# 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 $$(y_1,\ \dotsc\ y_{n-1})$$, $$(y_{i+2}-y_{i+1})(y_{i+1}-y_i)<0$$, 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 $$p(\sum_{i=2}^{k}a_i^2)=y_k.$$

• Have you checked Wikipedia? The page says the polynomials occur first in a 1854 paper by Chebyshev. Is there any reason to doubt this? Sep 9, 2016 at 20:05
• Would you mind describing your variant as an answer? I am curious to see it. Thank you.
– Hans
Sep 9, 2016 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. Sep 9, 2016 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. Sep 9, 2016 at 20:56
• This would have been better for hsm.stackexchange.com.
– user21349
Sep 9, 2016 at 22:45

The Chebyshev polynomials first appeared in his paper Théorie des mécanismes connus sous le nom de parallélogrammes (1854). The remarkable "mechanisms" described in this work can be seen in action here (click on each picture to activate it).

The context is described in MacTutor:

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.

• There is a bit more to the story, as far as Chebyshev's motivation goes: He was motivated by the open (at the time) problem of constructing a mechanical linkage converting circular motion to the linear one. He did not know how to solve the problem but, as an applied mathematician, looked for approximate solutions (good approximation with few joints, whose number is roughly the degree of the polynomial). Eventually, the initial problem was solved by Peucellier and, later, by Lipkin, Chebyshov's student. Sep 10, 2016 at 2:04
• @Mischa that indicates to me, that trigonometric functions probably played an important role in the discovery of the Chebyshev polynomials. Sep 10, 2016 at 8:37

An earlier, but different, approach to the questions posed and solved by next-to-be Tchebyshev Polynomials was given by Augustin-Louis Cauchy in his Cours d'Analyse (1821). See p. 230 and ff.

In the original paper the same Chebyshev admits that his min-max approximation by means of polynomials has been inspired by the followng paper of Poncelet:
Jean-Victor Poncelet, "Sur la valeur approchée linéaire et rationnelle des radicaux de la forme $$\sqrt{a^2+b^2}$$, $$\sqrt{a^2-b^2}$$ etc.", Journal für die Reine und Angewandte Mathematik, 13, 277-291 (1835), DOI:10.1515/crll.1835.13.277, ERAM 013.0495cj.