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References
E. Egerváry andP. Turán, Notes on interpolation. VI. (On the stability of the interpolation on an infinite interval),Acta Math. Acad. Sci. Hung.,10 (1959), pp. 55–62.
G. Szegő, Über gewisse Interpolationspolynome, die zu den Jacobischen und Laguerreschen Abszissen gehören,Math. Zeitschrift,35 (1932), pp. 579–602. See also his book,Orthogonal polynomials.
Note that\(\bar r_{vn} (x)\) and\(\bar q_{vn} (x)\) change sign in (0, ∞) and (−∞, ∞), resp., whereasr vn (x) andq vn (x) arenon-negative in the respective intervals.
It should not be difficult to replace the requirement of the boundedness bySzegő's conditionf(x)=O(x m). Similar remark holds for Theorem II.
И. П. Натансон, Конструктивная теория функций (Москва-Ленинград, 1949), p. 83.
The manipulation in (3.4) is for the factorx.
SeeG. Szegő,Orthogonal polynomials, p. 173. What we actually need, is the boundedness of the left-side in (3.6) only.
SeeSzegő's book, p. 102 and 173.
SeeSzegő's book, p. 344.
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Balázs, J., Turán, P. Notes on interpolation. VII. Acta Mathematica Academiae Scientiarum Hungaricae 10, 63–68 (1959). https://doi.org/10.1007/BF02063290
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DOI: https://doi.org/10.1007/BF02063290