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124 Polynomial Approximation of Differential Equations
with different techniques, shows a faster convergence behavior. Although the inequality
(6.6.7) ensures the same kind of convergence, it requires f to be more regular. In
nevai(1976) numerous extensions are provided, and in maday(1991) an estimate like
(6.6.4) is proven for Legendre Gauss-Lobatto points.
For analytic functions, the error estimates for the interpolation operator are expected
to exhibit an exponential decay. Results in this direction are given in tadmor(1986)
for Chebyshev expansions. Various improvements and estimates in different weighted
spaces are considered in ditzian and totik(1987).
To give an idea of the difficulties one may face when working with the interpolation
operators, we note for instance that an inequality like (6.2.7) is not verified, when
replacing Πw,n by Iw,n. This shows that the operator Iw,n is not continuous in
the L2 w(I) norm, even if f ∈ C0 ( Ī). To check this fact, we construct a sequence of
functions fk ∈ C0 (n)
( Ī), k ∈ N, such that fk(ξ j ) = 1, 1 ≤ j ≤ n, ∀k ∈ N. Therefore,
one has Iw,nfk ≡ 1, ∀k ∈ N, which implies that Iw,nfkL2 w (I) = 0 is constant with
respect to k. On the other hand, every fk can be defined at the remaining points in
such a way that limk→+∞ fk L 2 w (I) = 0. This is in contrast with the claim that Iw,n
is continuous. Nevertheless, Iw,n is continuous in the norm of C0 ( Ī), i.e., for any n ≥ 1,
we can find γ ≡ γ(n) such that
(6.8.2) Iw,nf C 0 (Ī) ≤ γ f C 0 (Ī), ∀f ∈ C 0 ( Ī).
In fact, for any f ∈ C0 ( Ī), (3.9.5) yields
(6.8.3) Iw,nf C 0 (Ī) ≤ γ |||Iw,nf|||w,n = γ max
1≤j≤n |f(ξ(n)
j )| ≤ γ f C 0 (Ī).
This is why the natural domain for Iw,n is the space of continuous functions. Unfor-
tunately, the constant γ grows with n.
* * * * * * * * * * * *
With the conclusion of this chapter, we have finished the first part of the book. Having
established the structure of the approximating functions we intend to use, as well as their
properties, we now turn our efforts toward the discretization of derivative operators, and
their associated problems.