I have a function $f(x)$ sampled at 11 $x$ positions:
I want to approximate the function by a Chebyshev expansion:
$ \ f(x) \simeq \sum\limits_{i=0}^m c_i T_i(y) - \frac{1}{2}c_0,\qquad y=2(x-x_1)/(x_{11}-x_1) - 1 $
The first $3$ Chebyshev polynomials are:
$ T_0(y) = 1\qquad T_1(y) = y\qquad T_2(y) = 2y^2-1 $
The coefficents $ c_i $ are given by: $ c_i = \frac{2}{\pi}\int_{-1}^{1}\frac{f(y)T_i(y)}{\sqrt{1-y^2}}dy $
In practice I think it may be easiest to change integration variable to $t=\arccos(y) $ followed by numerical integration?
My main goal with this question however is to understand the coefficients. From what I understand the coefficients can be viewed as "projections" that tell how similar the function is to the basis functions.
I would therefore think that $c_i = 0$ means completely different from the $i$-th basis function: $ \frac{T_i(y)}{\sqrt{1-y^2}}. $
A high $ c_i $ value would mean very similar to the i'th basis function and a large negative $ c_i $ value would mean that $-f(x)$ is very similar to the $i$-th basis function.
Is this correct?
Is it possible to normalize the coefficients so that: $ -1 \leq c_i \leq 1 $?
Do you have any tips on how to "interpret" the polynomials (I am a physicist)?
Thanks in advance for any answers!