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An Analogue of a Problem of J. Balázs and P. Turán

Published online by Cambridge University Press:  20 November 2018

J. Prasad  and
A. K. Varma
J. Prasad
Affiliation:
University of Florida, Gainesville, Florida
A. K. Varma
Affiliation:
University of Florida, Gainesville, Florida
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In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.

Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 54 - 63
Copyright
Copyright © Canadian Mathematical Society 1969

References

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