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98 Polynomial Approximation of Differential Equations
By virtue of this theorem, the operator Πw,n takes the name of orthogonal projector,
since the error f − Πw,nf is orthogonal to the space Pn. Choosing in particular
φ = Πw,nf in (6.2.5), application of the Schwarz inequality leads to
(6.2.7) Πw,nf L 2 w (I) ≤ f L 2 w (I), ∀n ∈ N, ∀f ∈ L 2 w(I).
The counterpart of relation (6.1.5) is deduced from the next result.
Theorem 6.2.3 - Let I be bounded, then for any f ∈ L 2 w(I) , we have
(6.2.8) lim
n→+∞ f − Πw,nf L 2 w (I) = 0.
Proof - If f ∈ C0 ( Ī), (6.2.4) and (5.3.3) together imply that
(6.2.9) f −Πw,nf L 2 w (I) ≤ f −Ψ∞,n(f) L 2 w (I) ≤ f −Ψ∞,n(f) C 0 (Ī)
The last term in (6.2.9) tends to zero in view of (6.1.5).
I
1
2
w dx .
Now, we prove the statement for a general f ∈ L 2 w(I). We can find a sequence of
functions {fm}m∈N, with fm ∈ C 0 ( Ī), m ∈ N, converging to f in the L2 w(I) norm.
Therefore, using the triangle inequality, we first note that
(6.2.10) f − Πw,nf L 2 w (I) ≤ f − fm L 2 w (I)
+fm − Πw,nfm L 2 w (I) + Πw,n(fm − f) L 2 w (I), ∀m ∈ N.
Finally, each one of the three terms on the right-hand side of (6.2.10) tends to zero
when m and n grow (for the last term we can recall (6.2.7)).
When I is not bounded, the proof of the previous theorem does not apply anymore.
Nevertheless, the expression (6.2.8) holds also in the case of Laguerre and Hermite
polynomials. The proof of this fact is more delicate and we refer to courant and
hilbert(1953), Vol.1, page 95, for the details. This means that, for all the orthogonal
systems of polynomials here considered, one is allowed to write