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On Tchebycheff Quadrature

Published online by Cambridge University Press:  20 November 2018

Paul Erdös  and
A. Sharma
Paul Erdös
Affiliation:
University of Alberta, Calgary
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Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula

(1)

is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 652 - 658
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Bernstein, S., Über die Quadraturformeln von Cotes und Tschebyscheff, Dokl. Akad. Nauk, 14 (1937), 323–327 (Russian).Google Scholar
2. Erdös, P. and Turán, P., On Interpolation III ﹛Interpolatory Theory of Polynomials), Ann. Math., 41 (3) (1940), 510–553.Google Scholar
3. Fejér, L., Mechanische Quadraturen mit Cotesschen Zahlen, Math. Z., 37 (1933), 287–310.Google Scholar
4. Natanson, L. P., Konstruktive Funktionentheorie (Berlin, 1955), pp. 466–479.Google Scholar