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Mathematics of the USSR-Sbornik, 1981, Volume 40, Issue 4, Pages 567–582
DOI: https://doi.org/10.1070/SM1981v040n04ABEH001854
(Mi sm2740)
 

This article is cited in 5 scientific papers (total in 5 papers)

On integration by parts in Burkill's $SCP$-integral

V. A. Sklyarenko
References:
Abstract: A number of properties of generalized integrals are proved. The main result is
Theorm 3. {\it Suppose that $f$ is $SCP$-integrable on $[a,b]$ with base $B$ and $SCP$-primitive function $\Phi$, and $G(x)=\int^x_ag\,dt$, where $g$ is a continuous function of bounded variation on $[a,b]$. Then the product $f\cdot G$ is $SCP$-integrable on $[a,b]$ with base $B$, and
$$ (SCP,B)\int^b_af\cdot G\,dx=\Phi\cdot G|^b_{x=a}-(D^*)\int^b_a\Phi g\,dx. $$
}
Theorem 3 can be used to prove that if
$$ f(x)=\frac{a_0}2+\sum^\infty_{n=1}(a_n\cos nx+b_n\sin nx) $$
is finite everywhere on $[-\pi,\pi]$, then
$$ a_n=\frac1\pi(SCP,B)\int^\pi_{-\pi}f(x)\cos nx\,dx,\qquad b_n=\frac1\pi\int^\pi_{-\pi}f(x)\sin nx\,dx $$
for $n\geqslant1$.
Bibliography: 10 titles.
Received: 04.06.1979
Bibliographic databases:
UDC: 517.397
MSC: Primary 26A39; Secondary 42A16, 42A20
Language: English
Original paper language: Russian
Citation: V. A. Sklyarenko, “On integration by parts in Burkill's $SCP$-integral”, Math. USSR-Sb., 40:4 (1981), 567–582
Citation in format AMSBIB
\Bibitem{Skl80}
\by V.~A.~Sklyarenko
\paper On integration by parts in Burkill's $SCP$-integral
\jour Math. USSR-Sb.
\yr 1981
\vol 40
\issue 4
\pages 567--582
\mathnet{http://mi.mathnet.ru//eng/sm2740}
\crossref{https://doi.org/10.1070/SM1981v040n04ABEH001854}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=587041}
\zmath{https://zbmath.org/?q=an:0468.26004|0447.26007}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1981MW11700006}
Linking options:
  • https://www.mathnet.ru/eng/sm2740
  • https://doi.org/10.1070/SM1981v040n04ABEH001854
  • https://www.mathnet.ru/eng/sm/v154/i4/p630
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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