Untitled - Cdm.unimo.it

Results in Approximation Theory 101
Proof - For k = 0, (6.2.15) is a trivial consequence of (6.2.13). For k ≥ 1, let us set
v := (1 − x 2 ) k w, x ∈ I, and observe that L 2 w(I) ⊂ L 2 v(I), since v ≤ w. This implies
that f ′ ∈ L 2 w(I) ⊂ L 2 a(I) admits the representation
(6.2.16) f ′ =
∞
cj u ′ j, a.e. in I,
j=1
where the cj’s are the Fourier coefficients of f. To show (6.2.16), we recall that {u ′ j }j≥1 ,
up to normalizing factors, is an orthogonal basis of polynomials with respect to the
weight function a (see (1.3.6) and (2.2.8)). Therefore, the Fourier coefficients dj, j ≥ 1,
of f ′ with respect to this new basis, are
(6.2.17) dj :=
= λj
I f ′ u ′ j
a dx
u ′ j 2 L 2 a (I)
I fujw dx
u ′ j 2 L 2 a (I)
=
= −
I fujw dx
uj 2 L 2 w (I)
I f(u′ j a)′ dx
u ′ j 2 L 2 a (I)
= cj, j ≥ 1.
Hence we get (6.2.16). In (6.2.17) we first integrated by parts (recalling that a(±1) = 0),
then we used the fact that {uj}j∈N are the solutions of a Sturm-Liouville problem, and
finally we used relation (2.2.15). For a correct justification of the equalities in (6.2.17),
one should first argue with f ∈ C 1 ( Ī) and then approximate a general f ∈ H1 w(I) by a
sequence of functions in C 1 ( Ī).
In a similar way, we expand dk f
dx k ∈ L 2 v(I) in series of the polynomial basis
k
d uj
dxk
,
j≥k
orthogonal with respect to the weight function v. The elements of this basis are solutions
of a Sturm-Liouville problem with eigenvalues (j − k)(j + α + β + 1 + k), j ≥ k (see
(1.3.6)). This yields
(6.2.18)
dkf =
dxk ∞
j=k
cj
dkuj , a.e. in I.
dxk Finally, repeated application of (2.2.15) to successive derivatives of uj, j ≥ 1, gives for
n > k: