Jacobi polynomials

Orthogonal polynomials on the interval $ [- 1, 1] $ with the weight function

$$ h ( x) = ( 1 - x) ^ \alpha ( 1 + x) ^ \beta ,\ \ \alpha , \beta > - 1,\ \ x \in [- 1, 1]. $$

The standardized Jacobi polynomials are defined by the Rodrigues formula:

$$ P _ {n} ( x; \alpha , \beta ) = \ P _ {n} ^ {( \alpha , \beta ) } ( x) = $$

$$ = \ \frac{(- 1) ^ {n} }{n! 2 ^ {n} } ( 1 - x) ^ {- \alpha } ( 1 + x) ^ {- \beta } \frac{d ^ {n} }{dx ^ {n} } [( 1 - x) ^ \alpha ( 1 + x ) ^ \beta ( 1 - x ^ {2} ) ^ {n} ], $$

and the orthonormal Jacobi polynomials have the form

$$ \widehat{P} _ {n} ( x; \alpha , \beta ) = $$

$$ = \ \sqrt { \frac{n! ( \alpha + \beta + 2n + 1) \Gamma ( \alpha + \beta + n + 1) }{2 ^ {\alpha + \beta + 1 } \Gamma ( \alpha + n + 1) \Gamma ( \beta + n + 1) } } P _ {n} ( x; \alpha , \beta ). $$

The polynomial $ P _ {n} ( x; \alpha , \beta ) $ satisfies the differential equation

$$ ( 1 - x ^ {2} ) y ^ {\prime\prime} + [ \beta - \alpha - ( \alpha + \beta + 2) x] y ^ \prime + n ( n + \alpha + \beta + 1) y = 0. $$

When $ \alpha \geq - 1/2 $ and $ \beta \geq - 1/2 $, the orthonormal Jacobi polynomials satisfy the following weighted estimate:

$$ ( 1 - x) ^ {( 2 \alpha + 1)/4 } ( 1 + x) ^ {( 2 \beta + 1)/4 } | \widehat{P} _ {n} ( x; \alpha , \beta ) | \leq c _ {1} , $$

$$ x \in [- 1, 1], $$

where the constant $ c _ {1} $ does not depend on $ n $ and $ x $. At $ x = \pm 1 $ the sequence $ \{ \widehat{P} _ {n} ( x; \alpha , \beta ) \} $ grows at a rate $ n ^ {\alpha + 1/2 } $ and $ n ^ {\beta + 1/2 } $, respectively.

Fourier series in Jacobi polynomials (cf. Fourier series in orthogonal polynomials) inside the interval $ (- 1, 1) $ 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 $ x = \pm 1 $ the orthonormal Jacobi polynomials grow unboundedly. The Fourier–Jacobi series of a function $ f $ is uniformly convergent on $ [- 1, 1] $ if $ f $ is $ p $ times continuously differentiable on this segment and $ f ^ { ( p) } \in \mathop{\rm Lip} \gamma $ with $ p + \gamma > q + 1/2 $, where

$$ q = \max \{ \alpha , \beta \} > - { \frac{1}{2} } . $$

Under these conditions the following inequality holds:

$$ \left | f ( x) - \sum _ {k = 0 } ^ { n } a _ {k} \widehat{P} _ {k} ( x; \alpha , \beta ) \right | \leq \ \frac{c _ {2} }{n ^ {p + \gamma } } n ^ {( 2q + 1)/2 } , $$

$$ x \in [- 1, 1], $$

where the constant $ c _ {2} $ does not depend on $ n $ and $ x $. On the other hand, when $ \alpha \geq - 1/2 $ and $ \beta \geq - 1/2 $, the remainder in the Fourier–Jacobi series for $ f $ satisfies the following weighted estimate:

$$ ( 1 - x ^ {2} ) ^ {1/4} \sqrt {h ( x) } \left | f ( x) - \sum _ {k = 0 } ^ { n } a _ {k} \widehat{P} _ {k} ( x; \alpha , \beta ) \right | \leq $$

$$ \leq \ c _ {3} E _ {n} ( f ) \mathop{\rm ln} n,\ \ x \in [- 1, 1], $$

where $ n \geq 2 $, the constant $ c _ {3} $ does not depend on $ n $ and $ x $, and $ E _ {n} ( f ) $ is the best uniform approximation error (cf. Best approximation) of the continuous function $ f $ on $ [- 1, 1] $ by polynomials of degree not exceeding $ n $.

The Jacobi polynomials were introduced by C.G.J. Jacobi [1] in connection with the solution of the hypergeometric equation. Special cases of the Jacobi polynomials are: the Legendre polynomials (when $ \alpha = \beta = 0 $); the Chebyshev polynomials of the first kind (when $ \alpha = \beta = - 1/2 $); the Chebyshev polynomials of the second kind (when $ \alpha = \beta = 1/2 $); and the ultraspherical polynomials (when $ \alpha = \beta $).

See also Classical orthogonal polynomials.

References

[1]	C.G.J. Jacobi, "Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe" J. Reine Angew. Math. , 56 (1859) pp. 149–165
[2]	P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1978) (In Russian)

Comments

See also [a4], [a1] and Fourier series in orthogonal polynomials.

Let $ \alpha , \beta > - 1 $ and $ - 1 < x, y < 1 $. Then there is a product formula of the form

$$ \frac{P _ {n} ^ {( \alpha , \beta ) } ( x) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) } \frac{P _ {n} ^ {( \alpha , \beta ) } ( y) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) } = \ \int\limits _ { - } 1 ^ { 1 } \frac{P _ {n} ^ {( \alpha , \beta ) } ( z) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) } \ d \mu _ {x,y} ( z),\ \ n = 0, 1 \dots $$

with positive measure $ d \mu _ {x,y} ( z) = d \mu _ {x,y} ^ {\alpha , \beta } ( z) $ if and only if $ \alpha \geq \beta $ and either $ \beta \geq - 1/2 $ or $ \alpha + \beta \geq 0 $. This yields a positive convolution structure for Jacobi series. For $ \alpha \geq \beta \geq - 1/2 $ the above measure can be computed explicitly from the addition formula for Jacobi polynomials. See [a1], Lecture 4.

For the dual problem one has

$$ P _ {n} ^ {( \alpha , \beta ) } ( x) P _ {m} ^ {( \alpha , \beta ) } ( x) = \ \sum _ {k = | n - m | } ^ { {n } + m } C ( k, m, n) P _ {k} ^ {( \alpha , \beta ) } ( x) , $$

with $ C ( k, m, n) \geq 0 $ if $ \alpha \geq \beta > - 1 $, $ \alpha + \beta \geq - 1 $. This yields a positive dual convolution structure for Jacobi series. See [a1], Lecture 5.

Jacobi polynomials admit many different group-theoretic interpretations. The three most important ones are as matrix elements of the irreducible representations of $ \mathop{\rm SU} ( 2) $( cf. [a5], Chapt. 3), as $ O ( p) \times O ( q) $- invariant spherical harmonics on the unit sphere in $ \mathbf R ^ {p + 1 } $( cf. [a2]) and as zonal spherical functions on the compact symmetric spaces of rank one (cf. [a3], Chapt. 5, §4.3).

References

[a1]	R. Askey, "Orthogonal polynomials and special functions" , Reg. Conf. Ser. Appl. Math. , 21 , SIAM (1975)
[a2]	B.L.J. Braaksma, B. Meulenbeld, "Jacobi polynomials as spherical harmonics" Nederl. Akad. Wetensch. Proc. Ser. A , 71 (1968) pp. 384–389
[a3]	S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4
[a4]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[a5]	N.Ya. Vilenkin, "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian)