Graphics Reference
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b
<>=
()() ()
Ú
fg
,
fxgxwxdx
,
(F.4b)
a
for some nonnegative
weight function
w(x). To keep formulas simple, let us assume
that [a,b] = [-1,1]. (It is easy to change the domain of functions by a linear change of
variables.)
Note that the sequence 1, x, x
2
,...is not a sequence of orthogonal polynomials
over [-1,1] with respect to the inner product defined by (F.4a).
F.2.2. Theorem.
The polynomials P
i
(x) in the sequence of orthogonal polynomials
associated to the inner product defined by (F.4a) are defined by the following recur-
sion formulas:
()
=
Px
1
0
21
1
i
i
+
+
i
()
=
()
-
()
Px
xP x
Px i
,
≥
0
.
(F.5)
i
+
1
i
i
-
1
i
+
1
Proof.
See [ConD72].
Definition.
The orthogonal polynomials P
i
(x) defined by formulas (F.5) are called
the
Legendre polynomials
.
The first few elements in the sequence of orthogonal polynomials P
i
(x) are easily
seen to be
1
2
31
1
2
(
)
(
)
2
3
1
,
x
,
x
-
,
53
x
-
x
, ....
One could use a simple Gram-Schmidt type algorithm applied to the sequence 1, x,
x
2
,...to find these polynomials.
F.2.3. Theorem.
The polynomials T
i
(x) in the sequence of orthogonal polynomials
associated to the inner product defined by (F.4b) with
1
wx
()
=
2
1
-
x
are defined by
(
)
i
()
=
-
1
()
Tx
cos
i
cos
x
(F.6a)
or by the following recursion formulas:
()
=
()
=
Tx
Tx TxTx i
1
2
0
()
-
()
,
≥
0
.
(F.6b)
i
+
1
i
i
-
1
Proof.
See [ConD72].