Geodesic
On integration by parts in Burkill's $SCP$-integral
Sbornik. Mathematics, Tome 40 (1981) no. 4, pp. 567-582 
V. A. Sklyarenko. On integration by parts in Burkill's $SCP$-integral. Sbornik. Mathematics, Tome 40 (1981) no. 4, pp. 567-582. http://geodesic.mathdoc.fr/item/SM_1981_40_4_a5/
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     author = {V. A. Sklyarenko},
     title = {On integration by parts in {Burkill's} $SCP$-integral},
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     volume = {40},
     number = {4},
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     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_4_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] J. G. Burkill, “Integral and trigonometrical series”, Proc. London Math. Soc., 1:1 (1951), 46–57 | DOI | MR | Zbl

[2] P. S. Bullen, S. N. Mukhopadhyay, “Integration by parts formulae for trigonometric integrals”, Proc. London Math. Soc., 3:1 (1974), 159–173 | DOI | MR

[3] H. Burkill, “A note on trigonometric series”, J. Math. Anal. and Appl., 40:1 (1972), 39–44 | DOI | MR | Zbl

[4] S. Saks, Teoriya integrala, IL, Moskva, 1949

[5] A. V. Efimov, “Lineinye metody priblizheniya nepreryvnykh periodicheskikh funktsii”, Matem. sb., 54(96) (1961), 51–90 | MR | Zbl

[6] W . H. Gage, R. D. James, “A generalized integral”, Proc Roy. Soc. Canada, HI, 40 (1946), 25–36 | MR

[7] G. Cross, “The relation between two symmetric integrals”, Proc Amer. Math. Soc., 14:1 (1963), 185–190 | DOI | MR | Zbl

[8] V. A. Skvortsov, “Po povodu opredelenii $P^2$- i $SCP$-integralov”, Vestnik MGU, matem., mekhan., 1966, no. 6, 12–19 | Zbl

[9] R. O. James, “A generalized integral. II”, Canad. J. Math., 3 (1950), 297–306 | MR

[10] V. A. Sklyarenko, “Nekotorye svoistva $P^2$-primitivnoi”, Matem. zametki, 12:6 (1972), 693–700 | Zbl