On Finite Part Integrals and Hadamard-Type Fractional Derivatives

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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Riemann derivatives and general integrals

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013174 ◽

1987 ◽

Vol 35 (2) ◽

pp. 187-211 ◽

Cited By ~ 2

Author(s):

S. De Sarkar ◽

A. G. Das

Keyword(s):

Continuous Functions ◽

Divided Differences ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Fundamental Properties ◽

Type Integral ◽

Definition Of

Sargent and later Bullen and Mukhopadhyay obtained a definition of absolutely continuous functions, functions, that is related to kth Peano derivatives. The generalised notions of ACkG*, [ACkG*], ACkG* above, etcetera functions led Bullen and Mukhopadhyay to define certain general integrals of the kth order.The present work is concerned with a further simplification of the definitions of such functions by the use of divided differences but still retaining similar fundamental properties. These concepts lead to the introduction of Denjoy and Ridder type integrals which are shown to be equivalent to a Perron type integral that corresponds to kth Riemann* derivatives. All three of these integrals are shown to be equivalent to the three integrals of Bullen and Mukhopadhyay.

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The spectral theorem for well-bounded operators

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700031827 ◽

1993 ◽

Vol 54 (3) ◽

pp. 334-351 ◽

Cited By ~ 1

Author(s):

Ian Doust ◽

Qiu Bozhou

Keyword(s):

Functional Calculus ◽

Continuous Functions ◽

Weak Compactness ◽

Compact Interval ◽

Spectral Theorem ◽

Type B ◽

Bounded Operators ◽

Absolutely Continuous Functions ◽

Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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