# §29.15 Fourier Series and Chebyshev Series

## §29.15(i) Fourier Coefficients

### ¶ Polynomial

When , , the Fourier series (29.6.1) terminates:

A convenient way of constructing the coefficients, together with the eigenvalues, is as follows. Equations (29.6.4), with , (29.6.3), and can be cast as an algebraic eigenvalue problem in the following way. Let

be the tridiagonal matrix with , , as in (29.3.11), (29.3.12). Let the eigenvalues of be with

29.15.3

and also let

be the eigenvector corresponding to and normalized so that

and

Then

and (29.15.1) applies, with again defined as in (29.2.5).

## §29.15(ii) Chebyshev Series

The Chebyshev polynomial of the first kind (§18.3) satisfies . Since (29.2.5) implies that , (29.15.1) can be rewritten in the form

This determines the polynomial of degree for which ; compare Table 29.12.1. The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an tridiagonal matrix; see Arscott and Khabaza (1962).

Using also , with denoting the Chebyshev polynomial of the second kind (§18.3), we obtain

For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).