Geodesic
An Analogue of a Problem of J. Balázs and P. Turán
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 54-63

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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 — x 2)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 theoryof a differential equation of the form y′ + A(x)y= 0. 
Prasad, J.; Varma, A. K. An Analogue of a Problem of J. Balázs and P. Turán. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 54-63. doi: 10.4153/CJM-1969-006-2
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