{"product_id":"gabor-analysis-and-algorithms-theory-and-applications-von-hans-g-feichtinger-thomas-strohmer-hrsg","title":"Gabor Analysis and Algorithms","description":"In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is \"reasonably\" smooth and localized. One is thus led naturally to considering a higher time-frequency density.\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9781461273820\"\u003e\u003ch3\u003eTheory and Applications\u003c\/h3\u003e\u003c\/div\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9780817639594\"\u003e\u003ch3\u003eTheory and Applications\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9781461273820","offer_id":39415715299421,"sku":"9781461273820","price":171.19,"currency_code":"EUR","in_stock":true},{"title":"Hardcover - 9780817639594","offer_id":576146669574,"sku":"9780817639594","price":171.19,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/7bc0b531-c8b5-4fb4-8574-beeb26915606.jpg?v=1771997008","url":"https:\/\/shop.autorenwelt.de\/en\/products\/gabor-analysis-and-algorithms-theory-and-applications-von-hans-g-feichtinger-thomas-strohmer-hrsg","provider":"Autorenwelt Shop","version":"1.0","type":"link"}