{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:58:01Z","timestamp":1760245081952},"reference-count":12,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2007,9]]},"abstract":"<jats:p>Let the operators D and T be the dilation-by-2 and translation-by-1 on [Formula: see text], which are both bilateral shifts of infinite multiplicity. If \u03c8(\u00b7) in [Formula: see text] is a wavelet, then {D<jats:sup>m<\/jats:sup>T<jats:sup>n<\/jats:sup>\u03c8(\u00b7)}<jats:sub>(m,n)\u2208\u2124<jats:sup>2<\/jats:sup><\/jats:sub>is an orthonormal basis for the Hilbert space [Formula: see text] but the reversed set {T<jats:sup>n<\/jats:sup>D<jats:sup>m<\/jats:sup>\u03c8(\u00b7)}<jats:sub>(n,m)\u2208\u2124<jats:sup>2<\/jats:sup><\/jats:sub>is not. In this paper we investigate the role of the reversed functions T<jats:sup>n<\/jats:sup>D<jats:sup>m<\/jats:sup>\u03c8(\u00b7) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of [Formula: see text] into T-reducing subspaces upon which part of the bilateral shift T consists of a countably infinite direct sum of bilateral shifts of multiplicity one, which mirrors a well-known decomposition of the bilateral shift D.<\/jats:p>","DOI":"10.1142\/s0219691307001999","type":"journal-article","created":{"date-parts":[[2007,9,4]],"date-time":"2007-09-04T11:33:08Z","timestamp":1188905588000},"page":"699-707","source":"Crossref","is-referenced-by-count":3,"title":["REVERSED WAVELET FUNCTIONS AND SUBSPACES"],"prefix":"10.1142","volume":"05","author":[{"given":"NHAN","family":"LEVAN","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, University of California in Los Angeles, Los Angeles, CA 90024-1594, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CARLOS S.","family":"KUBRUSLY","sequence":"additional","affiliation":[{"name":"Catholic University of Rio de Janeiro, 22453-900, Rio de Janeiro, RJ, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4754(99)00009-9"},{"key":"rf2","first-page":"vii + 68","volume":"134","author":"Dai X.","journal-title":"Mem. Amer. Math. Soc."},{"key":"rf3","first-page":"102","volume":"208","author":"Halmos P. R.","journal-title":"J. Reine Angew. Math."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1993-1117215-0"},{"key":"rf5","first-page":"41","volume":"14","author":"Kubrusly C. S.","journal-title":"Adv. Math. Sci. Appl."},{"key":"rf6","first-page":"643","volume":"16","author":"Kubrusly C. S.","journal-title":"Adv. Math. Sci. Appl."},{"key":"rf7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4171\/rmi\/22","volume":"2","author":"Lemari\u00e9 P.-G.","journal-title":"Rev. Mat. Iberoamericana"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4754(03)00037-5"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1142\/S0219691304000494"},{"key":"rf10","first-page":"69","volume":"315","author":"Mallat S. G.","journal-title":"Trans. Amer. Math. Soc."},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1965-0174977-5"},{"key":"rf12","volume-title":"Harmonic Analysis of Operators on Hilbert Space","author":"Nagy B. Sz.","year":"1970"}],"container-title":["International Journal of Wavelets, Multiresolution and Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219691307001999","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,17]],"date-time":"2024-02-17T18:51:53Z","timestamp":1708195913000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0219691307001999"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":12,"journal-issue":{"issue":"05","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2007,9]]}},"alternative-id":["10.1142\/S0219691307001999"],"URL":"https:\/\/doi.org\/10.1142\/s0219691307001999","relation":{},"ISSN":["0219-6913","1793-690X"],"issn-type":[{"value":"0219-6913","type":"print"},{"value":"1793-690X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}