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Denjoy-Bochner almost periodic functions

Part of: Real functions Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

B. K. Pal  and
S. N. Mukhopadhyay
B. K. Pal
Affiliation:
Department of Mathematics The University of BurdwanBurdwan 713104, West Bengal, India
S. N. Mukhopadhyay
Affiliation:
Department of Mathematics The University of BurdwanBurdwan 713104, West Bengal, India
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Abstract

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The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

MSC classification

Secondary: 26A99: None of the above, but in this section 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 2 , October 1984 , pp. 205 - 222
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Amerio, L. and Prouse, G., Almost periodic functions and functional equations (Von Nostrand Reinhold, New York, 1971).CrossRefGoogle Scholar
[2]Besicovitch, A. S., Almost periodic functions (Dover Publications, New York, 1958).Google Scholar
[3]Burkill, H., ‘Almost periodicity and non-absolutely integrable functions’, Proc. London Math. Soc. (2) 53 (1951), 32–42.CrossRefGoogle Scholar
[4]Corduneanu, C., Almost periodic functions (Interscience, New York, 1968).Google Scholar
[5]Pal, B. K., ‘Integration by parts formulae for Denjoy-Bochner and Denjoy Pettis integrals’, to appear.Google Scholar
[6]Solomon, D. W., Denjoy integration in abstract spaces (Memoirs of the Amer. Math. Soc. 85, 1969).Google Scholar
[7]Thomson, B. S., ‘Constructive definition for non-absolutely convergent integrals’, Proc. London Math. Soc. (3) 20 (1970), 699–716.CrossRefGoogle Scholar