Propriétés des matrices ``bien localisées'' près de leur diagonale et quelques applications. (Properties of matrices ``well localized'' near the diagonal and some applications)
Let (T,d) be a discrete metric space with the property that for every \(\epsilon >0\sum_{t\in T}\exp [-\epsilon d(s,t)]\) is bounded as s varies over T. For any positive real number \(\gamma\), let \({\mathcal E}_{\gamma}\) be the set of all matrices \((A_{s,t})_{s,t\in T}\) such that for all \(\gamma '<\gamma| A_{s,t}| \exp [\gamma 'd(s,t)]\) is bounded as s and t vary over T, i.e., the entries in A decay exponentially away from the diagonal. Such matrices define bounded operators on \(l_ 2(T).\) The author proves that \({\mathcal E}_{\gamma}\) is an algebra, and that if \(A\in {\mathcal E}_{\gamma}\) is invertible as an operator on \(l_ 2(T)\), then \(A^{-1}\in {\mathcal E}_{\gamma '}\) for some \(\gamma '\). He also proves a ``window'' lemma that shows that changes in the entries in one part of a matrix in \({\mathcal E}_{\gamma}\) do not affect the entries in \(A^{-1}\) that are located far away from the changed entries. More precisely, if \(\Omega\subset T\) and A and B are two invertible matrices in \({\mathcal E}_{\gamma}\) such that \(| A_{s,t}-B_{s,t}| \exp \{\gamma [d(s,\Omega)+d(t,\Omega)]\}\) is bounded as s and t vary over T, then for some positive \(\gamma '\), \(| A^{-1}_{s,t}-B^{- 1}_{s,t}| \exp \{\gamma '[d(s,\Omega)+d(t,\Omega)]\}\) is also bounded as s and t vary over T. Now assume (T,d) is a discrete metric space with the property that for some N \(\sum_{t\in D}| 1+d(s,t)|^{-N}\) is bounded as s varies over T. For any \(\alpha >N\), let \(Q_{\alpha}\) be the set of all matrices \((A_{s,t})_{s,t\in T}\) such that \(| A_{s,t}[1+d(s,t)]^{\alpha}|\) is bounded as s and t vary over T. The author proves the same type of results for \(Q_{\alpha}\) as for \({\mathcal E}_{\gamma}.\) Two applications concerning wavelets are given.
scientific article; zbMATH DE number 4189914