Chebyshev polynomials
December 7, 2016
1 First kind
On the one hand,
On the other hand,
Therefore
Now,
Hence
For
(1) |
Note
Theorem 1.
and
Proof.
For
Then
That is, for
Theorem 2.
Proof.
Using
and
Then
Therefore
That is, for
and by analytic continuation this is true for all
2 Second kind
Define
(2) |
Theorem 3.
and
Proof.
On the one hand,
On the other hand,
Hence
Now,
and
Hence
and then
By analytic continuation,
∎
Theorem 4.
Proof.
Therefore by analytic continuation,
∎
Theorem 5.
Proof.
Therefore by analytic continuation,
∎
Theorem 6.
Theorem 7.
Theorem 8.
3 Inner products
For
Theorem 9.
Proof.
Let
Using
Then
Doing the substitution
Therefore
For
∎