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Anal."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We prove that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency $e^{int}$ ensures that the anti-holomorphic part is much smaller than the holomorphic part $ \\| P_{-}(f)\\|_{L^2} \\ll \\|P_{+}(f)\\|_{L^2}.$ This enables us to use techniques from complex analysis, in particular the unwinding series. We study its stability and convergence properties and show that the unwinding series can provide a high-resolution, noise-robust time-frequency representation.<\/jats:p>","DOI":"10.1137\/16m1081087","type":"journal-article","created":{"date-parts":[[2017,11,30]],"date-time":"2017-11-30T12:12:50Z","timestamp":1512043970000},"page":"4838-4864","source":"Crossref","is-referenced-by-count":29,"title":["Carrier Frequencies, Holomorphy, and Unwinding"],"prefix":"10.1137","volume":"49","author":[{"given":"Ronald R.","family":"Coifman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Steinerberger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hau-tieng","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,11,30]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.116.061102"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.2012.6288767"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/78.382394"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2011.09.011"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1111\/rssb.12039"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2015.01.003"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2016.03.001"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970104"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.08.002"},{"key":"atypb12","first-page":"527","volume-title":"Wavelets Medicine Biol.","author":"Daubechies I.","year":"1996"},{"key":"atypb13","first-page":"374","author":"Daubechies I.","year":"2016","journal-title":"Philos. 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