# What is the Story behind the Chebyshev Polynomials?

Is there anything reliable known about who actually discovered the Chebyshev polynomials and what the motivation and circumstances were?

The reason why I am interested in knowing, is that I needed a solution for a variant of those polynomials: instead of all extrema having the same magnitude, I wanted to have them attain predefined values in a fixed order (I have found a solution for that problem, but involves a system of polynomial equations) and I wonder, whether the definition of the Chebyshev polynomials has been "guessed" or developed for a specific problem.

Edit:

at the request of @Hans, here is formal definition of my problem: given a sequence $(y_1,\ ...\ y_{n-1}), (y_{i+2}-y_{i+1})(y_{i+1}-y_i)<0$ of values, determine a polynomial $p(x)$ of degree $n$ and, $\ n$-$1$ abszissas $\ \xi_1 <,\ ...,\ <\xi_{n-1}$, so that $\ p(\xi_i)=y_i, p'(\xi_i)=0$

It should be noted that the polynomials that I am looking for, have no special properties, except for the predefined values in the extrema. The leading coefficient can be set to $1$ and the constant term to $0$.
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Construction of polynomials with predefined sequence of function-values for its local extrema:

we can w.l.o.g. assume that the sought polynomial has leading coefficient $1$, a local extremum in the origin and, that all other local extrema are located at positive abszissas. Then polynomial is
$$p(x) =\frac{1}{n}\int x\prod_{i=2}^{n-1}(x-\xi_i)$$ and $$p(\xi_i)=y_i$$ would a be system of polynomial equations for determining the $\xi_i$ and thus $p(x)$; the only problem being that, because of the symmetry, in the current formulation there is no control over the ordering of the $y(\xi_i)$.
That can however easily be fixed by defining $$\xi_k=\sum_{i=2}^{k}a_i^2$$ and solving the system of polynomial equations $$p(\sum_{i=2}^{k}a_i^2)=y_k$$

• Have you checked Wikipedia? The page says the polynomials occur first in a 1854 paper by Chebyshev. Is there any reason to doubt this? – Fabian Wirth Sep 9 '16 at 20:05
• Would you mind describing your variant as an answer? I am curious to see it. Thank you. – Hans Sep 9 '16 at 20:46
• @Hans of course I can do so; by the way my interest for those polynomials was motivated by thoughts about parametric interpolation. – Manfred Weis Sep 9 '16 at 20:50
• @FabianWirth actually not, I think that Wikipedia is as trustworthy as Arxiv, but apart from that, the article doesn't answer my question about the circumstances of discovery. – Manfred Weis Sep 9 '16 at 20:56
• This would have been better for hsm.stackexchange.com. – Ben Crowell Sep 9 '16 at 22:45