LEADER 03855nam 22005895 450 001 9910890173803321 005 20240926130225.0 010 $a3-031-70909-8 024 7 $a10.1007/978-3-031-70909-8 035 $a(MiAaPQ)EBC31691830 035 $a(Au-PeEL)EBL31691830 035 $a(CKB)36200513800041 035 $a(MiAaPQ)EBC31691112 035 $a(Au-PeEL)EBL31691112 035 $a(DE-He213)978-3-031-70909-8 035 $a(EXLCZ)9936200513800041 100 $a20240926d2024 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAnalysis and Partial Differential Equations /$fby Thomas Alazard 205 $a1st ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2024. 215 $a1 online resource (439 pages) 225 1 $aUniversitext,$x2191-6675 311 $a3-031-70908-X 327 $aPart I Functional Analysis -- 1 Topological Vector Spaces -- 2 Fixed Point Theorems -- 3 Hilbertian Analysis, Duality and Convexity -- Part II Harmonic Analysis -- 4 Fourier Series -- 5 Fourier Transform -- 6 Convolution -- 7 Sobolev Spaces -- 8 Harmonic Functions -- Part III Microlocal Analysis -- 9 Pseudo-Differential Operators -- 10 Symbolic Calculus -- 11 Hyperbolic Equations -- 12 Microlocal Singularities -- Part IV Analysis of Partial Differential Equations -- 13 The Calderón Problem -- 14 De Giorgi?s Theorem -- 15 Schauder?s Theorem -- 16 Dispersive Estimates -- Part V Recap and Solutions to the Exercises -- 17 Recap on General Topology -- 18 Inequalities in Lebesgue Spaces -- 19 Solutions. 330 $aThis textbook provides a modern introduction to advanced concepts and methods of mathematical analysis. The first three parts of the book cover functional analysis, harmonic analysis, and microlocal analysis. Each chapter is designed to provide readers with a solid understanding of fundamental concepts while guiding them through detailed proofs of significant theorems. These include the universal approximation property for artificial neural networks, Brouwer's domain invariance theorem, Nash's implicit function theorem, Calderón's reconstruction formula and wavelets, Wiener's Tauberian theorem, Hörmander's theorem of propagation of singularities, and proofs of many inequalities centered around the works of Hardy, Littlewood, and Sobolev. The final part of the book offers an overview of the analysis of partial differential equations. This vast subject is approached through a selection of major theorems such as the solution to Calderón's problem, De Giorgi's regularity theorem for elliptic equations, and the proof of a Strichartz?Bourgain estimate. Several renowned results are included in the numerous examples. Based on courses given successively at the École Normale Supérieure in France (ENS Paris and ENS Paris-Saclay) and at Tsinghua University, the book is ideally suited for graduate courses in analysis and PDE. The prerequisites in topology and real analysis are conveniently recalled in the appendix. 410 0$aUniversitext,$x2191-6675 606 $aDifferential equations 606 $aFunctional analysis 606 $aFourier analysis 606 $aDifferential Equations 606 $aFunctional Analysis 606 $aFourier Analysis 615 0$aDifferential equations. 615 0$aFunctional analysis. 615 0$aFourier analysis. 615 14$aDifferential Equations. 615 24$aFunctional Analysis. 615 24$aFourier Analysis. 676 $a515.35 700 $aAlazard$b Thomas$0845497 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910890173803321 996 $aAnalysis and Partial Differential Equations$94249052 997 $aUNINA