Orthogonal polynomials

A system of polynomials $ \{ P _ {n} \} $ which satisfy the condition of orthogonality

$$ \int\limits _ { a } ^ { b } P _ {n} ( x) P _ {m} ( x) h( x) dx = 0,\ \ n \neq m, $$

whereby the degree of every polynomial $ P _ {n} $ is equal to its index $ n $, and the weight function (weight) $ h( x) \geq 0 $ on the interval $ ( a, b) $ or (when $ a $ and $ b $ are finite) on $ [ a, b] $. Orthogonal polynomials are said to be orthonormalized, and are denoted by $ \{ \widehat{P} _ {n} \} $, if every polynomial has positive leading coefficient and if the normalizing condition

$$ \int\limits _ { a } ^ { b } \widehat{P} {} _ {n} ^ {2} ( x) h( x) dx = 1 $$

is fulfilled. If the leading coefficient of each polynomial is equal to 1, then the system of orthogonal polynomials is denoted by $ \{ \widetilde{P} _ {n} \} $.

The system of orthogonal polynomials $ \{ \widehat{P} _ {n} \} $ is uniquely defined if the weight function (differential weight) $ h $ is Lebesgue integrable on $ ( a, b) $, is not equivalent to zero and, in the case of an unbounded interval $ ( a, b) $, has finite moments

$$ h _ {n} = \int\limits _ { a } ^ { b } x ^ {n} h( x) dx. $$

Instead of a differential weight $ h $, an integral weight $ d \sigma ( x) $ can be examined, where $ \sigma $ is a bounded non-decreasing function with an infinite set of points of growth (in this case, the integral in the condition of orthogonality is understood in the Lebesgue–Stieltjes sense).

For the polynomial $ P _ {n} $ of degree $ n $ to be part of the system $ \{ P _ {n} \} $ with weight $ h $, it is necessary and sufficient that, for any polynomial $ Q _ {m} $ of degree $ m < n $, the condition

$$ \int\limits _ { a } ^ { b } P _ {n} ( x) Q _ {m} ( x) h( x) dx = 0 $$

is fulfilled. If the interval of orthogonality $ ( a, b) $ is symmetric with respect to the origin and the weight function $ h $ is even, then every polynomial $ P _ {n} $ contains only those degrees of $ x $ which have the parity of the number $ n $, i.e. one has the identity

$$ P _ {n} (- x) \equiv (- 1) ^ {n} P _ {n} ( x). $$

The zeros of orthogonal polynomials in the case of the interval $ ( a, b) $ are all real, different and distributed within $ ( a, b) $, while between two neighbouring zeros of the polynomial $ P _ {n} $ there is one zero of the polynomial $ P _ {n-1} $. Zeros of orthogonal polynomials are often used as interpolation points and in quadrature formulas.

Any three consecutive polynomials of a system of orthogonal polynomials are related by a recurrence formula

$$ P _ {n+1} ( x) = \ ( a _ {n} x + b _ {n} ) P _ {n} ( x) - c _ {n} P _ {n-1} ( x),\ \ n = 1, 2 \dots $$

where

$$ P _ {0} ( x) = \mu _ {0} , $$

$$ P _ {1} ( x) = \mu _ {1} x + \nu _ {1} \dots $$

$$ P _ {n} ( x) = \mu _ {n} x ^ {n} + \nu _ {n} x ^ {n-1} + \dots , $$

$$ a _ {n} = \frac{\mu _ {n+1} }{\mu _ {n} } ,\ b _ {n} = a _ {n} \left ( \frac{\nu _ {n+1} }{\mu _ {n+1} } - \frac{\nu _ {n} }{\mu _ {n} } \right ) , $$

$$ c _ {n} = \frac{\mu _ {n+1} \mu _ {n-1} }{\mu _ {n} ^ {2} } \cdot \frac{d _ {n} ^ {2} }{d _ {n-1} ^ {2} } ,\ \ d _ {n} ^ {2} = \int\limits _ { a } ^ { b } P _ {n} ^ {2} ( x) h( x) dx. $$

The number $ d _ {n} ^ {-1} $ is a normalization factor of the polynomial $ P _ {n} $, such that the system $ \{ d _ {n} ^ {-1} P _ {n} \} $ is orthonormalized, i.e.

$$ d _ {n} ^ {-1} P _ {n} ( x) = \widehat{P} _ {n} ( x). $$

For orthogonal polynomials one has the Christoffel–Darboux formula:

$$ \sum _ { k=0} ^ { n } \frac{1}{d _ {k} ^ {2} } P _ {k} ( x) P _ {k} ( t) = $$

$$ = \ \frac{1}{d _ {n} ^ {2} } \frac{\mu _ {n} }{\mu _ {n+1} } \frac{P _ {n+1} ( x) P _ {n} ( t) - P _ {n} ( x) P _ {n+1} ( t) }{x-t} . $$

Orthogonal polynomials are represented in terms of the moments $ \{ h _ {k} \} $ of the weight function $ h $ by the formula

$$ \widehat{P} _ {n} ( x) = \frac{1}{\sqrt {\Delta _ {n-1} \Delta _ {n} } } \psi _ {n} ( x), $$

where

$$ \psi _ {n} ( x) = \left | \begin{array}{llll} h _ {0} &h _ {1} &\dots &h _ {n} \\ h _ {1} &h _ {2} &\dots &h _ {n+1} \\ \cdot &\cdot &\dots &\cdot \\ h _ {n-1} &h _ {n-2} &\dots &h _ {2n-1} \\ 1 & x &\dots &x ^ {n} \\ \end{array} \right | , $$

while the determinant $ \Delta _ {n-1} $ is obtained from $ \psi _ {n} ( x) $ by cancelling the last row and column and $ \Delta _ {n} $ is defined in the same way from $ \psi _ {n+1} ( x) $.

On a set of polynomials $ \widetilde{Q} _ {n} $ of degree $ n $ with leading coefficient equal to one, the minimum of the functional

$$ F( \widetilde{Q} _ {n} ) = \int\limits _ { a } ^ { b } \widetilde{Q} {} _ {n} ^ {2} ( x) h( x) dx $$

is achieved if and only if

$$ \widetilde{Q} _ {n} ( x) \equiv \widetilde{P} _ {n} ( x); $$

moreover, this minimum is equal to $ d _ {n} ^ {2} $.

If the polynomials $ \{ P _ {n} \} $ are orthonormal with weight $ h $ on the segment $ [ a, b] $, then when $ p > 0 $, the polynomials

$$ \widehat{Q} _ {n} ( t) = \sqrt p \widehat{P} _ {n} ( pt+ q),\ \ n = 0, 1 \dots $$

are orthonormal with weight $ h( pt+ q) $ on the segment $ [ A, B] $ which transfers to the segment $ [ a, b] $ as a result of the linear transformation $ x = pt + q $. For this reason, when studying the asymptotic properties of orthogonal polynomials, the case of the standard segment $ [- 1, 1] $ is considered first, while the results thus obtained cover other cases as well.

The most important orthogonal polynomials encountered in solving boundary problems of mathematical physics are the so-called classical orthogonal polynomials: the Laguerre polynomials $ \{ L _ {n} ( x; \alpha ) \} $( for which $ h( x) = x ^ \alpha e ^ {-x} $, $ \alpha > - 1 $, and with interval of orthogonality $ ( 0, \infty ) $); the Hermite polynomials $ \{ H _ {n} ( x) \} $( for which $ h( x) = \mathop{\rm exp} (- x ^ {2} ) $, and with interval of orthogonality $ (- \infty , \infty ) $); the Jacobi polynomials $ \{ P _ {n} ( x; \alpha , \beta ) \} $( for which $ h( x) = ( 1- x) ^ \alpha ( 1+ x) ^ \beta $, $ \alpha > - 1 $, $ \beta > - 1 $, and with interval of orthogonality $ [- 1, 1] $) and their particular cases: the ultraspherical polynomials, or Gegenbauer polynomials, $ \{ P _ {n} ( x, \alpha ) \} $( for which $ \alpha = \beta $), the Legendre polynomials $ \{ P _ {n} ( x) \} $( for which $ \alpha = \beta = 0 $), the Chebyshev polynomials of the first kind $ \{ T _ {n} ( x) \} $( for which $ \alpha = \beta = - 1/2 $) and of the second kind $ \{ U _ {n} ( x) \} $( for which $ \alpha = \beta = 1/2 $).

The weight function $ h $ of the classical orthogonal polynomials $ \{ K _ {n} \} $ satisfies the Pearson differential equation

$$ \frac{h ^ \prime ( x) }{h(x)} = \frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x ^ {2} } = \frac{A(x)}{B(x)} ,\ \ x \in ( a, b) , $$

whereby, at the ends of the interval of orthogonality, the conditions

$$ \lim\limits _ {x \downarrow a } h( x) B( x) = \lim\limits _ {x \uparrow b } h( x) B( x) = 0 $$

are fulfilled.

The polynomial $ y = K _ {n} ( x) $ satisfies the differential equation

$$ B( x) y ^ {\prime\prime} + [ A( x) + B ^ \prime ( x)] y ^ \prime - n[ p _ {1} + ( n+ 1) q _ {2} ] y = 0. $$

For classical orthogonal polynomials one has the generalized Rodrigues formula

$$ K _ {n} ( x) = \frac{c _ {n} }{h( x) } \frac{d ^ {n} }{dx ^ {n} } [ h( x) B ^ {n} ( x)], $$

where $ c _ {n} $ is a normalization coefficient, and the differentiation formulas

$$ \frac{d}{dx} L _ {n} ( x; \alpha ) = \ - L _ {n-1} ( x; \alpha + 1),\ \ \frac{d}{dx} H _ {n} ( x) = 2nH _ {n-1} ( x), $$

$$ \frac{d}{dx} P _ {n} ( x; \alpha , \beta ) = \frac{1}{2} ( \alpha + \beta + n + 1 ) P _ {n-1} ( x; \alpha + 1, \beta + 1). $$

For particular cases of the classical orthogonal polynomials one has representations using the hypergeometric function

$$ P _ {n} ( x; \alpha , \beta ) = \left ( \begin{array}{c} n+ a \\ n \end{array} \right ) F \left ( - n, n + \alpha + \beta + 1; \alpha + 1; 1- \frac{x}{2} \right ) , $$

$$ P _ {n} ( x) = F \left ( - n, n+ 1; 1; 1- \frac{x}{2} \right ) , $$

$$ T _ {n} ( x) = F \left ( - n, n; \frac{1}{2} ; 1- \frac{x}{2} \right ) , $$

$$ U _ {n} ( x) = ( n+ 1) F \left ( - n, n+ 2 ; \frac{3}{2} ; 1- \frac{x}{2} \right ) $$

and using the degenerate hypergeometric function

$$ L _ {n} ( x ; \alpha ) = \ \left ( \begin{array}{c} {n+ \alpha } \\ n \end{array} \right ) \Phi (- n ; \alpha + 1; x), $$

$$ H _ {2n} ( x) = (- 1) ^ {n} ( 2n)! over {n!} \Phi \left ( - n; \frac{1}{2} ; x ^ {2} \right ) , $$

$$ H _ {2n+1} ( x) = (- 1) ^ {n} ( 2n+ 1)! over {n!} 2 x \Phi \left ( - n; \frac{3}{2} ; x ^ {2} \right ) . $$

Historically, the first orthogonal polynomials were the Legendre polynomials. Then came the Chebyshev polynomials, the general Jacobi polynomials, the Hermite and the Laguerre polynomials. All these classical orthogonal polynomials play an important role in many applied problems.

The general theory of orthogonal polynomials was formulated by P.L. Chebyshev. The basic research apparatus used was the continued fraction expansion of the integral $$ \int\limits _ { a } ^ { b } \frac{h(t)}{x-t} dt ; $$ the denominators of the convergents of this continued fraction form a system of orthogonal polynomials on the interval $ ( a, b) $ with weight $ h $.

In the study of orthogonal polynomials, great attention is paid to their asymptotic properties, since the conditions of convergence of Fourier series in orthogonal polynomials depend on these properties.

The asymptotic properties of the classical orthogonal polynomials were first studied by V.A. Steklov in 1907 (see [8]). He used and perfected the Liouville method, which was previously used in the study of solutions of the Sturm–Liouville equation. The Liouville–Steklov method was subsequently widely used, as a result of which the asymptotic properties of the Jacobi, Hermite and Laguerre orthogonal polynomials have been studied extensively.

In the general case of orthogonality on $ [- 1, 1] $ with arbitrary weight satisfying certain qualitative conditions, asymptotic formulas for orthogonal polynomials were first discovered by G. Szegö in 1920–1924. He introduced polynomials which were orthogonal on the circle, studied their basic properties and found an extremely important formula, representing polynomials orthogonal on $ [- 1, 1] $ by polynomials orthogonal on the circle. In his study of the asymptotic properties of polynomials orthogonal on the circle, Szegö developed a method based on a special generalization of the Fejér theorem on the representation of non-negative trigonometric polynomials by using methods and results of the theory of analytic functions.

In 1930, S.N. Bernstein [S.N. Bernshtein] [2], in his research on the asymptotic properties of orthogonal polynomials, used methods and results of the theory of approximation of functions. He examined the case of a weight function of the form

$$ \tag{1 } h( x) = \frac{h _ {0} ( x) }{\sqrt {1- x ^ {2} } } ,\ \ x \in (- 1, 1), $$

where the function $ h _ {0} ( x) $, called a trigonometric weight, satisfies the condition

$$ 0 < c _ {1} \leq h _ {0} ( x) \leq c _ {2} < \infty . $$

If on the whole segment $ [- 1, 1] $ the function $ h _ {0} ( x) $ satisfies a Dini–Lipschitz condition of order $ \gamma = 1 + \epsilon $, where $ \epsilon > 0 $, i.e. if

$$ | h _ {0} ( x + \delta ) - h _ {0} ( x) | \leq \frac{M}{| \mathop{\rm ln} | \delta | | ^ \gamma } ,\ x, x+ \delta \in [- 1, 1], $$

then for the polynomials $ \{ \widehat{P} _ {n} \} $ orthonormal with weight (1) on the whole segment $ [- 1, 1] $, one has the asymptotic formula

$$ \widehat{P} _ {n} ( x) = \sqrt { \frac{2}{\pi h _ {0} ( x) } } \cos ( n \theta + q) + O \left [ \frac{1}{( \mathop{\rm ln} n ) ^ \epsilon } \right ] , $$

where $ \theta = \mathop{\rm arccos} x $ and $ q $ depends on $ \theta $.

In the study of the convergence of Fourier series in orthogonal polynomials the question arises of the conditions of boundedness of the orthogonal polynomials, either at a single point, on a set $ A \subset [- 1, 1] $ or on the whole interval of orthogonality $ [- 1, 1] $, i.e. conditions are examined under which an inequality of the type

$$ \tag{2 } | \widehat{P} _ {n} ( x) | \leq M,\ \ x \in A \subseteq [- 1, 1] , $$

occurs. Steklov first posed this question in 1921. If the trigonometric weight $ h _ {0} ( x) $ is bounded away from zero on a set $ A $, i.e. if

$$ \tag{3 } h _ {0} ( x) \geq c _ {3} > 0,\ \ x \in A \subseteq [- 1, 1], $$

and satisfies certain extra conditions, then the inequality (2) holds. In the general case,

$$ \tag{4 } | \widehat{P} _ {n} ( x) | \leq \epsilon _ {n} \sqrt n ,\ \ \epsilon _ {n} \rightarrow 0,\ \ x \in [- 1, 1] , $$

follows from (3), when $ A=[- 1, 1] $, without extra conditions.

The zeros of the weight function are singular points in the sense that the properties of the sequence $ \{ \widehat{P} _ {n} \} $ are essentially different at the zeros and at other points of the interval of orthogonality. For example, let the weight function have the form

$$ h( x) = \ \frac{h _ {1} ( x) }{\sqrt {1- x ^ {2} } } \prod _ { k=1 } ^ { m } | x - x _ {k} | ^ {\gamma _ {k} } ,\ \ \gamma _ {k} > 0,\ \ x _ {k} \in (- 1,1). $$

If the function $ h _ {1} ( x) $ is positive and satisfies a Lipschitz condition on $ [- 1, 1] $, then the sequence $ \{ \widehat{P} _ {n} \} $ is bounded on every segment $ [ a, b] \subset [- 1, 1] $ which does not contain the points $ \{ x _ {k} \} $, while the inequalities

$$ | \widehat{P} _ {n} ( x _ {k} ) | \leq c _ {4} ( n+ 1) ^ {\gamma _ {k} /2 } ,\ \ k = 1 \dots m , $$

hold at the zeros.

The case where the zeros of the weight function are positioned at the ends of the segment of orthogonality was studied by Bernstein [2]. One of the results is that if the weight function has the form

$$ h( x) = h _ {1} ( x)( 1- x) ^ \alpha ( 1+ x) ^ \beta ,\ \ x \in [- 1, 1], $$

where the function $ h _ {1} ( x) $ is positive and satisfies a Lipschitz condition, then for $ \alpha > - 1/2 $, $ \beta > - 1/2 $, the orthogonal polynomials permit the weighted estimation

$$ ( 1- x) ^ {\alpha /2+ 1/4 } ( 1+ x) ^ {\beta /2+ 1/4 } | \widehat{P} _ {n} ( x) | \leq c _ {5} ,\ x \in [- 1, 1], $$

while at the points $ x = \pm 1 $ they increase at a rate $ n ^ {\alpha + 1/2 } $ and $ n ^ {\beta + 1/2 } $, respectively.

In the theory of orthogonal polynomials, so-called comparison theorems are often studied. One such is the Korous comparison theorem: If the polynomials $ \{ \widehat \omega _ {n} \} $ are orthogonal with weight $ p $ on the segment $ [ a, b] $ and are uniformly bounded on a set $ A \subset [ a, b] $, then the polynomials $ \{ \widehat{P} _ {n} \} $, orthogonal with weight $ h = p \cdot q $, are also bounded on this set, provided $ q $ is positive and satisfies a Lipschitz condition of order $ \alpha = 1 $ on $ [ a, b] $. Similarly, given certain conditions on $ q $, asymptotic formulas or other asymptotic properties can be transferred from the system $ \{ \widehat \omega _ {n} \} $ to the system $ \{ \widehat{P} _ {n} \} $. Moreover, if $ q $ is a non-negative polynomial of degree $ m $ on $ [ a, b] $, then the polynomials $ \{ \widehat{P} _ {n} \} $ can be represented by the polynomials $ \{ \widehat \omega _ {n} \} $ using determinants of order $ m+ 1 $( see [8]). Effective formulas for orthogonal polynomials have also been obtained for weight functions of the form

$$ \frac{1}{Q _ {m} ( x) \sqrt {1- x ^ {2} } } ,\ \ \frac{\sqrt {1- x ^ {2} } }{Q _ {m} ( x) } ,\ \ \sqrt {1- \frac{x}{1+x} } \frac{1}{Q _ {m} ( x) } , $$

where $ Q _ {m} $ is an arbitrary positive polynomial on $ [- 1, 1] $( see [8]). In most cases, the calculation of orthogonal polynomials with arbitrary weight is difficult for large numbers $ n $.

References

[1]	P.L. Chebyshev, "Complete collected works" , 2 , Moscow-Leningrad (1947) pp. 103–126; 314–334; 335–341; 357–374 (In Russian)
[2]	S.N. Bernshtein, "Collected works" , 2 , Moscow (1954) pp. 7–106 (In Russian)
[3]	Ya.L. Geronimus, "Orthogonal polynomials" Transl. Amer. Math. Soc. , 108 (1977) pp. 37–130
[4]	P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian)
[5]	V.B. Uvarov, "Special functions of mathematical physics" , Birkhäuser (1988) (Translated from Russian)
[6]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[7]	D. Jackson, "Fourier series and orthogonal polynomials" , Carus Math. Monogr. , 6 , Math. Assoc. Amer. (1971)
[8]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[9]	, Guide to special functions , Moscow (1979) (In Russian; translated from English)
[10]	J.A. Shohat, E. Hille, J.L. Walsh, "A bibliography on orthogonal polynomials" , Nat. Acad. Sci. USA (1940)

Comments

See also Fourier series in orthogonal polynomials. Two other textbooks are [a3] and [a2]. See [a1] for some more information on the history of the classical orthogonal polynomials. Regarding the asymptotic properties of the classical orthogonal polynomials it should be observed that many workers (P.S. Laplace, E. Heine, G. Darboux, T.J. Stieltjes, E. Hilb, etc.) preceded Stekov, but he was the first to adapt Liouville's method.

See [a5] for state-of-the-art surveys of many aspects of orthogonal polynomials. In particular, the general theory of orthogonal polynomials with weight functions on unbounded intervals has made big progress, see also [a4].

References

[a1]	R.A. Askey, "Discussion of Szegö's paper "An outline of the history of orthogonal polynomials" " R.A. Askey (ed.) , G.P. Szegö: Collected Works , 3 , Birkhäuser (1982) pp. 866–869
[a2]	T.S. Chihara, "An introduction to orthogonal polynomials" , Gordon & Breach (1978)
[a3]	G. Freud, "Orthogonal polynomials" , Pergamon (1971) (Translated from German)
[a4]	D.S. Lubinsky, "A survey of general orthogonal polynomials for weights on finite and infinite intervals" Acta Applic. Math. , 10 (1987) pp. 237–296
[a5]	P. Nevai (ed.) , Orthogonal polynomials: theory and practice , Kluwer (1990)