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X.—Interpolated Derivatives

Published online by Cambridge University Press:  15 September 2014

B. Spain
B. Spain
Affiliation:
Mathematical Institute, University of Edinburgh
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Extract

I. Various writers (Ferrar, 1927) have started out with different definitions of generalised derivatives. Essentially, the problem of the generalised derivative is a problem in interpolation. The values of the derivatives are known for all integer values of n; for all positive integers, being the ordinary derivatives; for zero, being the function itself; for negative integers, being repeated integrals. Any function of n which has the above values at the integers (i.e. any cotabular function) is a solution of the problem. Out of the infinite number of cotabular functions, there exists one discovered by E. T. Whittaker (Whittaker, 1915; Ferrar, 1925; Whittaker, 1935), called the cardinal function, possessing rather remarkable properties. In particular, if the cardinal series defining the cardinal function is convergent, then it is equivalent to the Newton-Gauss formula of interpolation.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 60 , Issue 2 , 1940 , pp. 134 - 140
Copyright
Copyright © Royal Society of Edinburgh 1940

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References

References to Literature

Bromwich, T. J., 1926. Infinite Series, 2nd ed., London.Google Scholar
Copson, E. T., 1935. Functions of a Complex Variable, Oxford.Google Scholar
Ferrar, W. L., 1925. “On the Cardinal Function of Interpolation-Theory,” Proc. Roy. Soc. Edin., vol. xlv, pp. 269–282.Google Scholar
Ferrar, W. L., 1927. “Generalised Derivatives and Integrals,” Proc. Roy. Soc. Edin., vol. xlviii, pp. 92–105.Google Scholar
Whittaker, E. T., 1915. “On the Functions which are represented by the Expansions of the Interpolation-Theory,” Proc. Roy. Soc. Edin., vol. xxxv, pp. 181–194.CrossRefGoogle Scholar
Whittaker, E. T., and Watson, G. N., 1927. Modern Analysis, 4th ed., Cambridge, § 16.2.Google Scholar
Whittaker, J. M., 1935. Cambridge Tracts, No. 33.Google Scholar