How to determine coefficients of $p(x) = x^6$ with the Chebyshev processing

Question

I want to calculate the coefficients of $p(x) = x^6$ with the Chebyshev processing.

How to do that?

Following question would be, how to estimate the error in $[-1,1]$, if i only use terms until grade 4?

Thank you so much.

Answer 1

I'm not sure what you're asking. Specifically, I don't know what you mean by "the Chebyshev processing".

I think you are asking how to express $x^6$ in terms of Chebyshev polynomials. If so, the answer is in these slides. Slide #12 says that: $$ x^6 = \frac{1}{32}T_6 + \frac{3}{16}T_4 + \frac{15}{32}T_2 + \frac{5}{16}T_0 $$

If you omit the $T_6$ term, the error on $[-1,1]$ is related to the maximum value of $T_6$ on $[-1,1]$, which you probably know how to calculate.