}axb≤≤ is said to be orthogonal on I with respect to a
weight
function
()0px>
if
()()()0
b
mn
a
pxfxfxdx=
∫
for mn≠ (6.33)
The norm
n
f
of
n
f
is defi ned by
2
()()
b
nn
a
fpxfxdx=
∫
(6.34)
The functions are called orthonormal on I if they are orthogonal on I
and all have a norm equal to 1.
In respect of functions with ()1,px= we simply say ‘orthogonal’.
Thus, functions
123
,,,fff… are orthogonal on some interval I if
()()0
b
mn
a
fxfxdx=
∫
for mn≠ (6.35)
The norm
n
f
of
n
f
is then simply given by
2
()
b
nn
a
ffxdx=
∫
(6.36) ...
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