LEADER 05073nam 22006255 450 001 9910495182903321 005 20251107172139.0 010 $a3-030-81265-0 024 7 $a10.1007/978-3-030-81265-2 035 $a(CKB)4100000012007902 035 $a(MiAaPQ)EBC6708573 035 $a(Au-PeEL)EBL6708573 035 $a(PPN)257357440 035 $a(OCoLC)1264720632 035 $a(DE-He213)978-3-030-81265-2 035 $a(EXLCZ)994100000012007902 100 $a20210821d2021 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aTheory of Distributions /$fby Svetlin G. Georgiev 205 $a2nd ed. 2021. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2021. 215 $a1 online resource (270 pages) 225 1 $aMathematics and Statistics Series 311 08$a3-030-81264-2 327 $aIntro -- Preface to the Second Edition -- Preface to the First Edition -- Contents -- 1 Introduction -- 1.1 The Spaces C0? and S -- 1.2 The Lp Spaces -- 1.2.1 Definition -- 1.2.2 The Inequalities of Hölder and Minkowski -- 1.2.3 Some Properties -- 1.2.4 The Riesz-Fischer Theorem -- 1.2.5 Separability -- 1.2.6 Duality -- 1.2.7 General Lp Spaces -- 1.3 The Convolution of Locally Integrable Functions -- 1.4 Cones in Rn -- 1.5 Advanced Practical Problems -- 1.6 Notes and References -- 2 Generalities on Distributions -- 2.1 Definitions -- 2.2 Order of a Distribution -- 2.3 Change of Variables -- 2.4 Sequences and Series -- 2.5 Support -- 2.6 Singular Support -- 2.7 Measures -- 2.8 Multiplying Distributions by C? Functions -- 2.9 Advanced Practical Problems -- 2.10 Notes and References -- 3 Differentiation -- 3.1 Derivatives -- 3.2 The Local Structure of Distributions -- 3.3 The Primitive of a Distribution -- 3.4 Simple and Double Layers on Surfaces -- 3.5 Advanced Practical Problems -- 3.6 Notes and References -- 4 Homogeneous Distributions -- 4.1 Definition -- 4.2 Properties -- 4.3 Advanced Practical Problems -- 4.4 Notes and References -- 5 The Direct Product of Distributions -- 5.1 Definition -- 5.2 Properties -- 5.3 Advanced Practical Problems -- 5.4 Notes and References -- 6 Convolutions -- 6.1 Definition -- 6.2 Properties -- 6.3 Existence -- 6.4 The Convolution Algebras D'(?+) and D'(?) -- 6.5 Regularization of Distributions -- 6.6 Fractional Differentiation and Integration -- 6.7 Advanced Practical Problems -- 6.8 Notes and References -- 7 Tempered Distributions -- 7.1 Definition -- 7.2 Direct Product -- 7.3 Convolution -- 7.4 Advanced Practical Problems -- 7.5 Notes and References -- 8 Integral Transforms -- 8.1 The Fourier Transform in S(Rn) -- 8.2 The Fourier Transform in S'(Rn) -- 8.3 Properties of the Fourier Transform in S'(Rn). 327 $a8.4 The Fourier Transform of Distributions with Compact Support -- 8.5 The Fourier Transform of Convolutions -- 8.6 The Laplace Transform -- 8.6.1 Definition -- 8.6.2 Properties -- 8.7 Advanced Practical Problems -- 8.8 Notes and References -- 9 Fundamental Solutions -- 9.1 Definition and Properties -- 9.2 Fundamental Solutions of Ordinary Differential Operators -- 9.3 Fundamental Solution of the Heat Operator -- 9.4 Fundamental Solution of the Laplace Operator -- 9.5 Advanced Practical Problems -- 9.6 Notes and References -- 10 Sobolev Spaces -- 10.1 Definitions -- 10.2 Elementary Properties -- 10.3 Approximation by Smooth Functions -- 10.4 Extensions -- 10.5 Traces -- 10.6 Sobolev Inequalities -- 10.7 The Space H-s -- 10.8 Advanced Practical Problems -- 10.9 Notes and References -- References -- Index. 330 $aThis book explains many fundamental ideas on the theory of distributions. The theory of partial differential equations is one of the synthetic branches of analysis that combines ideas and methods from different fields of mathematics, ranging from functional analysis and harmonic analysis to differential geometry and topology. This presents specific difficulties to those studying this field. This second edition, which consists of 10 chapters, is suitable for upper undergraduate/graduate students and mathematicians seeking an accessible introduction to some aspects of the theory of distributions. It can also be used for one-semester course. 410 0$aMathematics and Statistics Series 606 $aFunctional analysis 606 $aMathematical analysis 606 $aFourier analysis 606 $aFunctional Analysis 606 $aAnalysis 606 $aAnalysis 606 $aFourier Analysis 615 0$aFunctional analysis. 615 0$aMathematical analysis. 615 0$aFourier analysis. 615 14$aFunctional Analysis. 615 24$aAnalysis. 615 24$aAnalysis. 615 24$aFourier Analysis. 676 $a515.782 700 $aGeorgiev$b Svetlin$01218802 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910495182903321 996 $aTheory of distributions$92843007 997 $aUNINA