Introduction to the Theory of DistributionsThe theory of distributions is an extension of classical analysis, an area of particular importance in the field of linear partial differential equations. Underlying it is the theory of topological vector spaces, but it is possible to give a systematic presentation without a knowledge of this. The material in this book, based on graduate lectures given over a number of years requires few prerequisites but the treatment is rigorous throughout. From the outset, the theory is developed in several variables. It is taken as far as such important topics as Schwartz kernels, the Paley-Wiener-Schwartz theorem and Sobolev spaces. In this second edition, the notion of the wavefront set of a distribution is introduced. It allows many operations on distributions to be extended to larger classes and gives much more precise understanding of the nature of the singularities of a distribution. This is done in an elementary fashion without using any involved theories. This account will be useful to graduate students and research workers who are interested in the applications of analysis in mathematics and mathematical physics. |
Contents
III | 4 |
IV | 5 |
V | 7 |
VI | 10 |
VII | 13 |
VIII | 15 |
IX | 17 |
XI | 18 |
XXXVIII | 80 |
XL | 81 |
XLI | 85 |
XLII | 88 |
XLIII | 90 |
XLIV | 93 |
XLV | 96 |
XLVI | 101 |
XII | 20 |
XIII | 22 |
XIV | 23 |
XV | 25 |
XVI | 27 |
XVII | 29 |
XVIII | 30 |
XIX | 34 |
XXI | 36 |
XXII | 39 |
XXIII | 40 |
XXIV | 42 |
XXV | 44 |
XXVI | 48 |
XXVII | 50 |
XXIX | 53 |
XXX | 55 |
XXXI | 59 |
XXXII | 65 |
XXXIII | 68 |
XXXV | 73 |
XXXVI | 76 |
XXXVII | 78 |
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Common terms and phrases
analytic function bounded called Co(R compact set compact subset compact support complex number compute constant coefficients continuous function continuous linear form continuous map converges convolution Corollary cut-off function deduce defined definition derivatives diffeomorphism Dirac distribution ECO(K example exercise EY(R finite Fourier transform Fourier-Laplace transform Fréchet space function ƒ fundamental kernel fundamental solution gives Hence homogeneous of degree identity implies inequality inverse L₁(R lal<m lal<N Lemma Let XCR linear differential operator metric multi-index neighbourhood norm obtain obvious open set partition of unity polynomial positive real number proof of Theorem Proposition pull-back pullback map push-forward rapidly decreasing restriction satisfies Schwartz kernel second member semi-norm estimate sequence sequentially continuous map Show sing supp singular support smooth function subspace Suppose suppu tempered distribution test functions topology u₁ unique wavefront set WF(Pu WF(u zero ФЕС