= 0 for all n ∈ Z. From the above convergence result we deduce, for all f ∈ C

⟨e

, g ⟩

= 0 for all n ∈ Z. From the above convergence result we deduce, for all f ∈ C

([0, 2π]),

⟨f , g⟩

= lim

N→∞

⟨

!

c

(f )e

, g ⟩

= 0.

Since C

([0, 2π]) is known to be dense in L

([0, 2π], dx) it follows that g = 0, by Corollary

of L

([0, 2π], dx). Therefore, every f ∈ L

([0, 2π], dx) has a Fourier expansion, which converges (in the sense of the L

-topology). Thus, convergence of the Fourier series in the L

-topology is “natural,” from the point of view of having convergence of 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 L

(I , dx) separable, but for any interval I = [a, b], −∞ ≤ a < b ≤ +∞, as the results of this section will show. Furthermore an orthonormal basis will be constructed explicitly and some interesting 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 assumed to 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 such that ρ(x) e

≤ C for all x ∈ I.

The strategy to prove that the Hilbert space L

(I , dx) is separable is quite simple. A first step shows that the countable set of functions ρ

(x) = x

ρ(x), n = 0, 1, 2, . . . is total in this Hilbert space. The Gram–Schmidt orthonormalization then produces easily an orthonormal basis.

Lemma 17.1 The system of functions {ρⁿ

: n = 0, 1, 2, . . . } is total in the Hilbert

(I , dx), for any interval I .

Proof

For the proof we have to show: If an element h ∈ L

(I , dx) satisfies ⟨ρ

, h ⟩

= 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 a function h ∈ L

( R, dx). On the strip S

= {p = u + iv ∈ C : u, v ∈ R, |v| < α}, introduce the auxiliary function

F (p) =

"

R

ρ(x)h(x) e

dx.

The growth restriction on the weight function implies that F is a well-defined holo-

morphic function on S

(see Exercises). Differentiation of F generates the functions

246 17 Separable Hilbert Spaces

ρ_n in this integral:

F⁽ⁿ⁾(p)= dⁿF

dpⁿ(p)= iⁿ

!

Rh(x)ρ(x)xⁿe^ipxdx

for n = 0, 1, 2, . . . , and we deduce F⁽ⁿ⁾(0) = iⁿ⟨ρⁿ, h⟩² = 0 for all n. Since F is holomorphic in the strip S_α it follows that F (p) = 0 for all p ∈ S^α (see Theorem 9.5) 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) > 0

for 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 space L²(I , ρdx), which is defined as the space of all equivalence classes of measurable functions f : I→K such that"

I |f (x)|²ρ(x) dx < ∞ equipped with the inner prod- uct ⟨f , g⟩^ρ = "

I f(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) and L²(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, Lemma17.1can be restated as saying that the system of powers of x,{xⁿ : n= 0, 1, 2, . . .} is total in the Hilbert space L²(I , ρdx).

We proceed by applying the Gram–Schmidt orthonormalization to the system of powers{xⁿ : n= 0, 1, 2, . . . } in the Hilbert space L²(I , ρdx). This gives a sequence of polynomials P_k of degree k such that ⟨P^k, P_m⟩^ρ = δ^km. These polynomials are defined recursively in the following way: Q₀(x) = x⁰ = 1, and when for k ≥ 1 the polynomials Q₀, . . ., Q_k−1 are defined, we define the polynomial Q_k by

Q_k(x)= x^k−

k−1

#

n=0

⟨Qⁿ, x^k⟩^ρ

⟨Qⁿ, Qn⟩^ρQ_n.

Finally, the polynomials Q_k are normalized and we arrive at an orthonormal system of polynomials Pk:

P_k = 1

∥Q^k∥^ρQ_k, k= 0, 1, 2, . . . .

Note that according to this construction, Pk is a polynomial of degree k with positive coefficient for the power x^k. Theorem 17.1and Lemma17.1imply that the system of polynomials {P^k : k = 0, 1, 2, . . . } is an orthonormal basis of the Hilbert space L²(I , ρdx). If we now introduce the functions

e_k(x)= P^k(x)$

ρ(x), x ∈ I

we obtain an orthonormal basis of the Hilbert space L²(I , dx). This shows Theorem 17.3.

Theorem 17.3 For any interval I = (a, b), −∞ ≤ a < b ≤ +∞ the Hilbert space L²(I , dx) is separable, and the above system{ek : k = 0, 1, 2, . . .} is an orthonormal basis.

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 function for any of these intervals is for instance the exponential function ρ(x) = e^−αx2,

x ∈ R, for some α > 0. ✷

Naturally, the orthonormal polynomials P_k depend on the interval and the weight function. After some general properties of these polynomials have been studied we will determine the orthonormal polynomials for some intervals and weight functions explicitly.

Lemma 17.2 If Q_mis a polynomial of degree m, then⟨Q^m, P_k⟩^ρ = 0 for all k > m.

Proof Since {P^k : k = 0, 1, 2, . . .} is an ONB of the Hilbert space L²(I , ρdx) the polynomial Q_mhas a Fourier expansion with respect to this ONB: Q_m =!_∞

n=0c_nP_n, cn = ⟨Pⁿ, Q_m⟩^ρ. Since the powers x^k, k = 0, 1, 2, . . . are linearly independent functions on the interval I and since the degree of Q_m is m and that of P_n is n, the coefficients c_n in this expansion must vanish for n > m, i.e., Q_m =!m

n=0c_nP_n and

thus⟨P^k, Q_m⟩^ρ = 0 for all k > m. ✷

Since, the orthonormal system {P^k : k = 0, 1, 2, . . . } is obtained by the Gram–

Schmidt orthonormalization from the system of powers x^k for k = 0, 1, 2, . . . with respect to the inner product⟨·, ·⟩^ρ, the polynomial Pn+1 is generated by multiplying the polynomial P_n 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 the complete system of orthonormal polynomials for this weight and this interval by {P^k : k = 0, 1, 2, . . . }. Then, for every n ≥ 1, there are constants Aⁿ, B_n, C_n such that

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

Proof We know P_k(x) = a^kx^k + Q^k−1(x) with some constant a_k > 0 and some polynomial Q_k−1of degree smaller than or equal to k− 1. Thus, if we define Aⁿ =

an+1

an , it follows that P_n+1− AⁿxP_n is a polynomial of degree smaller than or equal to n, hence there are constants c_n,k such that

P_n+1− AⁿxP_n =

"n k=0

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

⟨P^j, P_n+1− AⁿxP_n⟩^ρ =

"n k=0

c_n,k⟨P^j, P_k⟩^ρ = c^n,j.

248 17 Separable Hilbert Spaces Since the polynomial P_k is orthogonal to all polynomials Q_j of degree j ≤ k − 1 we deduce that c_n,j = 0 for all j < n − 1, c^n,n−1 = −Aⁿ⟨xPⁿ−1, P_n⟩^ρ, and c_n,n =

−Aⁿ⟨xPⁿ, P_n⟩^ρ. The statement follows by choosing B_n = c^n,nand C_n = c^n,n−1. ✷ Proposition 17.2 For any weight function ρ on the interval I , the kth orthonormal polynomial P_k has exactly k simple real zeroes.

Proof Per construction the orthonormal polynomials P_khave real coefficients, have the degree k, and the coefficient c_k is positive. The fundamental theorem of algebra (Theorem 9.4) implies: The polynomial P_k has a certain number m ≤ k of simple real roots x₁, . . ., x_m and the roots which are not real occur in pairs of complex conjugate numbers, (z_j, z_j), j = m + 1, . . ., M with the same multiplicity n^j, m+ 2!M

j=m+1n_j = k. Therefore the polynomial P^k can be written as P_k(x) = c^k

"m j=1

(x− x^j)

"M j=m+1

(x− z^j)^nj(x− z^j)^nj.

Consider the polynomial Q_m(x) = c^k#m

j=1(x − x^j). It has the degree m and ex- actly m real simple roots. Since P_k(x) = Q^m(x)#M

j=m+1|x − z^j|^2nj, it follows that Pk(x)Qm(x) ≥ 0 for all x ∈ I and P^kQm ̸= 0, hence ⟨P^k, Qm⟩^ρ > 0. If the degree mof the polynomial Q_mwould be smaller than k, we would arrive at a contradiction to the result of the previous lemma, hence m= k and the pairs of complex conjugate

roots cannot occur. Thus we conclude. ✷

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

Lemma 17.3 The polynomial Q_k(x, λ)= P^k(x)+λP^k−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 P^k(x0)= P_k−1(x₀) = 0.

Proof Suppose that for some k ≥ 0 the orthonormal polynomials P^kand P_k−1have a common root x₀ ∈ I: P^k(x₀) = P^k−1(x₀) = 0. Since we know that these orthonormal polynomials have simple real roots, we know in particular P_k^′₋₁(x₀) ̸= 0 and thus we can take the real number λ₀ = _P^−P′ ^k′(x0)

k−1(x₀) to form the polynomial Q_k(x, λ₀) = P_k(x)+ λ⁰P_k−1(x). It follows that Q(x₀, λ₀) = 0 and Q^′k(x₀) = 0, i.e., x⁰ is a root of Q_k(·, λ) with multiplicity at least two. But this contradicts the previous lemma.

Hence there is no common root of the polynomials P_k and P_k−1. ✷ Theorem 17.4 (Knotensatz) Let {Pk : k = 0, 1, 2, . . . } be the orthonormal basis for some interval I and some weight function ρ. Then the roots of Pk−1 separate the roots of P_k, i.e., between two successive roots of P_k there is exactly one root of P_k−1. Proof Suppose that α < β are two successive roots of the polynomial P_k so that P_k(x)̸= 0 for all x ∈ (α, β). Assume furthermore that P^k−1 has no root in the open interval (α, β). The previous lemma implies that P_k−1 does not vanish in the closed

interval [α, β]. Since the polynomials P_k−1and−P^k−1have the same system of roots, we can assume that P_k−1is positive in [α, β] and P_k is negative in (α, β). Define the function f (x) = _P^−P_k−1^k(x)_(x). It is continuous on [α, β] and satisfies f (α) = f (β) = 0 and 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 Q^k(x, λ) = P^k(x)+ λPk−1(x)= P^k−1(x)(λ− f (x)). Therefore, for all λ ≥ λ⁰, the polynomials Q_k(·, λ) are nonnegative on [α, β], in particular Q_k(x, λ₀) ≥ 0 for all x ∈ [α, β]. Since λ₀ = f (x⁰), it follows that Q_k(x₀, λ₀) = 0, thus Q^k(·, λ⁰) has a root x₀ ∈ (α, β).

Since f has a maximum at x₀, we know 0 = f^′(x₀). The derivative of f is easily calculated:

f^′(x) = −P_k^′(x)P_k−1(x)− P^k(x)P_k^′₋₁(x) Pk−1(x)² .

Thus f^′(x₀) = 0 implies Pk^′(x₀)P_k−1(x₀) − P^k(x₀)P_k^′₋₁(x₀) = 0, and therefore Q^′_k(x₀) = Pk^′(x₀)+f (x⁰)P_k^′₋₁(x₀) = 0. Hence the polynomial Q^k(·, λ⁰) has a root of multiplicity 2 at x₀. This contradicts Lemma17.3and therefore the polynomial P_k−1 has at least one root in the interval (α, β). Since P_k−1has exactly k − 1 simple real roots according to Proposition 17.2, we conclude that P_k−1 has exactly one simple

root 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

a_kx^k.

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

k=0c_kP_k, c_k = ⟨P^k, Q⟩^ρ the value of the function F can be expressed in terms of the coefficients ck as F (Q)=#n

k=0c_k²and it follows that the orthonormal polynomials P_k minimize the function Q '→ F (Q) under obvious constraints (see Exercises).

17.3 Examples of Complete Orthonormal Systems for L

(I , ρdx)

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

17.3.1 I = R, ρ(x) = e

: Hermite Polynomials

Evidently, the function ρ(x)= e^−x2 is a weight function for the real line. Therefore, by Lemma17.1, 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 orthonormal basis {hn : n= 0, 1, 2, . . . }. The elements of this basis have the form (Rodrigues’

formula)

h_n(x) = (−1)ⁿc_ne^x22 ! d dx

"n# e^−x2$

= cⁿH_n(x) e^−x22 (17.1) with normalization constants

cn = (2ⁿn!√

π)^−1/2 n= 0, 1, 2, . . . .

Here the functions H_nare polynomials of degree n, called Hermite polynomials and the functions h_nare 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 17.4 apply to the Hermite polynomials.

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

H_n+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 h_n are the eigenfunctions of the quantum harmonic oscillator with the Hamiltonian H = ¹₂(P²+ Q²) for the eigen- value n+ ¹₂, H hn = (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 functions form an orthonormal basis.

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

17.3.2 I = R

, ρ(x) = e

: Laguerre Polynomials

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

Theorem 17.6 The system of Laguerre functions {ℓn : n = 0, 1, 2, . . . } which is constructed by orthonormalization of the system {xⁿe^−x2} : n = 0, 1, 2, . . . in L²(R⁺, dx) is an orthonormal basis. These Laguerre functions have the following form (Rodrigues’ formula):

ℓ_n(x) = 1

n!L_n(x) e^−x2, L_n(x) = e^x

! d dx

"n

(xⁿe^−x, n= 0, 1, 2, . . . . (17.4) For the system {Lⁿ : n= 0, 1, 2, . . . } of Laguerre polynomials Theorem 17.4 applies.

In the Exercises we show that the Laguerre polynomials of different order are related 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 Schrödinger equation 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, Lemma17.1says 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 which says that every continuous function on [a, b] is the uniform limit of a sequence of polynomials.

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 P_n(x) = 1

2ⁿn!

! d dx

"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 Legendre polynomials are normalized according to the relation

⟨Pⁿ, Pm⟩² = 2

2n+ 1δnm.

Again one can show that these polynomials satisfy a recursion relation and a second order 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 systems of orthogonal polynomials on the interval [−1, 1]:

Jacobi Pn^ν,µ ρ(x)= (1 − x)^µ, ν, µ >−1, Gegenbauer C_n^λ ρ(x)= (1 − x²)^λ−12, 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 Legendre polynomials of low order. This graph clearly shows that the zeros of the polynomial P_k are separated by the zeros of the polynomial P_k+1, k = 3, 4. In addition the orthonormal polynomials are listed explicitly up to order n = 6.