{ "item_title" : "Gabor Analysis and Algorithms", "item_author" : [" Hans G. Feichtinger", "Thomas Strohmer "], "item_description" : "In his paper Theory of CommunicationGab46], 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 functionsBBGK71, Per71] (it even has one function to spareBGZ75] . . . ) 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.", "item_img_path" : "https://covers3.booksamillion.com/covers/bam/0/81/763/959/0817639594_b.jpg", "price_data" : { "retail_price" : "179.99", "online_price" : "179.99", "our_price" : "179.99", "club_price" : "179.99", "savings_pct" : "0", "savings_amt" : "0.00", "club_savings_pct" : "0", "club_savings_amt" : "0.00", "discount_pct" : "10", "store_price" : "" } }

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