An Introduction to Orthogonal Polynomials

By Theodore S Chihara

Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study.
Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some specific systems of orthogonal polynomials. Numerous examples and exercises, an extensive bibliography, and a table of recurrence formulas supplement the text.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 1, 2014
• ISBN: 9780486141411

Book preview

INDEX

CHAPTER I

Elementary Theory of Orthogonal Polynomials

1 Introduction

It is an elementary exercise in calculus to use the trigonometric identity

to obtain the integration formula

The fact that the integral in (1.2) vanishes is expressed by saying that cos mθ and cos nθ are orthogonal over the interval (0, π) for m ≠ n. We also say that {1, cos θ, cos 2θ, …, cos nθ, …} is an orthogonal sequence over (0, π).

We observe that the change of variable, x = cos θ, converts (1.2) into

where we have written

We have

and by elementary trigonometric identities,

Using the identity (1.1) with m = 1, it is an easy proof by induction to show that Tn(x) is a polynomial in x of degree n. These polynomials are called the Tchebichef polynomials of the first kind. Because of (1.3) we say that the Tn(x) are orthogonal polynomials with respect to (1 – x²)–1/2, – 1 < x is an orthogonal polynomial sequence with respect to the weight function (1 – x²)–1/2 on the interval (–1, 1).

Somewhat more generally, consider a function w which is non-negative and integrable on an interval (a, b). We also assume that w(x) > 0 on a sufficiently large subset of (a, b) so that

(that is, w(x) > 0 on a subset of positive Lebesgue measure). In the event that (a, b) is unbounded, we will also have to impose the additional requirement that the moments

are all finite.

, Pn(x) of degree n, such that

then {Pn(x)} is called an orthogonal polynomial sequence with respect to the weight function w on (a, b).

If we write, for any integrable function f

then (1.5) and (1.6) can be written

is linear; that is,

for arbitrary constants a and b and integrable functions f and g. Without reference to ([π(x)] for any polynomial π(x). Indeed,

The latter observation thus suggests a further generalization. In place of (. We then use (1.8) and (1.10) to define on the vector space of all polynomials in one real variable.

If there is a sequence {Pn(x)} of polynomials satisfying (1.9) and the additional condition

is defined by (. Not every moment sequence {μn} will give rise to an orthogonal polynomial sequence so we will be concerned with existence questions.

Before turning to a more formal and detailed study of this generalization, we consider a second example which will illustrate the greater generality we wish to encompass.

Consider the function of two variables x and w, with parameter a ≠ 0,

Forming the Cauchy product of the two series above, we obtain

where

Since

we see that Pn(x) is a polynomial of degree n. Pn(x), or a certain constant multiple of Pn(x), is a Charlier polynomial and G(x, w) is called a generating function for {Pn(x)}.

We will now show that these polynomials satisfy an orthogonality relation which, at first glance, appears to be of a different type than (1.6). We will proceed formally, leaving it to the reader to supply the justification for the various limit interchanges that are invoked.

Referring to (1.11), we see that

hence

At the same time, because of (1.12) we also have

Comparing coefficients of vm wn in the two resulting series, we conclude

We say that {Pn(x)} is an orthogonal polynomial sequence with respect to the discrete mass distribution which has mass ak/k! at the point k (k = 0, 1, 2, …). If we write

for all polynomials by linearity, then (1.14) can be written

where δmn is the Kronecker delta defined by

Thus both (.

Finally we note that if ψ denotes a step function which is constant on each of the open intervals, (– ∞,0) and (k, k + 1) (k = 0, 1, 2, …), and has a jump of magnitude ak/k! at k (k = 0, 1, 2, …), then (1.14) can be written in terms of a (Riemann) Stieltjes integral as

Exercises

1.1 Show that

1.2 The Tchebichef polynomials of the second kind are defined by

(a) Show that Un(x) is a polynomial in x of degree n.

(b) Prove that

1.3 Use DeMoivre’s theorem to express cos nθ and sin (n + 1)θ/sin θ as polynomials in cos θ and thus obtain explicit formulas for Tn(x) and Un(x).

1.4 Let F(x, w) = e–(x–w)².

(a) Show that

and that the latter is of the form e–x²Hn(x), where Hn(x) is a polynomial of degree n.

(b) Show that

where Hn(x) = (–1)n ex² Dn e–x² is called the Hermite polynomial of degree n.

, prove that

(d) Finally, prove that

1.5

(a) Show that

and thus obtain the relation

(b) Show that H′n(x) = 2xHn(x) – Hn+1(x) and conclude that

(c) Show that y = Hn(x) satisfies the differential equation

1.6 Let H(x, w) = (1 – 2xw + wPn(x)wn.

(a) Show that

and deduce that

Since P0(x) = 1 and P1 = x, it follows that Pn(x) is a polynomial of degree n called the Legendre polynomial.

(b) Use the generating function, H, to prove that

2 The Moment Functional and Orthogonality

and the corresponding orthogonal polynomials. Throughout this book, polynomial will mean a polynomial with complex coefficients in one variable while real polynomial will refer to a polynomial with real coefficients.

DEFINITION 2.1 be a complex valued function defined on the vector space of all polynomials by

for all complex numbers αi, and all polynomials πi(x) (i is called the moment functional determined by the formal moment sequence {μn}. The number μn is called the moment of order n.

It follows immediately that if π(xckxk, then

x is always considered a real variable in these formulas so we also have

where z denotes the complex conjugate of the complex number z.

DEFINITION 2.2 called an orthogonal polynomial sequence provided for all nonnegative integers m and n,

Orthogonal polynomial sequence will be abbreviated OPS and we will use such phrases as "{Pn(x." When there is no danger of ambiguity, we will speak loosely of the Pn(x) as orthogonal polynomials.

If {Pn(x[Pn²(x)] = 1 (n 0), then it will be called an orthonormal polynomial sequence. That is, {Pn(x)} is an orthonormal polynomial sequence if Pn(x) is a polynomial of degree n and

In the general case, conditions (ii) and (iii) of Definition 2.2 can be replaced by

(Here, δmn is Kronecker’s delta defined by (1.16).)

exists, then

[1] = 0. Somewhat less trivially it is easy to show that no OPS can exist if, for example, μ0 = μ1 = μ2 = 1. For in this case, we would have

and (ii) requires

Thus we must have b = –c and this yields

Before taking up existence questions in earnest, however, we first note some equivalents to Definition 2.2.

THEOREM 2.1 be a moment functional and let {Pn(x)} be a sequence of polynomials. Then the following are equivalent:

(a) (Pn(x;

[π(x)Pn(x)] = 0 for every polynomial π(x) of degree m < n [π(x) Pn(x)] ≠ 0 if m = n;

[xmPn(x)] = Knδmn where Kn ≠ 0, m = 0, 1, …, n.

Proof Let {Pn(x. Since each Pk(x) is of degree k, it is clear that {P0(x), P1(x), … , Pm(x)} is a basis for the vector subspace of polynomials of degree at most m. Thus if π(x) is a polynomial of degree m, there exist constants ck such that

,

THEOREM 2.2 Let {Pn(x. Then for every polynomial π(x) of degree n,

where

Proof As noted previously, if π(x) is a polynomial of degree n, then there are constants ck such that

Multiplying both sides of this equation by Pm(xwe obtain

COROLLARY If {Pn(x, then each Pn(x) is uniquely determined up to an arbitrary non-zero factor. That is, if {Qn(x, then there are constants cn ≠ 0 such that

Proof If {Qn(x, then by Theorem 1.1,

Thus taking π(x) = Qn(x) in

It is clear that if {Pn(x, then so is {cnPn(x)} for every sequence of non-zero constants cn. The above Corollary shows conversely that an OPS {Pn(x)} is uniquely determined if it satisfies an additional condition that fixes the leading coefficient (the coefficient of xn) of each Pn(x).

The simplest and most direct method of singling out a particular OPS for a given moment functional is to specify explicitly the value of each leading coefficient. We will usually standardize by requiring that each Pn(x) be a monic polynomial—that is, that Pn(x) have 1 as its leading coefficient. An OPS in which each Pn(x) is monic will be referred to as a monic OPS.

Clearly if {Pn(x)} is an OPS and kn denotes the leading coefficient of Pn(x), then

.

On the other hand,

yields {pn(x> 0 and in this case pn(x) can be uniquely determined by the usual additional requirement that its leading coefficient be positive.

Finally, we note the obvious fact that if {Pn(x, then {Pn(x′ such that for some fixed constant, c ≠ 0,

Exercises

[xn] = an (n exists.

2.2 Let Pn(x) = xn (n 0). Show that {Pn(x)} is not an OPS.

exists. Let Cn be arbitrary non-zero numbers and show that each of the following uniquely determines a corresponding OPS, {Pn(x)}.

(a) Pn(x0) = Cn where x0 is not a zero of any Pk(x);

[xnPn(x)] = Cn;

(c)

with moment sequence {μnbe defined by

If {Pn(x.

2.5 Find explicitly {Pn(x)} such that

Pm(x)Pn(x)x–1/2(1 – x)–1/2 dx = Knδmn where Kn = π/2 for n > 0 and K0 = π

Pm(x)Pn(x)e–x²/2dx n! δmn;

(c)

2.6 If {Pn(xand if kn denotes the leading coefficient of Pn(x[Pm(x)Pn(x)] = Ln δmn, where Ln = kn (x)]/kn (n 0).

2.7 With the notation of Theorem 2.2, prove that for every polynomial π(x) of degree n,

3 Existence of OPS

In order to discuss existence theorems for OPS, we introduce the determinants

THEOREM 3.1 be a moment functional with moment sequence {μnis

Proof Write

Recalling Theorem 2.1, we observe that the orthogonality conditions

are equivalent to the system

exists, it is uniquely determined by the constants Kn in (3.2) (Ex. 2.3). It then follows that (3.2) has a unique solution so that Δn ≠ 0 (n 0).

Conversely, if Δn ≠ 0, then for arbitrary Kn ≠ 0, (3.3) has a unique solution so Pn(x) satisfying (3.2) exists. We also have

which is valid for n = 0 also if we define Δ–1 = 1. It follows that Pn(x) is of degree n, hence {Pn(x

Formula (3.4) is sufficiently useful to note it formally.

THEOREM 3.2 Let {Pn(x. Then for any polynomial πn(x) of degree n,

where an denotes the leading coefficient of πn(x) and kn denotes the leading coefficient of Pn(x).

Proof Writing

where πn–1(x) is a polynomial of degree n – 1, we have

Thus (3.5) follows from (3.4) with kn == cnn

is defined by a non-negative weight function as in (is frequently defined in terms of a Stieltjes integral:

where ψ is a bounded, non-decreasing function such that the set

is an infinite set.

In such cases, if π(x) is a polynomial not identically zero which is nonnegative for all real x[π(x)] > 0. It will be shown in Chapter II that this property characterizes moment functionals that can be represented as in (3.6). For now, we make the following definition.

DEFINITION 3.1 is called positive-definite [π(x)] > 0 for every polynomial π(x) that is not identically zero and is non-negative for all real x.

is positive-definite, it follows immediately that

Since

it follows by induction that μ2k+1 is real.

is positive-definite, a step-by-step method of constructing a corresponding ortho normal polynomial sequence can be described. Known as the Gram-Schmidt process it produces real orthonormal polynomials as follows.

First define

.

Next let

Then

provided we choose a [xp0(x)]. With this choice of a, then, we define

and observe that

Note also that p0(x) and p1(x) are both real polynomials.

In general, suppose p0(x), p1(x), …, pn(x) have been constructed such that each pi(x) is a real polynomial of degree i and

We then define Pn+1(x) by

Then Pn+1(x) is a real polynomial of degree n + 1 and we have

We therefore set

Then pn+1(x) is real and

exists.

For convenience of reference, we summarize the above formally.

THEOREM 3.3 has real moments and a corresponding OPS consisting of real polynomials exists.

We next relate the concept of positive-definite moment functionals to the determinants (3.1). We will need the following classical result characterizing non-negative polynomials.

LEMMA Let π(x) be a polynomial that is non-negative for all real x. Then there are real polynomials p(x) and q(x) such that

Proof If π(x0 for real x, then π(x) is a real polynomial so its real zeros have even multiplicity and its non-real zeros occur in conjugate pairs. Thus we can