{"product_id":"9781461273820","title":"Applied and Numerical Harmonic Analysis: Theory and Applications","description":"\u003ch1\u003eApplied and Numerical Harmonic Analysis: Theory and Applications\u003c\/h1\u003e \u003ch2\u003eFeichtinger, Hans G.; Strohmer, Thomas\u003c\/h2\u003e \u003cp\u003eIn 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.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Birkhäuser\u003c\/p\u003e \u003cp\u003ePublication Date: 2012-10-11\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9781461273820\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-1-4612-2016-9\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 496\u003c\/p\u003e ","brand":"Birkhäuser Boston","offers":[{"title":"Default Title","offer_id":45417049358476,"sku":"9781461273820","price":161.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9781461273820.jpg?v=1775740848","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9781461273820","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}