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104 Polynomial Approximation of Differential Equations
A proof identical to that of theorem 6.2.4 holds for Hermite polynomials, where λj = 2j,
j ∈ N, and a(x) = w(x) = e−x2, x ∈ R.
Theorem 6.2.6 - Let k ∈ N. Then there exists a constant C > 0 such that, for any
f ∈ H k w(R), one has
(6.2.22) f − Πw,nf L 2 w (R) ≤ C
In particular, (6.2.20) takes the form
k 1
d
√n
kf dxk
L 2 w (R)
(6.2.23) (Πw,nf) ′ = Πw,n−1f ′ , ∀f ∈ H 1 w(R), ∀n ≥ 1.
, ∀n > k.
From the examples illustrated here, it is clear that the speed of convergence to zero
of the error is strictly related to the eigenvalues λj, j ∈ N (i.e., the spectrum of the
Sturm-Liouville problem (1.1.1)). Somehow, this justifies the adoption of the adjective
spectral, commonly used in the frame of these approximation methods.
6.3 Inverse inequalities
Inverse inequalities are an effective tool for the analysis of convergence in approximation
theory. They are based on the fact that, in finite dimensional spaces, two given norms
are always equivalent. Thus, we are allowed to give a bound to strong norms by using
weaker norms. The drawback is that the constants in the equivalence relation, depend
on the dimension of the space considered, so that it is not possible to extrapolate the
same inequalities at the limit. Classical examples in finite element method are given for
instance in ciarlet(1978), p.140.
Let us start with the Jacobi case (i.e., a(x) = (1 − x 2 )w(x), x ∈ I =] − 1,1[).