A sampling theory for infinite weighted graphs
We prove two sampling theorems for infinite (countable discrete) weighted graphs \(G\); one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum \(X\) containing \(G\), and there are Hilbert spaces of functions on...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2011-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | http://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3115.pdf |
| _version_ | 1818113262872428544 |
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| author | Palle E. T. Jorgensen |
| author_facet | Palle E. T. Jorgensen |
| author_sort | Palle E. T. Jorgensen |
| collection | DOAJ |
| description | We prove two sampling theorems for infinite (countable discrete) weighted graphs \(G\); one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum \(X\) containing \(G\), and there are Hilbert spaces of functions on \(X\) that allow interpolation by sampling values of the functions restricted only on the vertices in \(G\). We sample functions on \(X\) from their discrete values picked in the vertex-subset \(G\). We prove two theorems that allow for such realistic ambient spaces \(X\) for a fixed graph \(G\), and for interpolation kernels in function Hilbert spaces on \(X\), sampling only from points in the subset of vertices in \(G\). A continuum is often not apparent at the outset from the given graph \(G\). We will solve this problem with the use of ideas from stochastic integration. |
| first_indexed | 2024-12-11T03:32:03Z |
| format | Article |
| id | doaj.art-e109ddb0c4f74db58394f471775f9335 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-11T03:32:03Z |
| publishDate | 2011-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-e109ddb0c4f74db58394f471775f93352022-12-22T01:22:22ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742011-01-01312209236http://dx.doi.org/10.7494/OpMath.2011.31.2.2093115A sampling theory for infinite weighted graphsPalle E. T. Jorgensen0The University of Iowa, Department of Mathematics, Iowa City, IA 52242-1419, USAWe prove two sampling theorems for infinite (countable discrete) weighted graphs \(G\); one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum \(X\) containing \(G\), and there are Hilbert spaces of functions on \(X\) that allow interpolation by sampling values of the functions restricted only on the vertices in \(G\). We sample functions on \(X\) from their discrete values picked in the vertex-subset \(G\). We prove two theorems that allow for such realistic ambient spaces \(X\) for a fixed graph \(G\), and for interpolation kernels in function Hilbert spaces on \(X\), sampling only from points in the subset of vertices in \(G\). A continuum is often not apparent at the outset from the given graph \(G\). We will solve this problem with the use of ideas from stochastic integration.http://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3115.pdfweighted graphHilbert spaceLaplace operatorsamplingShannonwhite noiseWiener transforminterpolation |
| spellingShingle | Palle E. T. Jorgensen A sampling theory for infinite weighted graphs Opuscula Mathematica weighted graph Hilbert space Laplace operator sampling Shannon white noise Wiener transform interpolation |
| title | A sampling theory for infinite weighted graphs |
| title_full | A sampling theory for infinite weighted graphs |
| title_fullStr | A sampling theory for infinite weighted graphs |
| title_full_unstemmed | A sampling theory for infinite weighted graphs |
| title_short | A sampling theory for infinite weighted graphs |
| title_sort | sampling theory for infinite weighted graphs |
| topic | weighted graph Hilbert space Laplace operator sampling Shannon white noise Wiener transform interpolation |
| url | http://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3115.pdf |
| work_keys_str_mv | AT palleetjorgensen asamplingtheoryforinfiniteweightedgraphs AT palleetjorgensen samplingtheoryforinfiniteweightedgraphs |