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020 _a9783031709098
_9978-3-031-70909-8
024 7 _a10.1007/978-3-031-70909-8
_2doi
050 4 _aQA370-380
072 7 _aPBKJ
_2bicssc
072 7 _aMAT007000
_2bisacsh
072 7 _aPBKJ
_2thema
100 1 _aAlazard, Thomas.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_978950
245 1 0 _aAnalysis and Partial Differential Equations
_h[electronic resource] /
_cby Thomas Alazard.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer Nature Switzerland :
_bImprint: Springer,
_c2024.
300 _aXIV, 439 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aUniversitext,
_x2191-6675
505 0 _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.
520 _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.
650 0 _aDifferential equations.
650 0 _aFunctional analysis.
650 0 _aFourier analysis.
650 1 4 _aDifferential Equations.
650 2 4 _aFunctional Analysis.
650 2 4 _aFourier Analysis.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031709081
776 0 8 _iPrinted edition:
_z9783031709104
830 0 _aUniversitext,
_x2191-6675
_973095
856 4 0 _uhttps://doi.org/10.1007/978-3-031-70909-8
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c63726
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