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Birkhoff Interpolation at the nth Roots of Unity: Convergence

Published online by Cambridge University Press:  20 November 2018

S. D. Riemenschneider  and
A. Sharma
S. D. Riemenschneider
Affiliation:
University of Alberta, Edmonton, Alberta
A. Sharma
Affiliation:
University of Alberta, Edmonton, Alberta
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The first investigations on this type of problem were carried out by O. Kiš [2]. Kiš considered the problem of interpolating a function and its second derivative at the nth roots of unity (the (0, 2) problem) by 2n – 1 degree polynomials, and the convergence of such approximates. Later, Sharma [8], [9] extended the existence and uniqueness results to (0, m) interpolation, and essentially to (0, m1, m2) interpolation. In the latter case, Sharma established the convergence results for (0, 2, 3), (0, 1, 3) and (0, 1, 4) interpolation as well. Although some further special cases were considered [10] these were the essential results until very recently. Now Cavaretta, Sharma and Varga [1] have established the existence and uniqueness theorem for all possible interpolations of this type (see Theorem 2.1 below). Motivated by the work of Cavaretta, Sharma and Varga and the earlier work of Sharma [9], a different proof of this result is provided, and this proof is used to establish a convergence theorem in the general case.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 362 - 371
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Cavaretta, A. S., Jr., Sharma, A. and Varga, R. S., Hermite-Birkhoff interpolation in the nth roots of unity. Trans. Amer. Math. Soc, to appear.Google Scholar
2. Kis, O., Remarks on interpolation (Russian), Acta Math. Acad. Sci. Hungar. 11 (1960), 49–64.Google Scholar
3. Lorentz, G. G., Approximation of functions (Holt, New York, 1966).Google Scholar
4. Lorentz, G. G. and Riemenschneider, S. D., Recent progress in Birkhoff interpolation, in Approximation theory and functional analysis (North-Holland Publishing Co., 1979), 187-236.Google Scholar
5. Lorentz, G. G. and Riemenschneider, S. D., Birkhoff interpolation:’ Some applications of coalescence in Quantitative approximation (Academic Press, 1980), 197–208.Google Scholar
6. Lozinski, S. M., Fejer's interpolating process, Dokl. Acad. Nauk. SSSR 24 (1939), 318–321.Google Scholar
7. Riesz, M., Eine trigonometrische Inter polationsfor met und einige Ungleichungen fur Polynôme, Jahresbericht d. Deutschen Math. Ver. 23 (1914), 354–368.Google Scholar
8. Sharma, A., Lacunary interpolation in the roots of unity, Z. Angew. Math. Mech. 1+6 (1966), 127–133.Google Scholar
9. Sharma, A., Some remarks on lacunary interpolation in the roots of unity, Israel J. Math. 2 (1964), 41–49.Google Scholar
10. Sharma, A., Some poised and nonpoised problems on interpolation, SI AM Rev. 1J+ (1972), 129–151.Google Scholar
11. Smirnov, V. I. and Lebedev, N. A., Functions of a complex variable, Constructive theory (Iliffe Books Ltd., London, 1968).Google Scholar
12. Zia-Uddin, , Note on an u alternant1 ‘ with factorial elements, Proceedings Edin. Math. Soc. 3 (1933), 296–299.Google Scholar