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30 Polynomial Approximation of Differential Equations
If α = −1 2 , by (1.5.6) we get
(2.3.11) TkTj = 1
2
Tk+j + T |k−j| , k,j ∈ N.
Besides, for Hermite polynomials we have
(2.3.12) HkHj =
min(k,j)
m=0
2 m m!
k j
Hk+j−2m, k,j ∈ N.
m m
Taking p = q in (2.3.4), we can finally evaluate the expansion coefficients of p 2 in
terms of those of p. The reader can find more results, suggestions and references in
askey(1975), lecture 5.
2.4 The projection operator
More generally, in analogy with (2.3.7), if f is a given function such that fw is integrable
in I (when I is not bounded fw will be also required to decay to zero at infinity), then
we define its Fourier coefficients by setting
(2.4.1) ck :=
1
fukw dx, k ∈ N.
uk 2 w
I
Let us introduce now the operator Πw,n : C 0 ( Ī) → Pn, n ∈ N. This operator acts as
follows. For any continuous function f, we compute its coefficients according to (2.4.1).
Then, Πw,nf ∈ Pn is defined to be the polynomial pn := n
k=0 ckuk. It turns out that
Πw,n is a linear operator. We call pn the orthogonal projection of f onto Pn, through
the inner product (·, ·)w. Of course, one has
(2.4.2) Πw,np = p, ∀p ∈ Pn,
(2.4.3) Πw,num = 0, ∀m > n.