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These approximations are usually not very good because the line or parabola are
poor approximations to f. The key to improving the accuracy is to look for better
approximating polynomial functions. The reason for sticking with polynomials or
piecewise polynomial functions is that they are easy to integrate. This leads us of
course to another big subject, namely, finding the best piecewise polynomial approx-
imations to a function.
Assume that the function f(x) is known at points x
0
, x
1
,..., x
n
. If one looks over
some of the simple standard approaches to finding an approximating polynomial,
such as Lagrange or Hermite interpolation, one will see that they all try to approxi-
mate f(x) by a polynomial g(x) of the form
n
Â
()
=
()()
gx
fx p x
i
,
(F.2)
i
i
=
0
where p
i
(x) are suitably chosen polynomials. (In the Hermite case one actually adds
another similar sum but one using the values of the derivative of f at x
i
.) Since we
want an approximation to f(x), we want the error function
()
=
()
-
()
Ex
fx
gx
(F.3)
to be “small” over [a,b]. Now, we can think of formula (F.2) as representing a linear
combination of polynomials p
i
(x). Continuing this line of thought, the problem then
becomes one of finding a basis p
0
(x), p
1
(x),...for the subspace of polynomials in the
vector space C
r
([a,b]) so that the linear combination in (F.2) best approximates this
arbitrary function f(x). Orthonormal bases of vector spaces always have many
advantages and so one is lead to looking for sequences p
0
(x), p
1
(x),...of “orthogo-
nal” polynomials.
Definition.
Given an inner product <,> on C
r
([a,b]), a sequence of polynomials p
0
(x),
p
1
(x),...in C
r
([a,b]) is called a
sequence of orthogonal polynomials over [a,b]
with
respect to <,> if the polynomials p
i
(x) have degree i and are pairwise orthogonal with
respect to <,>, that is, <p
i
(x),p
j
(x)>=0, for i π j.
Let <,> be an inner product on C
r
([a,b]).
F.2.1. Theorem.
(1) Any sequence of orthogonal polynomials [a,b] with respect to <,> forms a basis
for the space of all polynomials over [a,b].
(2) The condition p
0
(x) = 1 and p
i
(1) = 1, i ≥ 1, define a unique sequence of
orthogonal polynomials p
i
(x) called the
sequence of orthogonal polynomials
associated to
<
,
>. (We assume that 1 belongs to [a,b] here.)
Proof.
Easy.
The inner product on C
r
([a,b]) that we have in mind here is
b
<>=
()()
Ú
fg
,
fxgxdx
,
(F.4a)
a
or, more generally,
