<s>
In	O
mathematics	O
,	O
orthogonal	B-Algorithm
functions	I-Algorithm
belong	O
to	O
a	O
function	B-Algorithm
space	I-Algorithm
that	O
is	O
a	O
vector	O
space	O
equipped	O
with	O
a	O
bilinear	O
form	O
.	O
</s>
<s>
When	O
the	O
function	B-Algorithm
space	I-Algorithm
has	O
an	O
interval	O
as	O
the	O
domain	B-Algorithm
,	O
the	O
bilinear	O
form	O
may	O
be	O
the	O
integral	O
of	O
the	O
product	O
of	O
functions	O
over	O
the	O
interval	O
:	O
</s>
<s>
As	O
with	O
a	O
basis	O
of	O
vectors	O
in	O
a	O
finite-dimensional	O
space	O
,	O
orthogonal	B-Algorithm
functions	I-Algorithm
can	O
form	O
an	O
infinite	O
basis	O
for	O
a	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Suppose	O
is	O
a	O
sequence	O
of	O
orthogonal	B-Algorithm
functions	I-Algorithm
of	O
nonzero	O
L2-norms	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
sequence	O
is	O
of	O
functions	O
of	O
L2-norm	O
one	O
,	O
forming	O
an	O
orthonormal	B-Algorithm
sequence	I-Algorithm
.	O
</s>
<s>
To	O
have	O
a	O
defined	O
L2-norm	O
,	O
the	O
integral	O
must	O
be	O
bounded	O
,	O
which	O
restricts	O
the	O
functions	O
to	O
being	O
square-integrable	B-Algorithm
.	O
</s>
<s>
Several	O
sets	O
of	O
orthogonal	B-Algorithm
functions	I-Algorithm
have	O
become	O
standard	O
bases	O
for	O
approximating	O
functions	O
.	O
</s>
<s>
Together	O
with	O
cosine	O
functions	O
,	O
these	O
orthogonal	B-Algorithm
functions	I-Algorithm
may	O
be	O
assembled	O
into	O
a	O
trigonometric	O
polynomial	O
to	O
approximate	O
a	O
given	O
function	O
on	O
the	O
interval	O
with	O
its	O
Fourier	O
series	O
.	O
</s>
<s>
If	O
one	O
begins	O
with	O
the	O
monomial	O
sequence	O
on	O
the	O
interval	O
and	O
applies	O
the	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
process	I-Algorithm
,	O
then	O
one	O
obtains	O
the	O
Legendre	O
polynomials	O
.	O
</s>
<s>
The	O
study	O
of	O
orthogonal	O
polynomials	O
involves	O
weight	B-Algorithm
functions	I-Algorithm
that	O
are	O
inserted	O
in	O
the	O
bilinear	O
form	O
:	O
</s>
<s>
For	O
Laguerre	O
polynomials	O
on	O
the	O
weight	B-Algorithm
function	I-Algorithm
is	O
.	O
</s>
<s>
Both	O
physicists	O
and	O
probability	O
theorists	O
use	O
Hermite	O
polynomials	O
on	O
,	O
where	O
the	O
weight	B-Algorithm
function	I-Algorithm
is	O
or	O
.	O
</s>
<s>
Walsh	O
functions	O
and	O
Haar	O
wavelets	O
are	O
examples	O
of	O
orthogonal	B-Algorithm
functions	I-Algorithm
with	O
discrete	O
ranges	O
.	O
</s>
<s>
This	O
procedure	O
results	O
in	O
families	O
of	O
rational	O
orthogonal	B-Algorithm
functions	I-Algorithm
called	O
Legendre	O
rational	O
functions	O
and	O
Chebyshev	O
rational	O
functions	O
.	O
</s>
<s>
Solutions	O
of	O
linear	O
differential	O
equations	O
with	O
boundary	O
conditions	O
can	O
often	O
be	O
written	O
as	O
a	O
weighted	B-Algorithm
sum	I-Algorithm
of	O
orthogonal	O
solution	O
functions	O
(	O
a.k.a.	O
</s>
<s>
eigenfunctions	B-Algorithm
)	O
,	O
leading	O
to	O
generalized	O
Fourier	O
series	O
.	O
</s>
