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An Interpolation Series for Integral Functions

Published online by Cambridge University Press:  20 January 2009

Sheila Scott Macintyre
Sheila Scott Macintyre
Affiliation:
The University, Aberdeen.
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1. The Gontcharoff interpolation series

where

has been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

the constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

and that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 9 , Issue 1 , October 1953 , pp. 1 - 6
Copyright
Copyright © Edinburgh Mathematical Society 1953

References

REFERENCES

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