“Chebyshev Polynomial of the First Kind”的概念、定义、习题解题方法及翻译-数学辞典

单词

Chebyshev Polynomial of the First Kind

释义

Chebyshev Polynomial of the First Kind

A set of Orthogonal Polynomials defined as the solutions to the Chebyshev Differential Equation and denoted. They are used as an approximation to a Least Squares Fit, and are a specialcase of the Ultraspherical Polynomial with . The Chebyshev polynomials of the first kind areillustrated above for and , 2, ..., 5.

The Chebyshev polynomials of the first kind can be obtained from the generating functions

(1)

and

(2)

for

and

(Beeler et al. 1972, Item 15). (A closely related Generating Function is the basisfor the definition of Chebyshev Polynomial of the Second Kind.) They are normalized such that

. Theycan also be written

(3)

or in terms of a Determinant

(4)

In closed form,

(5)

where

is a Binomial Coefficient and

is the Floor Function.Therefore, zeros occur when

(6)

for

, 2, ...,

. Extrema occur for

(7)

where

. At maximum,

, and at minimum,

. The Chebyshev Polynomials areOrthonormal with respect to the Weighting Function

(8)

where

is the Kronecker Delta.Chebyshev polynomials of the first kind satisfy the additional discrete identity

(9)

where

for

, ...,

are the

zeros of

. They also satisfy the RecurrenceRelations

(10)

(11)

for

. They have a Complex integral representation

(12)

and a Rodrigues representation

(13)

Using a Fast Fibonacci Transform with multiplication law

(14)

gives

(15)

Using Gram-Schmidt Orthonormalization in the range (,1) with Weighting Function gives

			(16)

			(17)


			(18)

etc. Normalizing such that

gives

The Chebyshev polynomial of the first kind is related to the Bessel Function of the First Kind andModified Bessel Function of the First Kind by the relations

(19)

(20)

Letting allows the Chebyshev polynomials of the first kind to be written as

(21)

The second linearly dependent solution to the transformed differential equation

(22)

is then given by

(23)

which can also be written

(24)

where

is a Chebyshev Polynomial of the Second Kind. Note that

is therefore not a Polynomial.

The Polynomial

(25)

(of degree

) is the Polynomial of degree

which stays closest to

in the interval

. The maximumdeviation is

at the

points where

(26)

for

, 1, ...,

(Beeler et al. 1972, Item 15).

See also Chebyshev Approximation Formula, Chebyshev Polynomial of the Second Kind

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. ``Chebyshev (Tschebyscheff) Polynomials'' and ``Chebyshev Polynomials--Numerical Applications.'' §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731-748, 1985.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Iyanaga, S. and Kawada, Y. (Eds.). ``Cebysev (Tschebyscheff) Polynomials.'' Appendix A, Table 20.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1478-1479, 1980.

Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990.

Spanier, J. and Oldham, K. B. ``The Chebyshev Polynomials and .'' Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.

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