Geodesic
An analog of the Jordan--Dirichlet theorem for an operator with involution on a graph
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 17-24

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In this paper, we examine the convergence of eigenfunction expansions of a functional-differential operator with involution $\nu(x)=1-x$, which is defined on a geometric graph consisting of two edges, one of which is a loop. Sufficient conditions are obtained for the uniform convergence of the Fourier series in the eigenfunctions of the operator (an analog of the Jordan–Dirichlet theorem).
Keywords: functional-differential operator, involution, geometric graph, Fourier series. 
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     author = {E. I. Biryukova},
     title = {An analog of the {Jordan--Dirichlet} theorem for an operator with involution on a graph},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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E. I. Biryukova. An analog of the Jordan--Dirichlet theorem for an operator with involution on a graph. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 17-24. http://geodesic.mathdoc.fr/item/INTO_2021_193_a2/