Establish that the Chebyshev polynomial form a basis tor P3 PO(t)=1. Pl (t)=t. P2(t)=2t^2-1. P3(t)=4t^3-3t Solution The Chebyshev polynomials of the first kind are defined by the recurrence relation. P0(t) = 1 P1(t) = t and further by Pn+1(t) = 2t*Pn(t) - Pn-1(t) so P3(t) = 2t*P2(t) - P1(t) = 2t*(2t^2 - 1) - t = 4t^3 - 3t.

1. Establish that the Chebyshev polynomial form a basis tor P3 PO(t)=1. Pl (t)=t. P2(t)=2t^2-1.
P3(t)=4t^3-3t
Solution
The Chebyshev polynomials of the first kind are defined by the recurrence relation.
P0(t) = 1
P1(t) = t
and further by
Pn+1(t) = 2t*Pn(t) - Pn-1(t)
so
P3(t) = 2t*P2(t) - P1(t)
= 2t*(2t^2 - 1) - t
= 4t^3 - 3t