Since C 1 ([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 basis
of 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 Fourier
series in the L2 -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 L2 (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) eα|x| ≤ C for all x ∈ I.
The strategy to prove that the Hilbert space L2 (I , dx) is separable is quite simple. A
first step shows that the countable set of functions ρn (x) = x n 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
space L2 (I , dx), for any interval I .
Proof For the proof we have to show: If an element h ∈ 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 a
function h ∈ L2 (R, dx). On the strip Sα = {p = u + iv ∈ C : u, v ∈ R, |v| < α},
introduce the auxiliary function
"
F (p) =
ρ(x)h(x) eipx dx.
R
The growth restriction on the weight function implies that F is a well-defined holomorphic function on Sα (see Exercises). Differentiation of F generates the functions
246
17 Separable Hilbert Spaces
ρn in this integral:
F
(n)
dn F
(p) =
(p) = in
dp n
!
R
h(x)ρ(x)x n eipx dx
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 Sα it follows that F (p) = 0 for all p ∈ Sα√(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
of all equivalence classes of measurable
" the space
2
functions f : I →K such that I |f (x)| ρ(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, Lemma 17.1 can be restated
as saying that the system of powers of x, {x n : 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 {x n : n = 0, 1, 2, . . . } in the Hilbert space L2 (I , ρdx). This gives a sequence
of polynomials Pk of degree k such that ⟨Pk , Pm ⟩ρ = δkm . These polynomials are
defined recursively in the following way: Q0 (x) = x 0 = 1, and when for k ≥ 1 the
polynomials Q0 , . . ., Qk−1 are defined, we define the polynomial Qk by
k−1
#
⟨Qn , x k ⟩ρ
Qk (x) = xk −
Qn .
⟨Qn , Qn ⟩ρ
n=0
Finally, the polynomials Qk are normalized and we arrive at an orthonormal system
of polynomials Pk :
Pk =
1
Qk ,
∥Qk ∥ρ
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 17.1 and Lemma 17.1 imply that the system
of polynomials {Pk : k = 0, 1, 2, . . . } is an orthonormal basis of the Hilbert space
L2 (I , ρdx). If we now introduce the functions
$
x∈I
ek (x) = Pk (x) ρ(x),
we obtain an orthonormal basis of the Hilbert space L2 (I , dx). This shows Theorem
17.3.
17.2 Weight Functions and Orthogonal Polynomials
247
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
2
for any of these intervals is for instance the exponential function ρ(x) = e−αx ,
x ∈ R, for some α > 0.
✷
Naturally, the orthonormal polynomials Pk 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 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) the
polynomial Qm has a Fourier expansion with respect to this ONB: Qm = ∞
n=0 cn Pn ,
k
cn = ⟨Pn , Qm ⟩ρ . Since the powers x , k = 0, 1, 2, . . . are linearly independent
functions on the interval I and since the degree of Qm is m and that!
of Pn is n, the
coefficients cn in this expansion must vanish for n > m, i.e., Qm = m
n=0 cn Pn and
thus ⟨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 x k for k = 0, 1, 2, . . . with
respect to the inner product ⟨·, ·⟩ρ , the polynomial Pn+1 is generated by multiplying
the 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 the
complete 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 such
that
Pn+1 (x) = (An x + Bn )Pn (x) + Cn Pn−1 (x)
∀ x ∈ I.
Proof We know Pk (x) = ak x k + 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+1
, it follows that Pn+1 − An xPn is a polynomial of degree smaller than or equal
an
to n, hence there are constants cn,k such that
Pn+1 − An xPn =
n
"
cn,k Pk .
k=0
Now calculate the inner product with Pj , j ≤ n:
⟨Pj , Pn+1 − An xPn ⟩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 − 1
we 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 orthonormal
polynomial Pk has exactly k simple real zeroes.
Proof Per construction the orthonormal polynomials Pk have real coefficients, have
the 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 simple
real roots x1 , . . ., xm and the roots which are not real occur in pairs of complex
conjugate numbers, (zj , zj ), j = m + 1, . . ., M with the same multiplicity nj , m +
!
2 M
j =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 m
j =1 (x − xj ). It has the degree m and ex#
2nj
actly m real simple roots. Since Pk (x) = Qm (x) M
, it follows that
j =m+1 |x − zj |
Pk (x)Qm (x) ≥ 0 for all x ∈ I and Pk Qm ̸ = 0, hence ⟨Pk , Qm ⟩ρ > 0. If the degree
m of the polynomial Qm would 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 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 a
common root x0 ∈ I : Pk (x0 ) = Pk−1 (x0 ) = 0. Since we know that these orthonormal
′
polynomials have simple real roots, we know in particular Pk−1
(x0 ) ̸ = 0 and thus
−P ′ (x )
we can take the real number λ0 = P ′ k (x00 ) to form the polynomial Qk (x, λ0 ) =
k−1
Pk (x) + λ0 Pk−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 for
some interval I and some weight function ρ. Then the roots of Pk−1 separate the
roots 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 that
Pk (x) ̸ = 0 for all x ∈ (α, β). Assume furthermore that Pk−1 has no root in the open
interval (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 the
k (x)
function f (x) = P−P
. It is continuous on [α, β] and satisfies f (α) = f (β) = 0
k−1 (x)
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 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]. Since
λ0 = 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 easily
calculated:
′
(x)
Pk′ (x)Pk−1 (x) − Pk (x)Pk−1
′
.
f (x) = −
Pk−1 (x)2
′
(x0 ) = 0, and therefore
Thus f ′ (x0 ) = 0 implies Pk′ (x0 )Pk−1 (x0 ) − Pk (x0 )Pk−1
′
′
′
Qk (x0 ) = Pk (x0 ) + f (x0 )Pk−1 (x0 ) = 0. Hence the polynomial Qk (·, λ0 ) has a root of
multiplicity 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 real
roots according to Proposition 17.2, we conclude that Pk−1 has exactly one simple
root in (α, β) which proves the theorem.
✷
Remark 17.1 Consider the function
!
F (Q) = Q(x)2 ρ(x) dx,
I
Q(x) =
n
"
ak x k .
k=0
Since #
we can expand Q in terms of the orthonormal basis {Pk : k = 0, 1, 2, . . . },
Q = nk=0 ck Pk , ck = ⟨Pk , Q⟩ρ the value
#n of 2the function F can be expressed in
terms of the coefficients ck as F (Q) = k=0 ck and it follows that the orthonormal
polynomials Pk minimize the function Q '→ F (Q) under obvious constraints (see
Exercises).
17.3
Examples of Complete Orthonormal Systems for
L2 (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
2
I = R, ρ(x) = e−x : Hermite Polynomials
2
Evidently, the function ρ(x) = e−x is a weight function for the real line. Therefore,
x2
by Lemma 17.1, the system of functions ρn (x) = x n e− 2 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)
! "n #
$
d
x2
x2
n
−x 2
2
hn (x) = (−1) cn e
e
= cn Hn (x) e− 2
(17.1)
dx
with normalization constants
√
cn = (2n n! π)−1/2
n = 0, 1, 2, . . . .
Here the functions Hn are polynomials of degree n, called Hermite polynomials and
the functions hn are the Hermite functions of order n.
Theorem 17.5 The system of Hermite functions {hn : n = 0, 1, 2, . . . } is an orthonormal 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
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 the
quantum harmonic oscillator with the Hamiltonian H = 21 (P 2 + Q2 ) for the eigenvalue n + 21 , H hn = (n + 21 )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−x : 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
x
is constructed by orthonormalization of the system {x n e− 2 } : n = 0, 1, 2, . . . in
L2 (R+ , dx) is an orthonormal basis. These Laguerre functions have the following
form (Rodrigues’ formula):
! "n
d
1
− x2
x
ℓn (x) = Ln (x) e , Ln (x) = e
(x n e−x , n = 0, 1, 2, . . . . (17.4)
n!
dx
For the system {Ln : n = 0, 1, 2, . . . } of Laguerre polynomials Theorem 17.4
applies.
17.3 Examples of Complete Orthonormal Systems for L2 (I , ρdx)
251
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 positive constant as a weight function. Thus, Lemma 17.1 says that the system of powers
{x n : 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 polynomials, 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 orthonormalization of the system of powers leads to a well-known system of polynomials.
Theorem 17.7 The system of Legendre polynomials
! "n
1
d
Pn (x) = n
(x 2 − 1)n , x ∈ [−1, 1], n = 0, 1, 2, . . .
2 n! dx
(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
δnm .
2n + 1
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,
(1 − x 2 )y ′′ − 2xy ′ + n(n + 1)y = 0,
where y = Pn (x).
(17.8)
(17.9)
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]:
ν,µ
Jacobi Pn
Gegenbauer Cnλ
Tschebyschew 1st kind
Tschebyschew 2nd kind
ρ(x) = (1 − x)µ ,
1
ν, µ > −1,
ρ(x) = (1 − x 2 )λ− 2 , λ > −1/2,
ρ(x) = (1 − x)−1/2 ,
ρ(x) = (1 − x 2 )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
Pk are separated by the zeros of the polynomial Pk+1 , k = 3, 4. In addition the
orthonormal polynomials are listed explicitly up to order n = 6.