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

V. A. Sklyarenko

Sbornik. Mathematics, Tome 40 (1981) no. 4, pp. 567-582

Résumé

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.