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On the Peano Derivatives

  • P. S. Bullen (a1) and S. N. Mukhopadhyay (a1)

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Let f be a real valued function defined in some neighbourhood of a point x. If there are numbers α 1, α 2, … α r-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 α 0 = f 0(x) = f(x). The definition is such that if the mth Peano derivative exists so does the nth for 0 ≦ nm.

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References

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1. Bruckner, A. M., An affirmative answer to a problem of Zahorski and some consequences, Michigan Math. J. 18 (1966), 1526.
2. Bullen, P. S., A criterion for n-convexity, Pacific J. Math. 86 (1971), 8198.
3. Burkill, J. C., The Cesáro-Perron scale of integration, Proc. London Math. Soc. 39 (1935), 543552.
4. Ellis, H. W., Darboux properties and applications to nonabsolutely convergent integrals, Can. J. Math. 3 (1951), 471484.
5. James, R. D., Generalised nth primitives, Trans. Amer. Math. Soc. 76 (1954), 149176.
6. Marcinkiewicz, J. and Zygmund, A., On the differentiability of functions and summability of trigonometric series, Fund. Math. 26 (1936), 143.
7. Mukhopadhyay, S. N., On a certain property of the derivative, Fund. Math. 67 (1970), 279284.
8. Oliver, H. W., The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444456.
9. Saks, S., Theory of the integral (Hafner, Warsaw, 1937).
10. Sargent, W. L. C., On the Cesáro derivates of a function, Proc. London Math. Soc. 40 (1936), 235254.
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), 148153.
12. Verblunsky, S., On the Peano derivatives, Proc. London Math. Soc. 22 (1971), 313324.
13. Weil, C. E., On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363376.
14. Zahorski, Z., Sur la première derivée, Trans. Amer. Math. Soc. 69 (1950), 154.
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On the Peano Derivatives

  • P. S. Bullen (a1) and S. N. Mukhopadhyay (a1)

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