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100 Polynomial Approximation of Differential Equations
Figure 6.2.1 shows the behavior of the approximating polynomials Πw,nf, when w
is the Legendre weight function, n = 4, 6, 8, 10, and f is defined by f(x) = |x| − 1
2 .
A slower rate of convergence is measured at the point x = 0, where f presents a singu-
larity in the derivative. In figure 6.2.2, a typical discontinuous function, i.e., f(x) =
−1/2 if x < 0
1/2 if x ≥ 0 , is approximated by Πw,nf, with w ≡ 1 and n = 7, 9, 11, 13. In situations
like this one, the Gibbs phenomenon develops (see courant and hilbert(1953),
Vol.1). As a consequence, the oscillations of the approximating polynomials, in the
neighborhood of the point x = 0, stay bounded but do not damp for increasing n.
Next, we evaluate the rate of convergence of the limit in (6.2.8). This depends on
the regularity of the function f. The smoother the function, the faster is the convergence.
This problem has been widely studied by many authors. The classical approach relates
the speed of convergence to the so called modulus of continuity of f (see timan(1963)).
A more recent approach expresses the smoothness of f in terms of its norm in appro-
priate Sobolev spaces. We prefer the latter, since it is more suitable for the analysis of
differential equations. Results in this direction are given for instance in babuˇska, sz-
abo and katz(1981) for the Legendre case. Extensions are considered in canuto and
quarteroni(1980) and canuto and quarteroni(1982a). Other references are listed
in canuto, hussaini, quarteroni and zang(1988). The next theorem summarizes
the results for Jacobi type approximations. Thus, {λj = j(j + α + β + 1)}j∈N denotes
the sequence of eigenvalues corresponding to the Sturm-Liouville problem (1.3.1). We
recall that a and w are related by the formula a(x) = (1 − x 2 )w(x), x ∈ I =] − 1,1[.
According to definition (5.7.1), f ∈ H k w(I) when f and its derivatives of order up to k
belong to L 2 w(I).
Theorem 6.2.4 - Let k ∈ N, then there exists a constant C > 0 such that, for any
f ∈ H k w(I), one has
(6.2.15) f − Πw,nf L 2 w (I) ≤ C
k 1
(1
2 k/2
− x )
n
dkf dxk
L 2 w (I)
, ∀n > k.