Untitled - Cdm.unimo.it

Results in Approximation Theory 119
≤ sup
x∈Ī
|f − pn| √
v (x)
I
v −1 1
2
w dx
⎛
n
+ ⎝
v
j=1
−1 (ξ (n)
j ) w (n)
j
⎞ 1
2
where v(x) := e−δx , x ∈ Ī. Due to theorem 6.5.2, the summation on the right hand
side of (6.6.15) converges to
I v−1 wdx < +∞. Hence, it is bounded by a constant
independent of n. According to theorem 6.1.4, we can choose the sequence {pn}n≥1 to
obtain (6.6.14).
On the determination of the rate of convergence, we just mention a result given in
maday, pernaud-thomas and vandeven(1985), for the case α = 0 (w(x) = e −x ).
Theorem 6.6.4 - Let k ≥ 1 and δ ∈]0,1[. Then we can find a constant C > 0 such
that, for any f ∈ H k v (I), with v(x) := e −δx , one has
(6.6.16) f − Iw,nf L 2 w (I) ≤ C
k−1 1
√n
The same relation holds for the error f − Ĩw,nf.
⎠
f H k v (I), ∀n ≥ 1.
We argue similarly for the Hermite weight function (w(x) = e−x2). A theorem like 6.6.3
is derived from theorems 6.1.5 and 6.5.3. Further results are considered in the next
section.
6.7 Laguerre and Hermite functions
In the analysis of differential equations in unbounded domains, the solution will be
required to decay at infinity with an exponential rate. Therefore, approximation by a
,