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On the Peano Derivatives

Published online by Cambridge University Press:  20 November 2018

P. S. Bullen  and
S. N. Mukhopadhyay
P. S. Bullen
Affiliation:
University of British Columbia, Vancouver, British Columbia
S. N. Mukhopadhyay
Affiliation:
University of British Columbia, Vancouver, British Columbia
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Let f be a real valued function defined in some neighbourhood of a point x. If there are numbers α1, α2, … αr-1, independent of h such that

then the number αk is called the kth Peano derivative (also called kth de la Vallée Poussin derivative [6]) of f at x and we write αk = fk(x). It is convenient to write α0 = f0(x) = f(x). The definition is such that if the mth Peano derivative exists so does the nth for 0 ≦ n ≦ m.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 1 , February 1973 , pp. 127 - 140
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bruckner, A. M., An affirmative answer to a problem of Zahorski and some consequences, Michigan Math. J. 18 (1966), 15–26.Google Scholar
2. Bullen, P. S., A criterion for n-convexity, Pacific J. Math. 86 (1971), 81–98.Google Scholar
3. Burkill, J. C., The Cesáro-Perron scale of integration, Proc. London Math. Soc. 39 (1935), 543–552.Google Scholar
4. Ellis, H. W., Darboux properties and applications to nonabsolutely convergent integrals, Can. J. Math. 3 (1951), 471–484.Google Scholar
5. James, R. D., Generalised nth primitives, Trans. Amer. Math. Soc. 76 (1954), 149–176.Google Scholar
6. Marcinkiewicz, J. and Zygmund, A., On the differentiability of functions and summability of trigonometric series, Fund. Math. 26 (1936), 1–43.Google Scholar
7. Mukhopadhyay, S. N., On a certain property of the derivative, Fund. Math. 67 (1970), 279–284.Google Scholar
8. Oliver, H. W., The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444–456.Google Scholar
9. Saks, S., Theory of the integral (Hafner, Warsaw, 1937).Google Scholar
10. Sargent, W. L. C., On the Cesáro derivates of a function, Proc. London Math. Soc. 40 (1936), 235–254.Google Scholar
11. Sargent, W. L. C., On sufficient conditions for a function integrable in the Cesáro-Perron sense to be monotonie, Quart. J. Math. Oxford Ser. 12 (1941), 148–153.Google Scholar
12. Verblunsky, S., On the Peano derivatives, Proc. London Math. Soc. 22 (1971), 313–324.Google Scholar
13. Weil, C. E., On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363–376.Google Scholar
14. Zahorski, Z., Sur la première derivée, Trans. Amer. Math. Soc. 69 (1950), 1–54.Google Scholar