Download PDF

Chebyshev polynomial∗
drini†
2013-03-21 13:34:56
The Chebyshev polynomials of first kind are defined by the simple formula
Tn (x) = cos(nt),
where x = cos t.
It is an example of a trigonometric polynomial.
This can be seen to be a polynomial by expressing cos(kt) as a polynomial
of cos(t), by using the formula for cosine of angle-sum:
cos(1t)
=
cos(t)
cos(2t)
=
cos(t) cos(t) − sin(t) sin(t) = 2(cos(t))2 − 1
cos(3t)
=
..
.
4(cos(t))3 − 3 cos(t)
So we have
T0 (x)
=
1
T1 (x)
=
x
T2 (x)
=
2x2 − 1
T3 (x)
=
..
.
4x3 − 3x
These polynomials obey the recurrence relation:
Tn+1 (x) = 2xTn (x) − Tn−1 (x)
for n = 1, 2, . . .
∗ hChebyshevPolynomiali
created: h2013-03-21i by: hdrinii version: h32161i Privacy
setting: h1i hDefinitioni h42C05i h42A05i h33C45i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
Related are the Chebyshev polynomials of the second kind that are defined
as
Un−1 (cos t) =
sin(nt)
,
sin(t)
which can similarly be seen to be polynomials through either a similar process
as the above or by the relation Un−1 (t) = nTn0 (t).
The first few are:
U0 (x)
=
1
U1 (x)
=
2x
U2 (x)
=
4x2 − 1
U3 (x)
=
..
.
8x3 − 4x
The same recurrence relation also holds for U :
Un+1 (x) = 2xUn (x) − Un−1 (x)
for n = 1, 2, . . ..
2