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178 Polynomial Approximation of Differential Equations
of references is provided in chatelin(1983), where hints for the treatment of problems
similar to the one considered here are given in chapter 4B.
We can say something more about the Legendre case. Combining the proof of theo-
rem 2.2.1 with that of theorem 8.2.2, one checks that, for α = β = 0, the eigenfunctions
pn,m, 1 ≤ m ≤ n − 1, as well as their derivatives, are orthogonal with respect to the
inner product (·, ·)w,n (see section 3.8). This is true if we assume that the eigenvalues
λn,m, 1 ≤ m ≤ n − 1, are distinct.
Let Xm, 1 ≤ m ≤ n − 1, denote a subspace of P 0 n of dimension m. Then, by
virtue of the orthogonality of the eigenfunctions, the following characterization, known
as Min-Max principle, holds:
(8.6.7) λn,m = min
Xm⊂P 0 n
max
p∈Xm
p≡0
p ′ 2 w
p 2 w,n
, 1 ≤ m ≤ n − 1.
Applications of formula (8.6.7), in the field of finite-element approximations, are pre-
sented for instance in strang and fix(1973), chapter six.
Other situations may be considered. The problem
⎧
⎨ −φ
(8.6.8)
⎩
′′ + µφ = λ φ in I =] − 1,1[, µ > 0,
φ ′ (−1) = φ ′ (1) = 0,
has a countable set of eigenvalues λm := π2
4 m2 + µ, m ∈ N. These are effectively
approximated by the eigenvalues of problem (8.2.13). We only remark that two of the
λn,m’s are proportional to the parameter γ and are out of interest.
Different arguments apply to problem (8.2.12). Here, the λn,m’s do not approach
the eigenvalues of problem (8.6.8). If we replace the second equation in (8.2.12) by
p ′ n,m(η (n)
0 ) = λn,mpn,m(η (n)
0 ), we get a negative eigenvalue, say λn,0. Another eigen-
value, say λn,1, is such that limn→+∞(λn,0 + λn,1) = −1. The remaining eigenvalues
approximate those of problem (8.6.8), with the exception of λ0 = µ.
These considerations demonstrate the importance of the treatment of boundary
conditions in spectral methods. A non correct specification may give raise to unexpected
results. Though this is true in all numerical methods, the situation seems more crucial