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Generalised Hölderian functions

Published online by Cambridge University Press:  17 April 2009

S. De Sarkar  and
S. Panda
S. De Sarkar
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani – 741 235, Dist. Nadia, West Bengal, India
S. Panda
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani – 741 235, Dist. Nadia, West Bengal, India
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Abstract

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The concept of kth Hölderian functions on an interval [a, b] which generalises the notion of Hölderian (Lipschitzian) functions of positive order on [a, b] is introduced. The relationship of such functions to functions of bounded kth variation and absolutely kth continuous functions is examined. Properties induced by higher order derivatives in this new class of functions are investigated.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 1 , August 1989 , pp. 147 - 155
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bullen, P.S. and Mukhopadhya, S.N., ‘Relations between some general nth order derivatives’, Fund. Math. 80 (1974), 257–276.CrossRefGoogle Scholar
[2]Das, A.G. and Lahiri, B.K., ‘On absolutely kth continuous functions’, Fund. Math. 105 (1980), 159–169.CrossRefGoogle Scholar
[3]Das, U. and Das, A.G., ‘Convergence in kth variation and RSk integrals’, J. Austral. Math. Soc. Ser. A 31 (1981), 163–174.CrossRefGoogle Scholar
[4]De Sarkar, S. and Das, A.G., ‘On functions of bounded kth variation’, J. Indian Inst. Sci. 64(B) (1983), 299–309.Google Scholar
[5]De Sarkar, S. and Das, A.G., ‘A convergence theorem in essential kth variation’, Soochow J. Math. 11 (1985), 51–57.Google Scholar
[6]De Sarkar, S. and Das, A.G., ‘On functions of bounded essential kth variation’, Bull. Calcutta Math. Soc. 78 (1986), 249–258.Google Scholar
[7]De Sarkar, S. and Das, A.G., ‘Riemann derivatives and general integrals’, Bull. Austral. Math. Soc. 35 (1987), 187–211.CrossRefGoogle Scholar
[8]De Sarkar, S., Das, A.G. and Lahiri, B.K., ‘Approximate Riemann* derivative and approximate pk –, Dk – integrals’, Indian J. Math. 27 (1985), 1–32.Google Scholar
[9]Hardy, G.H., ‘Weierstrass's non-differeaitiable function’, Trans. Amer. Math. Soc. 17 (1916), 301–325.Google Scholar
[10]Milne-Thomson, L.M., The calculus of finite differences (Chelsea, New York, 1981).Google Scholar
[11]Oliver, H.V., ‘The exact Peano derivative’, Trans. Amer. Math. Soc. 76 (1954), 444–456.CrossRefGoogle Scholar
[12]Russell, A.M., ‘Functions of bounded kth variation’, Proc. London Math. Soc. (3)26 (1973), 547–563.CrossRefGoogle Scholar
[13]Russell, A.M., ‘An integral representation for a generahised variation of a function’, Bull. Austral. Math. Soc. 11 (1974), 225–229.CrossRefGoogle Scholar
[14]Russell, A.M., ‘Further results on an integral representation of functions of generalised variation’, Bull. Austral. Math. Soc. 18 (1978), 407–420.CrossRefGoogle Scholar