# Jacobi polynomials

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Orthogonal polynomials on the interval with the weight function The standardized Jacobi polynomials are defined by the Rodrigues formula:  and the orthonormal Jacobi polynomials have the form  The polynomial satisfies the differential equation When and , the orthonormal Jacobi polynomials satisfy the following weighted estimate:  where the constant does not depend on and . At the sequence grows at a rate and , respectively.

Fourier series in Jacobi polynomials (cf. Fourier series in orthogonal polynomials) inside the interval are similar to trigonometric Fourier series. But in neighbourhoods of the end points of this interval, the orthogonality properties of Fourier–Jacobi series are different, because at the orthonormal Jacobi polynomials grow unboundedly. The Fourier–Jacobi series of a function is uniformly convergent on if is times continuously differentiable on this segment and with , where Under these conditions the following inequality holds:  where the constant does not depend on and . On the other hand, when and , the remainder in the Fourier–Jacobi series for satisfies the following weighted estimate:  where , the constant does not depend on and , and is the best uniform approximation error (cf. Best approximation) of the continuous function on by polynomials of degree not exceeding .

The Jacobi polynomials were introduced by C.G.J. Jacobi  in connection with the solution of the hypergeometric equation. Special cases of the Jacobi polynomials are: the Legendre polynomials (when ); the Chebyshev polynomials of the first kind (when ); the Chebyshev polynomials of the second kind (when ); and the ultraspherical polynomials (when ).