VI.45 Pafnuty Chebyshev

b. Okatovo, Russia, 1821; d. Saint Petersburg, Russia, 1894
Assistant, Extraordinary then full Professor of Mathematics,
Saint Petersburg (1847-82); Artillery Committee (1856);
Scientific Committee of the Ministry of Education (1856)

Fascinated by Watt’s parallelogram (the linkage used in steam engines) and the problem of converting circular motion into rectilinear motion, Chebyshev embarked on a deep study of the theory of hinge mechanisms. In particular, he sought the linkage that would produce the minimum deviation from a straight line over a given range. This corresponds to the mathematical problem of finding, from among the class of functions chosen to approximate a given function, the one with the smallest absolute error for all specified values of the argument. It was in this context, in particular considering the approximation of functions by polynomials, that Chebyshev discovered the polynomials now named after him (see [III.85]). These polynomials were first published in his memoir “Théorie des mécanismes connus sous le nom de parallélogrammes” (1854), and they marked the beginning of his important contributions to the theory of orthogonal polynomials.

Chebyshev polynomials of the first kind are defined by T_n(cos θ) = cos (nθ), for n = 0, 1, 2, . . . . These polynomials also satisfy the recurrence relation T_n+1 (x) = 2xT_n(2) – T_n–1(x), where T₀(x) = 1 and T₁(x) = x. Chebyshev polynomials of the second kind satisfy U_n(cos θ) = sin ((n + 1)θ)/sin θ and the recurrence relation U_n+1 (x) = 2xU_n(x)— U_n–1(x), where U₀(x) = 1 and U₁ (x) = 2x.

Chebyshev also had a significant impact on number theory, coming close to proving the PRIME NUMBER THEOREM [V.26]. In probability he is remembered for Chebyshev’s inequality, a result that is simple but has innumerable applications.