to functions which satisfy suitable boundary conditions. More precisely, it is
appropriate to impose two independent boundary conditions of the form
B
1
(
f
)
=
α
1
f
(
a
) +
α
1
f
(
a
) +
β
1
f
(
b
) +
β
1
f
(
b
) = 0
,
(3)
B
2
(
f
)
=
α
2
f
(
a
) +
α
2
f
(
a
) +
β
2
f
(
b
) +
β
2
f
(
b
) = 0
,
(4)
where the
α
’s and
β
’s are constants. We say that the boundary operators
B
1
,
B
2
in (3),(4) are
selfadjoint
(relative to the operator
L
), if
r
(
f
g

f
g
)
b
a
= 0
for all
f, g
satisfying (3),(4).
Most boundary conditions that arise in practice are
separated
, meaning that
each one of (3),(4) involves a condition at only one endpoint,
αf
(
a
) +
α f
(
a
) = 0
,
βf
(
b
) +
β f
(
b
) = 0
,
(5)
where (
α, α
) = (0
,
0), and (
β, β
) = (0
,
0).
It is easy to check that separated boundary conditions are always selfadjoint
(relative to
any
operator
L
).
There is also one set of nonseparated boundary conditions that is commonly
used, the
periodic
boundary conditions
f
(
a
) =
f
(
b
)
,
f
(
a
) =
f
(
b
)
.
(6)
These are selfadjoint relative to
L
provided that
r
(
a
) =
r
(
b
)
.
We are now ready to formulate the boundary value problems that lead to
orthogonal bases for
R
(
a, b
).
Definition 4.2.
Let a formally selfadjoint DOP
L
and two boundary
operators
B
1
, B
2
, which are selfadjoint for
L
, be given by (1) and (3),(4),
respectively, where
r, r
and
p
are real and continuous on
[
a, b
]
and
r >
0
on
[
a, b
]
, and let
w
be a positive, continuous function on
[
a, b
]
.
26