The solution of the generalized convolution equation

The following equation is considered:
$$ \sum_{k=0}^p\int_a^b f^{(k)}(x+t)\,d\sigma_k(t)=0, $$
where the functions $\sigma_k(t)$ are of bounded variation on $[a,b]$, the function $\sigma_p(t)$ having jumps at the end points. A series of elementary solutions is associated with the solution by a certain rule (RZhMat., 1966, 4B106). The convergence of this series is investigated. The results of Sedletskii (RZhMat., 1971, 6B114) for the case $p=0$ are used.
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