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The work of Chebyshev on orthogonal polynomials. (English) Zbl 0858.01016

Rassias, Th. M. (ed.) et al., Topics in polynomials of one and several variables and their applications. Volume dedicated to the memory of P. L. Chebyshev (1821-1894). Singapore: World Scientific. 495-512 (1993).
Summary: P. L. Chebyshev was probably the first mathematician to recognize 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. Other isolated instances of orthogonal polynomials occurring in the work of various mathematicians will be mentioned later. 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. He discovered the discrete analog of the Jacobi polynomials but their importance was not recognized until this century. They were rediscovered by Hahn and named after him upon their rediscovery. L. Ya. Geronimus, Theory of orthogonal polynomials (Russian), Moscow (1950) has pointed out that in his first paper on orthogonal polynomials, Chebyshev already had the Christoffel-Darboux formula. This paper appeared more than a decade before Christoffel’s and Darboux’s work (which they did independently of each other). In recent years, the extent of Chebyshev’s contributions has gained greater recognition.
In this paper, we discuss some of his fundamental papers on orthogonal polynomials, with an emphasis on the discrete ones. Some detail of his methods and techniques is also provided.
For the entire collection see [Zbl 0849.00029].

MSC:

01A55 History of mathematics in the 19th century
33-03 History of special functions

Biographic References:

Chebyshev, P. L.
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