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- W2950206413 abstract "Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $mathscr{H}_{E}$ we characterize the Friedrichs extension of the $mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=left(V,Eright)$, $V$ for vertices, and emph{E} for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair $left(mathscr{H}_{E},Deltaright)$ where $mathscr{H}_{E}$ in an energy Hilbert space, and $Deltaleft(=Delta_{c}right)$ is the $c$-Graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $mathscr{H}_{E}$ consisting of dipoles. Now $Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}left(Vright)$ and $mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}left(Vright)$-operator, but generally not as an $mathscr{H}_{E}$ operator. Hence as an $mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}left(Vright)$." @default.
- W2950206413 created "2019-06-27" @default.
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- W2950206413 date "2014-04-04" @default.
- W2950206413 modified "2023-09-24" @default.
- W2950206413 title "Frames and Factorization of Graph Laplacians" @default.
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- W2950206413 doi "https://doi.org/10.48550/arxiv.1404.1424" @default.
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