17.2 Weight Functions and Orthogonal Polynomials 245

⟨en, g⟩2 = 0 for all n ∈ Z. From the above convergence result we deduce, forall f ∈ C1([0, 2π ]),

⟨f , g⟩2 = limN→∞

⟨N∑

n=−N

cn(f )en, g⟩2 = 0.

Since C1([0, 2π ]) is known to be dense in L2([0, 2π ], dx) it follows that g = 0,by Corollary 17.2, hence by Theorem 17.2, this system is an orthonormal basisof L2([0, 2π ], dx). Therefore, every f ∈ L2([0, 2π ], dx) has a Fourier expansion,which converges (in the sense of the L2-topology). Thus, convergence of the Fourierseries in the L2-topology is “natural,” from the point of view of having convergenceof this series for the largest class of functions.

17.2 Weight Functions and Orthogonal Polynomials

Not only for the interval I = [0, 2π ] are the Hilbert spaces L2(I , dx) separable,but for any interval I = [a, b], −∞ ≤ a < b ≤ +∞, as the results of this sectionwill show. Furthermore an orthonormal basis will be constructed explicitly and someinteresting properties of the elements of such a basis will be investigated.

The starting point is a weight function ρ : I→R on the interval I which is assumedto have the following properties:

1. On the interval I , the function ρ is strictly positive: ρ(x) > 0 for all x ∈ I.

2. If the interval I is not bounded, there are two positive constants α and C suchthat ρ(x) eα|x| ≤ C for all x ∈ I.

The strategy to prove that the Hilbert space L2(I , dx) is separable is quite simple. Afirst step shows that the countable set of functions ρn(x) = xnρ(x), n = 0, 1, 2, . . .is total in this Hilbert space. The Gram–Schmidt orthonormalization then produceseasily an orthonormal basis.

Lemma 17.1 The system of functions {ρn : n = 0, 1, 2, . . . } is total in the Hilbertspace L2(I , dx), for any interval I .

Proof For the proof we have to show: If an elementh ∈ L2(I , dx) satisfies ⟨ρn, h⟩2 =0 for all n, then h = 0.

In the case I ̸= R we consider h to be be extended by 0 to R\I and thus get afunction h ∈ L2(R, dx). On the strip Sα = {p = u + iv ∈ C : u, v ∈ R, |v| < α},introduce the auxiliary function

F (p) =∫

Rρ(x)h(x) eipx dx.

The growth restriction on the weight function implies that F is a well-defined holo-morphic function on Sα (see Exercises). Differentiation of F generates the functions

246 17 Separable Hilbert Spaces

ρn in this integral:

F (n)(p) = dnF

dpn(p) = in

∫

Rh(x)ρ(x)xneipx dx

for n = 0, 1, 2, . . . , and we deduce F (n)(0) = in⟨ρn, h⟩2 = 0 for all n. Since F isholomorphic in the strip Sα it follows that F (p) = 0 for all p ∈ Sα (see Theorem9.5) and thus in particular F (p) = 0 for all p ∈ R. But F (p) =

√2πL(ρh)(p)

where L is the inverse Fourier transform (see Theorem 10.1), and we know⟨Lf , Lg⟩2 = ⟨f , g⟩2 for all f , g ∈ L2(R, dx) (Theorem 10.7). It follows that⟨ρh, ρh⟩2 = ⟨L(ρh), L(ρh)⟩2 = 0 and thus ρh = 0 ∈ L2(R, dx). Since ρ(x) > 0for x ∈ I this implies h = 0 and we conclude. ✷

Technically it is simpler to do the orthonormalization of the system of functions{ρn : n ∈ N} not in the Hilbert space L2(I , dx) directly but in the Hilbert spaceL2(I , ρdx), which is defined as the space of all equivalence classes of measurablefunctions f : I→K such that

∫I|f (x)|2ρ(x) dx < ∞ equipped with the inner prod-

uct ⟨f , g⟩ρ =∫If (x)g(x)ρ(x) dx. Note that the relation ⟨f , g⟩ρ = ⟨√ρf ,

√ρg⟩2

holds for all f , g ∈ L2(I , ρdx). It implies that the Hilbert spaces L2(I , ρdx) andL2(I , dx) are (isometrically) isomorphic under the map

L2(I , ρdx) ∋ f (→ √ρf ∈ L2(I , dx).

This is shown in the Exercises. Using this isomorphism, Lemma 17.1 can be restatedas saying that the system of powers of x, {xn : n = 0, 1, 2, . . .} is total in the Hilbertspace L2(I , ρdx).

We proceed by applying the Gram–Schmidt orthonormalization to the system ofpowers {xn : n = 0, 1, 2, . . . } in the Hilbert space L2(I , ρdx). This gives a sequenceof polynomials Pk of degree k such that ⟨Pk , Pm⟩ρ = δkm. These polynomials aredefined recursively in the following way: Q0(x) = x0 = 1, and when for k ≥ 1 thepolynomials Q0, . . ., Qk−1 are defined, we define the polynomial Qk by

Qk(x) = xk −k−1∑

n=0

⟨Qn, xk⟩ρ⟨Qn, Qn⟩ρ

Qn.

Finally, the polynomials Qk are normalized and we arrive at an orthonormal systemof polynomials Pk:

Pk = 1∥Qk∥ρ

Qk , k = 0, 1, 2, . . . .

Note that according to this construction, Pk is a polynomial of degree k with positivecoefficient for the power xk . Theorem 17.1 and Lemma 17.1 imply that the systemof polynomials {Pk : k = 0, 1, 2, . . . } is an orthonormal basis of the Hilbert spaceL2(I , ρdx). If we now introduce the functions

ek(x) = Pk(x)√

ρ(x), x ∈ I

we obtain an orthonormal basis of the Hilbert space L2(I , dx). This shows Theorem17.3.

17.2 Weight Functions and Orthogonal Polynomials 247

Theorem 17.3 For any interval I = (a, b), −∞ ≤ a < b ≤ +∞ the Hilbert spaceL2(I , dx) is separable, and the above system {ek : k = 0, 1, 2, . . .} is an orthonormalbasis.

Proof Only the existence of a weight function for the interval I has to be shown.Then by the preceding discussion we conclude. A simple choice of a weight functionfor any of these intervals is for instance the exponential function ρ(x) = e−αx2

,x ∈ R, for some α > 0. ✷

Naturally, the orthonormal polynomials Pk depend on the interval and the weightfunction. After some general properties of these polynomials have been studied wewill determine the orthonormal polynomials for some intervals and weight functionsexplicitly.

Lemma 17.2 If Qm is a polynomial of degree m, then ⟨Qm, Pk⟩ρ = 0 for all k > m.

Proof Since {Pk : k = 0, 1, 2, . . .} is an ONB of the Hilbert space L2(I , ρdx) thepolynomial Qm has a Fourier expansion with respect to this ONB: Qm = ∑∞

n=0 cnPn,cn = ⟨Pn, Qm⟩ρ . Since the powers xk , k = 0, 1, 2, . . . are linearly independentfunctions on the interval I and since the degree of Qm is m and that of Pn is n, thecoefficients cn in this expansion must vanish for n > m, i.e., Qm = ∑m

n=0 cnPn andthus ⟨Pk , Qm⟩ρ = 0 for all k > m. ✷

Since, the orthonormal system {Pk : k = 0, 1, 2, . . . } is obtained by the Gram–Schmidt orthonormalization from the system of powers xk for k = 0, 1, 2, . . . withrespect to the inner product ⟨·, ·⟩ρ , the polynomial Pn+1 is generated by multiplyingthe polynomial Pn with x and adding some lower order polynomial as correction.Indeed one has

Proposition 17.1 Let ρ be a weight for the interval I = (a, b) and denote thecomplete system of orthonormal polynomials for this weight and this interval by{Pk : k = 0, 1, 2, . . . }. Then, for every n ≥ 1, there are constants An, Bn, Cn suchthat

Pn+1(x) = (Anx + Bn)Pn(x) + CnPn−1(x) ∀ x ∈ I.

Proof We know Pk(x) = akxk + Qk−1(x) with some constant ak > 0 and some

polynomial Qk−1 of degree smaller than or equal to k − 1. Thus, if we define An =an+1an

, it follows that Pn+1 − AnxPn is a polynomial of degree smaller than or equalto n, hence there are constants cn,k such that

Pn+1 − AnxPn =n∑

k=0

cn,kPk.

Now calculate the inner product with Pj , j ≤ n:

⟨Pj , Pn+1 − AnxPn⟩ρ =n∑

k=0

cn,k⟨Pj , Pk⟩ρ = cn,j .

248 17 Separable Hilbert Spaces

Since the polynomial Pk is orthogonal to all polynomials Qj of degree j ≤ k − 1we deduce that cn,j = 0 for all j < n − 1, cn,n−1 = −An⟨xPn−1, Pn⟩ρ , and cn,n =−An⟨xPn, Pn⟩ρ . The statement follows by choosing Bn = cn,n and Cn = cn,n−1. ✷

Proposition 17.2 For any weight function ρ on the interval I , the kth orthonormalpolynomial Pk has exactly k simple real zeroes.

Proof Per construction the orthonormal polynomials Pk have real coefficients, havethe degree k, and the coefficient ck is positive. The fundamental theorem of algebra(Theorem 9.4) implies: The polynomial Pk has a certain number m ≤ k of simplereal roots x1, . . ., xm and the roots which are not real occur in pairs of complexconjugate numbers, (zj , zj ), j = m + 1, . . ., M with the same multiplicity nj , m +2

∑Mj=m+1 nj = k. Therefore the polynomial Pk can be written as

Pk(x) = ck

m∏

j=1

(x − xj )M∏

j=m+1

(x − zj )nj (x − zj )nj .

Consider the polynomial Qm(x) = ck

∏mj=1 (x − xj ). It has the degree m and ex-

actly m real simple roots. Since Pk(x) = Qm(x)∏M

j=m+1 |x − zj |2nj , it follows thatPk(x)Qm(x) ≥ 0 for all x ∈ I and PkQm ̸= 0, hence ⟨Pk , Qm⟩ρ > 0. If the degreem of the polynomial Qm would be smaller than k, we would arrive at a contradictionto the result of the previous lemma, hence m = k and the pairs of complex conjugateroots cannot occur. Thus we conclude. ✷

In the Exercises, with the same argument, we prove the following extension ofthis proposition.

Lemma 17.3 The polynomial Qk(x, λ) = Pk(x)+λPk−1(x) has k simple real roots,for any λ ∈ R.

Lemma 17.4 There are no points x0 ∈ I and no integer k ≥ 0 such that Pk(x0) =Pk−1(x0) = 0.

Proof Suppose that for some k ≥ 0 the orthonormal polynomials Pk and Pk−1 have acommon root x0 ∈ I : Pk(x0) = Pk−1(x0) = 0. Since we know that these orthonormalpolynomials have simple real roots, we know in particular P ′

k−1(x0) ̸= 0 and thus

we can take the real number λ0 = −P ′k (x0)

P ′k−1(x0) to form the polynomial Qk(x, λ0) =

Pk(x) + λ0Pk−1(x). It follows that Q(x0, λ0) = 0 and Q′k(x0) = 0, i.e., x0 is a root

of Qk(·, λ) with multiplicity at least two. But this contradicts the previous lemma.Hence there is no common root of the polynomials Pk and Pk−1. ✷

Theorem 17.4 (Knotensatz) Let {Pk : k = 0, 1, 2, . . . } be the orthonormal basis forsome interval I and some weight function ρ. Then the roots of Pk−1 separate theroots of Pk , i.e., between two successive roots of Pk there is exactly one root of Pk−1.

Proof Suppose that α < β are two successive roots of the polynomial Pk so thatPk(x) ̸= 0 for all x ∈ (α, β). Assume furthermore that Pk−1 has no root in the openinterval (α, β). The previous lemma implies that Pk−1 does not vanish in the closed

17.3 Examples of Complete Orthonormal Systems for L2(I , ρdx) 249

interval [α, β]. Since the polynomials Pk−1 and −Pk−1 have the same system of roots,we can assume that Pk−1 is positive in [α, β] and Pk is negative in (α, β). Define thefunction f (x) = −Pk (x)

Pk−1(x) . It is continuous on [α, β] and satisfies f (α) = f (β) = 0and f (x) > 0 for all x ∈ (α, β). It follows that λ0 = sup {f (x) : x ∈ [α, β]} = f (x0)for some x0 ∈ (α, β). Now consider the family of polynomials Qk(x, λ) = Pk(x) +λPk−1(x) = Pk−1(x)(λ − f (x)). Therefore, for all λ ≥ λ0, the polynomials Qk(·, λ)are nonnegative on [α, β], in particular Qk(x, λ0) ≥ 0 for all x ∈ [α, β]. Sinceλ0 = f (x0), it follows that Qk(x0, λ0) = 0, thus Qk(·, λ0) has a root x0 ∈ (α, β).Since f has a maximum at x0, we know 0 = f ′(x0). The derivative of f is easilycalculated:

f ′(x) = −P ′k(x)Pk−1(x) − Pk(x)P ′

k−1(x)Pk−1(x)2

.

Thus f ′(x0) = 0 implies P ′k(x0)Pk−1(x0) − Pk(x0)P ′

k−1(x0) = 0, and thereforeQ′

k(x0) = P ′k(x0)+f (x0)P ′

k−1(x0) = 0. Hence the polynomial Qk(·, λ0) has a root ofmultiplicity 2 at x0. This contradicts Lemma 17.3 and therefore the polynomial Pk−1

has at least one root in the interval (α, β). Since Pk−1 has exactly k − 1 simple realroots according to Proposition 17.2, we conclude that Pk−1 has exactly one simpleroot in (α, β) which proves the theorem. ✷

Remark 17.1 Consider the function

F (Q) =∫

I

Q(x)2ρ(x) dx, Q(x) =n∑

k=0

akxk.

Since we can expand Q in terms of the orthonormal basis {Pk : k = 0, 1, 2, . . . },Q = ∑n

k=0 ckPk , ck = ⟨Pk , Q⟩ρ the value of the function F can be expressed interms of the coefficients ck as F (Q) = ∑n

k=0 c2k and it follows that the orthonormal

polynomials Pk minimize the function Q '→ F (Q) under obvious constraints (seeExercises).

17.3 Examples of Complete Orthonormal Systems forL2(I , ρdx)

For the intervals I = R, I = R+ = [0, ∞), and I = [−1, 1] we are going toconstruct explicitly an orthonormal basis by choosing a suitable weight function andapplying the construction explained above. Certainly, the above general results applyto these concrete examples, in particular the “Knotensatz.”

17.3.1 I = R, ρ(x) = e−x2: Hermite Polynomials

Evidently, the function ρ(x) = e−x2is a weight function for the real line. Therefore,

by Lemma 17.1, the system of functions ρn(x) = xne− x22 generates the Hilbert space

250 17 Separable Hilbert Spaces

L2(R, dx). Finally the Gram–Schmidt orthonormalization produces an orthonormalbasis {hn : n = 0, 1, 2, . . . }. The elements of this basis have the form (Rodrigues’formula)

hn(x) = (−1)ncnex22

(d

dx

)n (e−x2

)= cnHn(x) e− x2

2 (17.1)

with normalization constants

cn = (2nn!√π )−1/2 n = 0, 1, 2, . . . .

Here the functions Hn are polynomials of degree n, called Hermite polynomials andthe functions hn are the Hermite functions of order n.

Theorem 17.5 The system of Hermite functions {hn : n = 0, 1, 2, . . . } is an or-thonormal basis of the Hilbert space L2(R, dx). The statements of Theorem 17.4apply to the Hermite polynomials.

Using Eq. (17.1) one deduces in the Exercises that the Hermite polynomials satisfythe recursion relation

Hn+1(x) − 2xHn(x) + 2nHn−1(x) = 0 (17.2)

and the differential equation (y = Hn(x))

y ′′ − 2xy ′ + 2ny = 0. (17.3)

These relations show that the Hermite functions hn are the eigenfunctions of thequantum harmonic oscillator with the Hamiltonian H = 1

2 (P 2 + Q2) for the eigen-value n + 1

2 , Hhn = (n + 12 )hn, n = 0, 1, 2. . . . . For more details we refer to [2–4].

In these references one also finds other methods to prove that the Hermite functionsform an orthonormal basis.

Note also that the Hermite functions belong to the Schwartz test function spaceS(R).

17.3.2 I = R+, ρ(x) = e−x: Laguerre Polynomials

On the positive real line the exponential function ρ(x) = e−x certainly is a weightfunction. Hence our general results apply here and we obtain

Theorem 17.6 The system of Laguerre functions {ℓn : n = 0, 1, 2, . . . } whichis constructed by orthonormalization of the system {xne− x

2 } : n = 0, 1, 2, . . . inL2(R+, dx) is an orthonormal basis. These Laguerre functions have the followingform (Rodrigues’ formula):

ℓn(x) = 1n!Ln(x) e− x

2 , Ln(x) = ex

(ddx

)n

(xne−x , n = 0, 1, 2, . . . . (17.4)

For the system {Ln : n = 0, 1, 2, . . . } of Laguerre polynomials Theorem 17.4applies.

17.3 Examples of Complete Orthonormal Systems for L2(I , ρdx) 251

In the Exercises we show that the Laguerre polynomials of different order arerelated according to the identity

(n + 1)Ln+1(x) + (x − 2n − 1)Ln(x) + nLn−1(x) = 0, (17.5)

and are solutions of the second order differential equation (y = Ln(x))

xy ′′ + (1 − x)y ′ + ny = 0. (17.6)

In quantum mechanics this differential equation is related to the radial Schrödingerequation for the hydrogen atom.

17.3.3 I = [−1, +1], ρ(x) = 1: Legendre Polynomials

For any finite interval I = [a, b], −∞ < a < b < ∞ one can take any posi-tive constant as a weight function. Thus, Lemma 17.1 says that the system of powers{xn : n = 0, 1, 2. . . . } is a total system of functions in the Hilbert space L2([a, b], dx).It follows that every element f ∈ L2([a, b], dx) is the limit of a sequence of polyno-mials, in the L2-norm. Compare this with the Theorem of Stone–Weierstrass whichsays that every continuous function on [a, b] is the uniform limit of a sequence ofpolynomials.

For the special case of the interval I = [−1, 1] the Gram–Schmidt orthonormal-ization of the system of powers leads to a well-known system of polynomials.

Theorem 17.7 The system of Legendre polynomials

Pn(x) = 12nn!

(ddx

)n

(x2 − 1)n, x ∈ [−1, 1], n = 0, 1, 2, . . . (17.7)

is an orthogonal basis of the Hilbert space L2 ([−1, 1], dx). The Legendrepolynomials are normalized according to the relation

⟨Pn, Pm⟩2 = 22n + 1

δnm.

Again one can show that these polynomials satisfy a recursion relation and a secondorder differential equation (see Exercises):

(n + 1)Pn+1(x) − (2n + 1)xPn(x) + nPn−1(x) = 0, (17.8)

(1 − x2)y ′′ − 2xy ′ + n(n + 1)y = 0, (17.9)

where y = Pn(x).

252 17 Separable Hilbert Spaces

Fig. 17.1 Legendre polynomials P3, P4, P5

Without further details we mention the weight functions for some other systemsof orthogonal polynomials on the interval [−1, 1]:

Jacobi Pν,µn ρ(x) = (1 − x)µ, ν, µ > −1,

Gegenbauer Cλn ρ(x) = (1 − x2)λ− 1

2 , λ > −1/2,

Tschebyschew 1st kind ρ(x) = (1 − x)−1/2,

Tschebyschew 2nd kind ρ(x) = (1 − x2)1/2.

We conclude this section by an illustration of the Knotensatz for some Legendrepolynomials of low order. This graph clearly shows that the zeros of the polynomialPk are separated by the zeros of the polynomial Pk+1, k = 3, 4. In addition theorthonormal polynomials are listed explicitly up to order n = 6.