CHEBYSHEV POLYNOMIALS

Chebyshev polynomials of the first kind are the trigonometric polynomials defined by:

(1)

whence

(2)

For Chebyshev polynomials, a generalized Rodrigues formula is valid

(3)

A recurrent relationship holds for Chebyshev polynomials

(4)

Chebyshev polynomials for a negative value of n are defined by the relationship:

(5)

Chebyshev polynomials of the first kind are orthogonal with respect to a weight function on the interval [-1, 1]. The orthogonality relationship is:

(6)

The roots of the polynomial T(x), defined by the equality , k = 1, 2, ..., n are often used as cusps of quadrature and interpolation formulas.

Chebyshev polynomials of the first kind with a unit coefficient of the higher term, i.e., are the polynomials least deviated from zero on the interval [-1, 1], i.e., for any other polynomial F_n(x) of degree n with unit heading coefficient the following relationship holds:

(7)

This property of Chebyshev polynomials is used for constructing optimal iteration algorithms in solving problems of heat transfer with the help of numerical methods.