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Results in Approximation Theory 103
the approximating polynomials, being analytic, do not affect the convergence behavior.
In comparison with other methods, smaller values of the parameter n are in general
sufficient to guarantee a satisfactory approximation error.
Derivatives of the error f −Πw,nf can be also estimated, which result in a similar con-
vergence behavior. Nevertheless, the rate of convergence is slower than that exhibited
by other projection operators which are introduced later on. The reader interested in
this field is referred to canuto and quarteroni(1982a), where other error estimates
concerning Chebyshev and Legendre weights are taken into account. With techniques
related to interpolation spaces (see section 5.7), formula (6.2.15) can be suitably gener-
alized to Sobolev spaces with real exponent. For other best approximation polynomials
in more general functional spaces (for instance L p w(I) spaces, p > 1) the reader is ad-
dressed to ditzian and totik(1987) and the references therein. Chebyshev expansions
for functions with singularities at the endpoints of the interval [−1,1] are studied in
boyd(1981).
The same kind of result can be established for other systems of orthogonal poly-
nomials. Laguerre polynomials require a little care. Here λj = j, j ∈ N, and
a(x) = xw(x) = x α+1 e −x , α > −1, x ∈ I =]0,+∞[. The inequality a(x) ≤ w(x) only
holds for x ∈]0,1] ⊂ I, which implies that L 2 w(I) ⊂ L 2 a(I). Nevertheless, the proof of
theorem 6.2.4 is still applicable and gives the next result.
Theorem 6.2.5 - Let k ∈ N. Then there exists a constant C > 0 such that, for any
f satisfying dm f
dx m x m/2 ∈ L 2 w(I), 0 ≤ m ≤ k, one has
(6.2.21) f − Πw,nf L 2 w (I) ≤ C
k 1
x
k/2 √n
dkf dxk
L 2 w (I)
, ∀n > k.
Further estimates are given in maday, pernaud-thomas and vandeven(1985) for
the case α = 0. In that paper, f is assumed to belong to the weighted space H k ω(I),
with ω := e −(1−ǫ)x , ǫ > 0 arbitrarily small. In this situation, we obtain the inclusion
H 1 ω(I) ⊂ L 2 w(I) ∩ H 1 a(I).