This first book on greedy approximation gives a systematic presentation of the fundamental results. It also contains an introduction to two hot topics in numerical mathematics: learning theory and compressed sensing. Nonlinear approximation is becoming increasingly important, especially since two types are frequently employed in applications: adaptive methods are used in PDE solvers, while m-term approximation is used in image/signal/data processing, as well as in the design of neural networks. The fundamental question of nonlinear approximation is how to devise good constructive methods (algorithms) and recent results have established that greedy type algorithms may be the solution. The author has drawn on his own teaching experience to write a book ideally suited to graduate courses. The reader does not require a broad background to understand the material. Important open problems are included to give students and professionals alike ideas for further research.
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analog approximation theory approximation with regard Assume assumption bases best m-term approximation coefﬁcients compact completes the proof condition consider constant convex Corollary cubature cubature formulas deﬁne denote DeVore dictionary discuss element f entropy numbers error exists FejÚr kernel ﬁnd ﬁnite ﬁrst following theorem formulate function f Gm(f Greedy Algorithm greedy approximation greedy basis Haar basis Hilbert space implies Kashin Konyagin and Temlyakov l2 norm Lebesgue measure Lemma linear linear subspace lower bound matrix measure modulus of smoothness norm notation obtain optimal orthonormal basis parameter problem Proof Let proof of Theorem Pure Greedy Algorithm quasi-greedy basis random variable rate of convergence satisﬁes satisfying Schauder basis Section smooth Banach space smoothness p(u space with modulus step subspace Temlyakov 2007 Thenfor trigonometric polynomials uniformly smooth Banach vector WCGA Weak Greedy Algorithm WRGA