Untitled - Cdm.unimo.it

96 Polynomial Approximation of Differential Equations
6.2 Estimates for the projection operator
As usual, we denote by {un}n∈N the sequence of orthogonal polynomials in Ī (where
Ī = [−1,1],
Ī = [0,+∞[ or Ī ≡ R), with respect to the weight function w (we refer in
particular to those introduced in chapter one).
Another best approximation problem can be formulated in terms of the norm · w
introduced in (2.1.9). In this case, we wish to find Ψw,n(f) ∈ Pn such that
(6.2.1) f − Ψw,n(f)w = inf
ψ∈Pn
where f is a given function in Ī.
f − ψw,
For the reader acquainted with the results of chapter five, the natural space to develop
the theory is L 2 w(I). Under this assumption, we can still define the Fourier coefficients
of f as
(6.2.2) ck := (f,uk) L2 w (I)
uk2 L2 w (I)
, k ∈ N.
We note that the integral
I fukwdx is finite in virtue of the Schwarz inequality (2.1.7).
In this formulation we can also include the cases when I is not bounded, corresponding
respectively to Laguerre and Hermite polynomials. Now, even if the functions in L 2 w(I)
are not required to be bounded, the exponential decay of the weight w is sufficient
to insure the existence of the coefficients in (6.2.2). The reader unfamiliar with these
functional spaces, can draw conclusions by thinking in terms of Riemann integrals and
continuous functions, and by replacing the norms · L 2 w (I) and · C 0 (Ī), by · w
and · ∞ respectively.
The projector Πw,n : L 2 w(I) → Pn, n ∈ N, is defined in the usual way (see section 2.4).
The next proposition fully characterizes the solution of problem (6.2.1).
Theorem 6.2.1 - For any f ∈ L 2 w(I) and any n ∈ N, there exists a unique polynomial
Ψw,n(f) ∈ Pn that satisfies (6.2.1). Moreover Ψw,n(f) = Πw,nf.