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Orthogonality 23
The function w is called weight function. If I is not bounded, we just have to be careful
that the integrals in (2.1.8) and (2.1.9) are finite. We will be more precise in chapter
five.
With the help of this short introduction, we are ready to analyze more closely the
solutions of Sturm-Liouville problems.
2.2 Orthogonal functions
Two functions u,v ∈ C 0 ( Ī) are said to be orthogonal when (u,v)w = 0 for some
weight function w.
Let us assume that the function a in (1.1.1) vanishes at the endpoints of the interval
I (if I is not bounded, limx→±∞ a(x) = 0). Then, we have the following fundamental
result.
Theorem 2.2.1 - Let {un} n∈N be a sequence of solutions of (1.1.1) corresponding to
the eigenvalues {λn} n∈N . Let us require that λn = λm if n = m. Then one has
(2.2.1)
I
unum w dx = 0, ∀n,m ∈ N with n = m.
Proof - Let us multiply the equation (1.1.1) by a differentiable function v and integrate
in I. Recalling the assumptions on a, after integration by parts, one gets
(2.2.2)
au ′ v ′
dx + buv dx = λ uvw dx.
I
I
Let n = m, thus relation (2.2.2) is satisfied either when u = un, v = um, λ = λn, or
when u = um, v = un, λ = λm. In both cases the left-hand sides coincide. Having
λn = λm, this implies (2.2.1).
I