{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T16:09:41Z","timestamp":1782230981654,"version":"3.54.5"},"reference-count":19,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2014,3]]},"abstract":"<jats:p> For a, b &gt; 0 and g \u2208 L<jats:sup>2<\/jats:sup>(\u211d), write \ud835\udca2(g, a, b) for the Gabor system: [Formula: see text] Let S be an a\u2124-periodic measurable subset of \u211d with positive measure. It is well-known that the projection \ud835\udca2(g\u03c7<jats:sub>S<\/jats:sub>, a, b) of a frame \ud835\udca2(g, a, b) in L<jats:sup>2<\/jats:sup>(\u211d) onto L<jats:sup>2<\/jats:sup>(S) is a frame for L<jats:sup>2<\/jats:sup>(S). However, when ab &gt; 1 and S \u2260 \u211d, \ud835\udca2(g, a, b) cannot be a frame in L<jats:sup>2<\/jats:sup>(\u211d) for any g \u2208 L<jats:sup>2<\/jats:sup>(\u211d), while it is possible that there exists some g such that \ud835\udca2(g, a, b) is a frame for L<jats:sup>2<\/jats:sup>(S). So the projections of Gabor frames in L<jats:sup>2<\/jats:sup>(\u211d) onto L<jats:sup>2<\/jats:sup>(S) cannot cover all Gabor frames in L<jats:sup>2<\/jats:sup>(S). This paper considers Gabor systems in L<jats:sup>2<\/jats:sup>(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L<jats:sup>2<\/jats:sup>(S) of the form \ud835\udca2(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II). <\/jats:p>","DOI":"10.1142\/s0219691314500131","type":"journal-article","created":{"date-parts":[[2013,12,16]],"date-time":"2013-12-16T02:43:06Z","timestamp":1387161786000},"page":"1450013","source":"Crossref","is-referenced-by-count":12,"title":["RATIONAL TIME-FREQUENCY GABOR FRAMES ASSOCIATED WITH PERIODIC SUBSETS OF THE REAL LINE"],"prefix":"10.1142","volume":"12","author":[{"given":"JEAN-PIERRE","family":"GABARDO","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"YUN-ZHANG","family":"LI","sequence":"additional","affiliation":[{"name":"College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. 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