Document Zbl 07905063

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Fourier Anal. Appl.</a> <a href="/?q=in%3A515153" title="Articles in this Issue">30, No. 4, Paper No. 43, 24 p. (2024)</a>. </div> <div class="abstract">Spectral factorization is a process by which a positive definite matrix function $S$ on the unit circle $\mathbb{T}$ is expressed in the form $S(t)=S_+(t)S_+^*(t)$, where $S_+$ can be extended analytically inside $\mathbb{T}$ and $S_+^*$ is its Hermitian conjugate. The main result of the paper says that if $S$ is an $r\times r$ polynomial matrix function which is positive definite on $\mathbb{T}$ and $S(z)=M(z)M^*(z)$ is its lower-upper factorization with \[ M(z)= \left(\begin{array}{ccccc} f_1^+(z) & 0 & \dots & 0 & 0 \\ \xi_{21}(z) & f_2^*(z) & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \xi_{r-1,1} (z) & \xi_{r-1,2}(z) & \dots & \xi_{r-1}^+(z) & 0 \\ \xi_{r1}(z) & \xi_{r2}(z) & \dots & \xi_{r,r-1}(z) & f_r^+(z) \end{array}\right). \] If the entries of the matrix function $M$ and the poles of the functions $\xi_{ij}$ inside $\mathbb{T}$ are known exactly, then the spectral factorization of $S$ can also be found exactly. Some applications of the obtained results in signal processing are discussed.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10736">Oleksiy Karlovych (Lisboa)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7905063">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A68" title="MSC2020">47A68</a> </td> <td class="space"> Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65T60" title="MSC2020">65T60</a> </td> <td class="space"> Numerical methods for wavelets </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A83" title="MSC2020">15A83</a> </td> <td class="space"> Matrix completion problems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amatrix+spectral+factorization">matrix spectral factorization</a>; <a href="/?q=ut%3Aparaunitary+matrix+functions">paraunitary matrix functions</a>; <a href="/?q=ut%3Amatrix+completion+problem">matrix completion problem</a></div> <div class="software"> <h3>Software:</h3><a href="/software/46540">ExactMPF</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 07905063" data-ciurl="/ci/07905063" data-biburl="/bibtex/07905063.bib" data-amsurl="/amsrefs/07905063.bib" data-xmlurl="/xml/07905063.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07905063.pdf" title="Zbl 07905063 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00041-024-10100-3" aria-label="DOI for “On the exact spectral factorization of rational matrix functions with applications to paraunitary filter banks”" title="10.1007/s00041-024-10100-3">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2312.03518"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adukov, VM; Adukova, NV; Mishuris, G., An explicit Wiener-Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package, Proc. 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