@article{Kovalev_2017, title={Point interactions on the line and Riesz bases of δ -functions}, volume={69}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/1808}, abstractNote={We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$.
Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf}
\bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime
ot = y\prime \prime \bigr\}$. By using this
relationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ \delta (x y_j), y_j \in Y \bigr\} $ and their derivatives
$\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ but, conversely, form these bases in the case where $d > 0$. We also present the description of the Friedrichs and
Krein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the description
of all nonnegative selfadjoint extensions of the operator $A\prime$ are proposed.}, number={12}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={KovalevYu. G.}, year={2017}, month={Dec.}, pages={1615-1624} }