Distribution theory : convolution, Fourier transform, and Laplace transform
著者
書誌事項
Distribution theory : convolution, Fourier transform, and Laplace transform
(De Gruyter graduate lectures)
De Gruyter, c2013
- : pbk
大学図書館所蔵 全4件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Includes bibliographical references and index
内容説明・目次
内容説明
The theory of distributions has numerous applications and is extensively used in mathematics, physics and engineering. There is however relatively little elementary expository literature on distribution theory. This book is intended as an introduction. Starting with the elementary theory of distributions, it proceeds to convolution products of distributions, Fourier and Laplace transforms, tempered distributions, summable distributions and applications. The theory is illustrated by several examples, mostly beginning with the case of the real line and then followed by examples in higher dimensions. This is a justified and practical approach, it helps the reader to become familiar with the subject. A moderate number of exercises are added.
It is suitable for a one-semester course at the advanced undergraduate or beginning graduatelevelor for self-study.
目次
Preface 2
1 Definition and first properties of distributions 7
1.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Support of a distribution . . . . . . . . . . . . . . . . . . . . . 10
2 Differentiating distributions 13
2.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . 13
2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The distributions x 1+ ( 6= 0, 1, 2, . . . )* . . . . . . . . . . 16
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Green's formula and harmonic functions . . . . . . . . . . . . 19
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Multiplication and convergence of distributions 27
3.1 Multiplication with a C1 function . . . . . . . . . . . . . . . 27
3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Convergence in D0 . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Distributions with compact support 31
4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . 31
4.2 Distributions supported at the origin . . . . . . . . . . . . . . 32
4.3 Taylor's formula for Rn . . . . . . . . . . . . . . . . . . . . . 33
4.4 Structure of a distribution* . . . . . . . . . . . . . . . . . . . 34
5 Convolution of distributions 36
5.1 Tensor product of distributions . . . . . . . . . . . . . . . . . 36
5.2 Convolution product of distributions . . . . . . . . . . . . . . 38
5.3 Associativity of the convolution product . . . . . . . . . . . . 44
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.5 Newton potentials and harmonic functions . . . . . . . . . . . 45
5.6 Convolution equations . . . . . . . . . . . . . . . . . . . . . . 47
5.7 Symbolic calculus of Heaviside . . . . . . . . . . . . . . . . . 50
5.8 Volterra integral equations of the second kind . . . . . . . . . 52
5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.10 Systems of convolution equations* . . . . . . . . . . . . . . . 55
5.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 The Fourier transform 57
6.1 Fourier transform of a function on R . . . . . . . . . . . . . . 57
6.2 The inversion theorem . . . . . . . . . . . . . . . . . . . . . . 60
6.3 Plancherel's theorem . . . . . . . . . . . . . . . . . . . . . . . 61
6.4 Differentiability properties . . . . . . . . . . . . . . . . . . . . 62
6.5 The Schwartz space S(R) . . . . . . . . . . . . . . . . . . . . 63
6.6 The space of tempered distributions S0(R) . . . . . . . . . . . 65
6.7 Structure of a tempered distribution* . . . . . . . . . . . . . 66
6.8 Fourier transform of a tempered distribution . . . . . . . . . 67
6.9 Paley Wiener theorems on R* . . . . . . . . . . . . . . . . . . 69
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.11 Fourier transform in Rn . . . . . . . . . . . . . . . . . . . . . 73
6.12 The heat or diffusion equation in one dimension . . . . . . . . 75
7 The Laplace transform 79
7.1 Laplace transform of a function . . . . . . . . . . . . . . . . . 79
7.2 Laplace transform of a distribution . . . . . . . . . . . . . . . 80
7.3 Laplace transform and convolution . . . . . . . . . . . . . . . 81
7.4 Inversion formula for the Laplace transform . . . . . . . . . . 84
8 Summable distributions* 87
8.1 Definition and main properties . . . . . . . . . . . . . . . . . 87
8.2 The iterated Poisson equation . . . . . . . . . . . . . . . . . . 88
8.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 89
8.4 Canonical extension of a summable distribution . . . . . . . . 91
8.5 Rank of a distribution . . . . . . . . . . . . . . . . . . . . . . 93
9 Appendix 96
9.1 The Banach-Steinhaus theorem . . . . . . . . . . . . . . . . . 96
9.2 The beta and gamma function . . . . . . . . . . . . . . . . . 103
Bibliography 108
Index 109
「Nielsen BookData」 より