Representation of arbitrary functions by certain special series

Let $M(x,t)$ be continuous for $0\leqslant t\leqslant x$, $0\leqslant x\leqslant1$ and let $g(x)$ be of bounded variation in $[0,1]$. Further, let $M(x,t,\lambda)=\sum_{k=0}^\infty\lambda^kM_k(x,t)$, where $M_1(x,t)=M(x,t)$, and $M_k(x,t)=\int_t^xM_{k-1}(x,\tau)M(\tau,t)\,d\tau$ for $k>1$. The paper studies the problem of the representation of a certain class of functions by series whose partial sums are given by
$$ P_n(x,f)=\frac1{2\pi i}\int_{C_n}\frac{\varphi(x,\lambda)}{L(\lambda)}\int_0^1\int_0^xM(x,t,\lambda)f(t)\,dt\,dg(x)\qquad(n=1,2,\dots), $$
where $f(x)$ is the expanded function, $\varphi(x,\lambda)=\psi(x)+\lambda\int_0^xM(x,t,\lambda)\psi(t)\,dt$, $\psi(x)\in C[0,1]$, $L(\lambda)=\int_0^1\varphi(x,\lambda)\,dg(x)$ and is $\{C_n\}^\infty_{n=1}$ a sequence of circles in the $\lambda$-plane with common center at zero and radii $r_n\uparrow+\infty$. This problem contains, in particular, the problem of expansion in the eigenfunctions of an ordinary differential equation in $[0,1]$ with certain irregular decomposing boundary conditions.
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