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Interpolation by Polynomials in z and z–1 in the Roots of Unity

Published online by Cambridge University Press:  20 November 2018

A. Sharma
A. Sharma*
Affiliation:
University of Alberta
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Given a function ƒ(z), continuous on C: |z| = 1 in the complex plane, there is a close analogy between approximation in the sense of least squares by polynomials on the unit circle and interpolation by polynomials in the nth roots of unity to the same function. For detailed discussion of the problem and its generalization for a suitable Jordan curve one can refer to Walsh (3) or to a recent paper by Curtiss (2). More recently, Curtiss (1) has considered the problem of interpolation by polynomials in non-equally spaced points on the unit circle and has pointed out the limitations inherent in the problem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 16 - 23
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Curtiss, J. H., Polynomial interpolation in points equidistributed on the unit circle, Pacific J. Math., 12 (1962), 863–877.Google Scholar
2. Curtiss, J. H., A stochastic treatment of some classical interpolation problems, Proc. Fourth Berkeley Symposium Math. Statistics and Probability, Vol. II (Berkeley, 1961), pp. 79–93.Google Scholar
3. Walsh, J. L., Interpolation and approximation by rational functions in the complex domain (Providence, 1960). pp. 153–156, 179-182.Google Scholar
4. Walsh, J. L., Interpolation and an analogue of the Laurent development, Proc. Nat. Acad. Sci., 19 (1) (1933), 203-20.Google Scholar
5. Walsh, J. L. and Sharma, A.. Least square approximation and interpolation in the roots of unity, Pacific J. Math., 14 (1964), 727–730.Google Scholar