{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T10:04:23Z","timestamp":1773396263801,"version":"3.50.1"},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2005,9]]},"abstract":"<jats:p> Let \u03c6 : \u211d<jats:sup>d<\/jats:sup> \u2192 \u2102 be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where \u0393 \u2282 \u211d<jats:sup>d<\/jats:sup> is a lattice, \u039b is a finite subset of \u0393, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the \u0393-translates of \u03c6, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [c<jats:sub>Ai-j<\/jats:sub>]<jats:sub>i,j\u2208\u0393<\/jats:sub> and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by \u03c6. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function \u03c6 has accuracy \u03ba, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than \u03ba. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A<jats:sup>-1<\/jats:sup>. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of \u03c6, and hence, every function in the shift-invariant space generated by \u03c6 can be written locally as a linear combination of translates of the homogeneous functions. <\/jats:p>","DOI":"10.1142\/s0219691305000889","type":"journal-article","created":{"date-parts":[[2005,8,22]],"date-time":"2005-08-22T07:56:45Z","timestamp":1124697405000},"page":"321-345","source":"Crossref","is-referenced-by-count":1,"title":["REFINABLE SHIFT INVARIANT SPACES IN \u211d<sup>d<\/sup>"],"prefix":"10.1142","volume":"03","author":[{"given":"CARLOS A.","family":"CABRELLI","sequence":"first","affiliation":[{"name":"Depto. de Matem\u00e1tica FCEyN, Univ. de Buenos Aires, Cdad. Univ., Pab. I, 1428 Capital Federal, Argentina and Conicet, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"SIGRID B.","family":"HEINEKEN","sequence":"additional","affiliation":[{"name":"Depto. de Matem\u00e1tica FCEyN, Univ. de Buenos Aires, Cdad. Univ., Pab. I, 1428 Capital Federal, Argentina and Conicet, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"URSULA M.","family":"MOLTER","sequence":"additional","affiliation":[{"name":"Depto. de Matem\u00e1tica FCEyN, Univ. de Buenos Aires, Cdad. Univ., Pab. I, 1428 Capital Federal, Argentina and Conicet, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","first-page":"171","volume":"17","author":"Aldroubi A.","journal-title":"Appl. Comput. Harmonic Anal."},{"key":"rf2","first-page":"549","volume":"112","author":"Bandt C.","journal-title":"Proc. Amer. Math. Soc."},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1109\/78.984733"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1006\/jath.1997.3211"},{"key":"rf5","unstructured":"C.\u00a0Cabrelli, C.\u00a0Heil and U.\u00a0Molter, Advances in Wavelets, ed. K. S.\u00a0Lau (Springer-Verlag, New York, 1999)\u00a0pp. 121\u2013163."},{"key":"rf6","first-page":"213","volume":"6","author":"Cabrelli C.","journal-title":"J. Fourier Anal. Appl."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0025-3_2"},{"key":"rf8","author":"Cabrelli C. A.","journal-title":"Proc. Amer. Math. Soc."},{"key":"rf9","series-title":"Memoirs of the American Mathematical Society","volume-title":"Self-Similarity and Multiwavelets in Higher Dimensions","volume":"170","author":"Cabrelli C.","year":"2004"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160410705"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1109\/18.119723"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1981.30.30055"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-98-00925-9"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1998.2353"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623790"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-002-0027-0"}],"container-title":["International Journal of Wavelets, Multiresolution and Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219691305000889","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T23:07:43Z","timestamp":1565132863000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0219691305000889"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9]]},"references-count":16,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2005,9]]}},"alternative-id":["10.1142\/S0219691305000889"],"URL":"https:\/\/doi.org\/10.1142\/s0219691305000889","relation":{},"ISSN":["0219-6913","1793-690X"],"issn-type":[{"value":"0219-6913","type":"print"},{"value":"1793-690X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,9]]}}}