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Description of  Analysis and Partial Differential Equations (Universitext)

303170908X rar This textbook provides a modern introduction to advanced concepts and methods of mathematical analysis. The first three parts of the book cover functional analysis, harmonic analysis, and microlocal analysis. Each chapter is designed to provide readers with a solid understanding of fundamental concepts while guiding them through detailed proofs of significant theorems. These include the universal approximation property for artificial neural networks, Brouwer's domain invariance theorem, Nash's implicit function theorem, Caldern's reconstruction formula and wavelets, Wiener's Tauberian theorem, Hrmander's theorem of propagation of singularities, and proofs of many inequalities centered around the works of Hardy, Littlewood, and Sobolev. The final part of the book offers an overview of the analysis of partial differential equations. This vast subject is approached through a selection of major theorems such as the solution to Caldern's problem, De Giorgi's regularity theorem for elliptic equations, and the proof of a StrichartzBourgain estimate. Several renowned results are included in the numerous examples. Read more