Optimal Sequential and Non-Sequential Procedures for Evaluating a Functional.

TL;DR: This book provides a comprehensive treatment of information-based complexity, the branch of computational complexity that deals with the intrinsic difficulty of the approximate solution of problems for which the information is partial, noisy, and priced.

TL;DR: Compressed sensing allows for faithful recovery of images and signals with a significantly smaller number of measurements than the size of the object.

TL;DR: The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability, depending on how to measure the lack of exponential dependence.

TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.

TL;DR: It is hoped that the material which will be presented in the sequel will be convincing enough to prove again that the theory of best approximation in normed linear spaces constitutes both a rigorous theoretical foundation for the existing classical and more recent results in various concrete spaces and a powerfull tool for obtaining new results, solving the new problems which appear.

TL;DR: In this article, an extension of the Golomb-Weinberger method for estimating a linear operator U from inaccurate data is presented, showing that smoothing of the data followed by U is an optimal procedure.