Sets of functions

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS
class of operators called Hermitian operators (the operator in the simple harmonic
oscillator equation is an example) have particularly useful properties and these
will be studied in detail. It turns out that many of the interesting differential
operators met within the physical sciences are Hermitian. Before continuing our
discussion of the eigenfunctions of Hermitian operators, however, we will consider
some properties of general sets of functions.
17.1 Sets of functions
In chapter 8 we discussed the definition of a vector space but concentrated on
spaces of finite dimensionality. We consider now the infinite-dimensional space
of all reasonably well-behaved functions f(x), g(x), h(x), . . . on the interval
a ≤ x ≤ b. That these functions form a linear vector space is shown by noting
the following properties. The set is closed under
(i) addition, which is commutative and associative, i.e.
f(x) + g(x) = g(x) + f(x),
[f(x) + g(x)] + h(x) = f(x) + [g(x) + h(x)] ,
(ii) multiplication by a scalar, which is distributive and associative, i.e.
λ [f(x) + g(x)] = λf(x) + λg(x),
λ [µf(x)] = (λµ)f(x),
(λ + µ)f(x) = λf(x) + µf(x).
Furthermore, in such a space
(iii) there exists a ‘null vector’ 0 such that f(x) + 0 = f(x),
(iv) multiplication by unity leaves any function unchanged, i.e. 1 × f(x) = f(x),
(v) each function has an associated negative function −f(x) that is such that
f(x) + [−f(x)] = 0.
By analogy with finite-dimensional vector spaces we now introduce a set
of linearly independent basis functions yn (x), n = 0, 1, . . . , ∞, such that any
‘reasonable’ function in the interval a ≤ x ≤ b (i.e. it obeys the Dirichlet conditions
discussed in chapter 12) can be expressed as the linear sum of these functions:
f(x) =
∞
cn yn (x).
n=0
Clearly if a different set of linearly independent basis functions un (x) is chosen
then the function can be expressed in terms of the new basis,
f(x) =
∞
n=0
556
dn un (x),
17.1 SETS OF FUNCTIONS
where the dn are a different set of coefficients. In each case, provided the basis
functions are linearly independent, the coefficients are unique.
We may also define an inner product on our function space by
b
f ∗ (x)g(x)ρ(x) dx,
(17.6)
f|g =
a
where ρ(x) is the weight function, which we require to be real and non-negative
in the interval a ≤ x ≤ b. As mentioned above, ρ(x) is often unity for all x. Two
functions are said to be orthogonal (with respect to the weight function ρ(x)) on
the interval [a, b] if
b
f ∗ (x)g(x)ρ(x) dx = 0,
(17.7)
f|g =
a
and the norm of a function is defined as
b
1/2 f = f|f1/2 =
f ∗ (x)f(x)ρ(x) dx
=
a
b
1/2
|f(x)|2 ρ(x) dx
.
(17.8)
a
It is also common practice to define a normalised function by f̂ = f/f, which
has unit norm.
An infinite-dimensional vector space of functions, for which an inner product
is defined, is called a Hilbert space. Using the concept of the inner product, we
can choose a basis of linearly independent functions φ̂n (x), n = 0, 1, 2, . . . that are
orthonormal, i.e. such that
b
(17.9)
φ̂∗i (x)φ̂j (x)ρ(x) dx = δij .
φ̂i |φ̂j =
a
If yn (x), n = 0, 1, 2, . . . , are a linearly independent, but not orthonormal, basis for
the Hilbert space then an orthonormal set of basis functions φ̂n may be produced
(in a similar manner to that used in the construction of a set of orthogonal
eigenvectors of an Hermitian matrix; see chapter 8) by the following procedure:
φ0 = y0 ,
φ1 = y1 − φ̂0 φ̂0 |y1 ,
φ2 = y2 − φ̂1 φ̂1 |y2 − φ̂0 φ̂0 |y2 ,
..
.
φn = yn − φ̂n−1 φ̂n−1 |yn − · · · − φ̂0 φ̂0 |yn ,
..
.
It is straightforward to check that each φn is orthogonal to all its predecessors
φi , i = 0, 1, 2, . . . , n − 1. This method is called Gram–Schmidt orthogonalisation.
Clearly the functions φn form an orthogonal set, but in general they do not have
unit norms.
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EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS
Starting from the linearly independent functions yn (x) = xn , n = 0, 1, . . . , construct three
orthonormal functions over the range −1 < x < 1, assuming a weight function of unity.
The first unnormalised function φ0 is simply equal to the first of the original functions, i.e.
φ0 = 1.
The normalisation is carried out by dividing by
φ0 |φ0 1/2 =
1/2
1
−1
1 × 1 du
=
√
2,
with the result that the first normalised function φ̂0 is given by
φ0
φ̂0 = √ = 12 .
2
The second unnormalised function is found by applying the above Gram–Schmidt orthogonalisation procedure, i.e.
φ1 = y1 − φ̂0 φ̂0 |y1 .
It can easily be shown that φ̂0 |y1 = 0, and so φ1 = x. Normalising then gives
φ̂1 = φ1
−1/2
1
u × u du
−1
=
3
x.
2
The third unnormalised function is similarly given by
φ2 = y2 − φ̂1 φ̂1 |y2 − φ̂0 φ̂0 |y2 = x2 − 0 − 13 ,
which, on normalising, gives
φ̂2 = φ2
1
−1
u2 −
1 2
3
−1/2
du
=
1
2
5
(3x2
2
− 1).
By comparing the functions φ̂0 , φ̂1 and φ̂2 with the list in subsection 18.1.1, we see that
this procedure has generated (multiples of) the first three Legendre polynomials. If a function is expressed in terms of an orthonormal basis φ̂n (x) as
f(x) =
∞
cn φ̂n (x)
(17.10)
n=0
then the coefficients cn are given by
b
cn = φ̂n |f =
a
φ̂∗n (x)f(x)ρ(x) dx.
Note that this is true only if the basis is orthonormal.
558
(17.11)