03943nam a22005295i 4500 978-3-031-70909-8 DE-He213 20251030113106.0 cr nn 008mamaa 240926s2024 sz | s |||| 0|eng d 9783031709098 978-3-031-70909-8 10.1007/978-3-031-70909-8 doi QA370-380 PBKJ bicssc MAT007000 bisacsh PBKJ thema Alazard, Thomas. author. aut http://id.loc.gov/vocabulary/relators/aut 78950 Analysis and Partial Differential Equations [electronic resource] / by Thomas Alazard. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Springer, 2024. XIV, 439 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Universitext, 2191-6675 Part 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. 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, 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. Differential equations. Functional analysis. Fourier analysis. Differential Equations. Functional Analysis. Fourier Analysis. SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031709081 Printed edition: 9783031709104 Universitext, 2191-6675 73095 https://doi.org/10.1007/978-3-031-70909-8 ZDB-2-SMA ZDB-2-SXMS 63726 63726