Geodesic
Operators Whose Resolvents Have Convolution Representations and Their Spectral Analysis
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex analysis, Tome 153 (2018), pp. 94-107

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In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors of second-order differential operators on the interval whose resolvents possess convolution representations. We obtain the convolution representation of resolvents of second-order differential operators on an interval with integral boundary conditions. Then, using the convolution generated by the initial differential operator, we construct the Fourier transform. A connection between the convolution operation in the original functional space and the multiplication operation in the space of Fourier transforms is established. Finally, the problem on the convergence of spectral expansions generated by the original differential operator is studied. Examples of convolutions generated by operators are also presented.
Mots-clés : convolution, spectral decomposition
Keywords: resolvent, boundary-value problem, differential operator, boundary form. 
@article{INTO_2018_153_a6,
     author = {B. E. Kanguzhin},
     title = {Operators {Whose} {Resolvents} {Have} {Convolution} {Representations} and {Their} {Spectral} {Analysis}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {94--107},
     publisher = {mathdoc},
     volume = {153},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_153_a6/}
}
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B. E. Kanguzhin. Operators Whose Resolvents Have Convolution Representations and Their Spectral Analysis. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex analysis, Tome 153 (2018), pp. 94-107. http://geodesic.mathdoc.fr/item/INTO_2018_153_a6/