03618nam a22004455i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001400153072001700167072002300184072001600207100007800223245009300301250001800394264006900412300003400481336002600515337002600541338003600567347002400603490002900627505071700656520144201373650002802815650002502843650002202868650002802890650002502918650002202943710003402965773002602999776003603025776003603061830002903097856004603126978-3-031-70909-8DE-He21320251030113106.0cr nn 008mamaa240926s2024    sz |    s    |||| 0|eng d  a97830317090987 a10.1007/978-3-031-70909-82doi 4aQA370-380 7aPBKJ2bicssc 7aMAT0070002bisacsh 7aPBKJ2thema1 aAlazard, Thomas.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut10aAnalysis and Partial Differential Equationsh[electronic resource] /cby Thomas Alazard.  a1st ed. 2024. 1aCham :bSpringer Nature Switzerland :bImprint: Springer,c2024.  aXIV, 439 p.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aUniversitext,x2191-66750 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.  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. 0aDifferential equations. 0aFunctional analysis. 0aFourier analysis.14aDifferential Equations.24aFunctional Analysis.24aFourier Analysis.2 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303170908108iPrinted edition:z9783031709104 0aUniversitext,x2191-667540uhttps://doi.org/10.1007/978-3-031-70909-8