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Results in Approximation Theory 99
(6.2.11) f =
∞
ckuk, ∀f ∈ L 2 w(I),
k=0
where the ck’s are the Fourier coefficients of f and the equality has to be intended
almost everywhere. By orthogonality, one also obtains the Parseval identity
∞
∀f ∈ L2w(I). (6.2.12) f 2 L2 w (I) =
k=0
c 2 k uk 2 L 2 w (I),
From (6.2.11) and (6.2.12), one easily deduces that
(6.2.13) f − Πw,nf L 2 w (I) ≤ f L 2 w (I), ∀f ∈ L 2 w(I).
We remark that the convergence of the series (6.2.11) is not in general uniform in
I. On the other hand, discontinuous functions can be also approximated. For instance,
when f is continuous with the exception of the point x0 ∈] − 1,1[, then for Legendre
expansions one has
(6.2.14)
∞
k=0
provided the two limits exist.
ckuk(x0) = 1
2
lim
x→x −
0
f(x) + lim
x→x +
f(x)
0
Figure 6.2.1 - Legendre orthogonal Figure 6.2.2 - Legendre orthogonal
projections for f(x) = |x| − 1/2. projections for f(x) = −1/2, x < 0,
f(x) = 1/2, x ≥ 0.
,