Let $n$ be a given positive integer and let $f(x) = \sum_{k=0}^{n} c_kT_k(x)$, where $c_i \in \mathbb{R}$, $0 \leq i \leq n$. If

$|f(x)| \leq 1$, for  $|x| \leq 1$, is it possible to get the maximum of $|c_i|$?

For polynomials, bounds were obtained by Markov in 1892 as explained in these previous questions [polynomial1][1] and [polynomial2][2], which show that there is a relation with Chebyshev polynomials. However, it is curretly unclear to me how to translate these Theorems to Chebyshev expansions.

I have done some research and found some interesting Theorems in a paper from Majidian:

> Majidian H. On the decay rate of Chebyshev coefficients. Appl Numer Math 2017;113:44–53. https://doi.org/10.1016/j.apnum.2016.11.004.

However, I'm not a mathematician and have problems interpreting the conditions of these Theorems. A simple explanation in layman's terms or written out example for let's say $c_6$ would help me a lot! Thanks.



  [1]: https://mathoverflow.net/questions/257909/find-the-maximum-of-a-p-if-a-0a-1x-dotsa-nxn-1-1-mapsto-1-1?noredirect=1&lq=1
  [2]: https://mathoverflow.net/questions/281115/a-bounded-polynomial-having-bounded-coefficients-several-variables?noredirect=1&lq=1