condition
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On the other hand, for any polynomial of degree
or less and satisfying
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one has, for any , the inequality
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If a function is continuous on the interval
and if its modulus of continuity
satisfies the Dini condition
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then this function can be expanded in a Fourier–Chebyshev series,
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which converges uniformly on . The coefficients in this series are defined by the formula
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If the function is
-times continuously differentiable on
and if its
-th derivative
satisfies a Lipschitz condition of order
, i.e.
, then one has the inequality
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where the constant does not depend on
and
.
Chebyshev polynomials of the second kind are defined by
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These polynomials are orthogonal on the interval with weight function
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For any polynomial with leading coefficient 1 one has the inequality
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The Chebyshev polynomials were introduced in 1854 by P.L. Chebyshev (cf. [1]). Both systems of Chebyshev polynomials are special cases of ultraspherical polynomials and Jacobi polynomials.
References
[1] | P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 23–51 (In Russian) |
[2] | G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) |
Chebyshev polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_polynomials&oldid=46330