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](https://mathoverflow.net/questions/249480/what-is-the-story-behind-the-chebyshev-polynomials#comment613286_249480) of @Hans, here is formal definition of my problem:
given a sequence $(y\_1,\ \dotsc\ y\_{n-1})$, $(y\_{i+2}-y\_{i+1})(y\_{i+1}-y\_i)<0$, of values, determine a polynomial $p(x)$ of degree $n$, and $\ n{-}1$ abscissæ $\ \xi\_1 <\ \dotsb\ <\xi\_{n-1}$, 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 $0$.  

$$\ $$


**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 abscissæ. 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 be a 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
$$p(\sum\_{i=2}^{k}a\_i^2)=y\_k.$$



What is the story behind the Chebyshev polynomials?
The Chebyshev polynomials first appeared in his paper [Théorie des mécanismes connus sous le nom de parallélogrammes](http://www.math.technion.ac.il/hat/fpapers/cheb11.pdf) (1854). The remarkable "mechanisms" described in this work can be seen in action [here](http://www.tcheb.ru) (click on each picture to activate it).


The context is described in [MacTutor:](http://www-history.mcs.st-and.ac.uk/history/Biographies/Chebyshev.html)



> 
> Chebyshev was probably the first mathematician to recognise the
> general concept of orthogonal polynomials. A few particular orthogonal
> polynomials were known before his work. Legendre and Laplace had
> encountered the Legendre polynomials in their work on celestial
> mechanics in the late eighteenth century. Laplace had found and
> studied the Hermite polynomials in the course of his discoveries in
> probability theory during the early nineteenth century.
> It was Chebyshev who saw the
> possibility of a general theory and its applications. His work arose
> out of the theory of least squares approximation and probability; he
> applied his results to interpolation, approximate quadrature and other
> areas.
> 
> 
> 


For a more extensive account of the history of this discovery, see [The theory of best approximation of functions.](https://www.sciencedirect.com/science/article/pii/S0021904500934762)


