Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2020, 8(2), 34-38
DOI: 10.12691/tjant-8-2-3
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A Sobolev Space Inroad to Riemann Integrability

Nassar H. S. Haidar1,

1CRAMS: Center for Research in Applied Mathematics & Statistics, AUL, Beirut, Lebanon

Pub. Date: July 13, 2020

Cite this paper:
Nassar H. S. Haidar. A Sobolev Space Inroad to Riemann Integrability. Turkish Journal of Analysis and Number Theory. 2020; 8(2):34-38. doi: 10.12691/tjant-8-2-3


A conditioned equivalence is proved for a certain weighted Sobolev space to the space of Riemann integrable functions. An equivalence representing a new result that not only asserts the sufficiency (but non-necessity) nature of bounded variation of functions for their Riemann integrability, but also reveals a potential for some novel computational findings.

bounded variation Sobolev space Stieltjes integrals continuity of functions Riemann integrability

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[1]  Royden, H. R, Real analysis, Macmillan, London, 1988.
[2]  Graves, L. M, “Riemann Integration and Taylor's theorem in general analysis,” Transactions of the AMS, 27. 163-177. 1927.
[3]  Gordon, R, “Riemann integrability in Banach spaces,” Rocky Mountain Journal of Mathematics, 21(3). 923-949. 1991.
[4]  Rejouani, R, “On the question of the Riemann integrability of functions with values in a Banach space,” Vestnik Moskov. Univ., Ser. I, Mat. Mech. 26(4). 75-79.1971.
[5]  McShane, E, A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals, Vol.88, AMS, Providence, RI, 1969.
[6]  Diestel, J. and Uhl, J. J, Vector measures, AMS, Providence, RI, 1977.
[7]  Kufner, A., John, O. and Fučík, S, Function spaces, Noordhoff, Leyden, 1977.
[8]  Rudin,W, Principles of mathematical analysis, McGraw-Hill, Auckland, 1964.
[9]  Titchmarsh,E. C, The theory of functions, Oxford University Press, London,1939.
[10]  Haidar, N. H. S, “A geometric note on integration,” Computers & Mathematics With Applications, 59(9). 3130-3136. 2010.
[11]  Hille, E and Philips, R. S, Functional analysis on semigroups, AMS, Providence, RI, 1974.
[12]  Young, L. C, “An equality of the Hölder type, connected with Stieltjes integration,” Acta Mathematica, 67(1). 251-282. 1936.
[13]  Kolmogorov, A. N and Fomin, S. V, Introductory real analysis, Dover Publications, 1975.
[14]  Janke, E., Emde, F. and Lösch, F, Taflen hoherer funktionen, Tubner, Stuttgart, 1960.
[15]  Haidar, N. H. S, “A rational triplic form for the exponential,” Journal of Computational Analysis and Applications, 4(4). 389-404. 2002.