Starting from a sequence
{
p
n
(
x
;
μ
0
)
}
\{ {p_n}(x;\,{\mu _0})\}
of orthogonal polynomials with an orthogonality measure
μ
0
{\mu _0}
supported on
E
0
⊂
[
−
1
,
1
]
{E_0} \subset [ - 1,\,1]
, we construct a new sequence
{
p
n
(
x
;
μ
)
}
\{ {p_n}(x;\,\mu )\}
of orthogonal polynomials on
E
=
T
−
1
(
E
0
)
E = {T^{ - 1}}({E_0})
(
T
T
is a polynomial of degree
N
N
) with an orthogonality measure
μ
\mu
that is related to
μ
0
{\mu _0}
. If
E
0
=
[
−
1
,
1
]
{E_0} = [ - 1,\,1]
, then
E
=
T
−
1
(
[
−
1
,
1
]
)
E = {T^{ - 1}}([ - 1,\,1])
will in general consist of
N
N
intervals. We give explicit formulas relating
{
p
n
(
x
;
μ
)
}
\{ {p_n}(x;\,\mu )\}
and
{
p
n
(
x
;
μ
0
)
}
\{ {p_n}(x;\,{\mu _0})\}
and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses
T
T
to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.