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A variant of Carathéodory's problem *

Published online by Cambridge University Press:  20 January 2009

Gilbert Strang
Gilbert Strang
Affiliation:
Massachusetts Institute of Technology.
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In this note we ask two questions and answer one. The questions can be combined as follows:

Does there exist a polynomial of the form

which starts with prescribed complex coefficients c0, …, cr–1; and satisfies

These differ from the classical problems of Carathéodory in one essential respect: the values of p and its first r–1 derivatives are given at the point z = 1 on the circumference of the unit circle, while in the original problem they were given at z = 0. Carathéodory's own answer was in terms of his “moment curve”, but the forms studied a few years later by Toeplitz yield a more convenient statement of the solution.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 16 , Issue 1 , June 1968 , pp. 43 - 48
Copyright
Copyright © Edinburgh Mathematical Society 1968

References

REFERENCES

(1) Grenander, U. and Szegö, G., Toeplitz Forms and their Applications (University of California Press, Berkeley and Los Angeles, 1958).CrossRefGoogle Scholar
(2) Strang, G., Accurate partial difference methods II: Non-linear problems, Numerische Math. 6 (1964), 37–46.CrossRefGoogle Scholar
(3) Strang, G, Unbalanced polynomials and difference methods for mixed problems, SIAM J. Numer. Anal. 2 (1964), 46–51.Google Scholar
(4) Strang, G., Implicit difference methods for initial boundary value problems, Math. Anal, and Applications, 16 (1966), 188–198.CrossRefGoogle Scholar