Fourier transforms and the theory of distributions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Englewood Cliffs, NJ
Prentice-Hall
1966
|
| Series: | Prentice-Hall applied mathematics series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 318 S. |
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MARC
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| 100 | 1 | |a Arsac, Jacques |d 1929- |e Verfasser |0 (DE-588)118937030 |4 aut | |
| 240 | 1 | 0 | |a Transformations de Fourier et théorie des distributions |
| 245 | 1 | 0 | |a Fourier transforms and the theory of distributions |c J. Arsac |
| 264 | 1 | |a Englewood Cliffs, NJ |b Prentice-Hall |c 1966 | |
| 300 | |a XV, 318 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
PART ONE THE FOURIER TRANSFORM
1 MATHEMATICAL REVIEW 3
I Review of Algebra 3
1.1 Sets 3
1.2 Functions 5
1.3 Laws of algebra. Internal laws 6
1.4 Laws of algebra. External laws 8
1.5 Vector space (for the field C of complex numbers) 8
1.6 Linear functions 9
1.7 Order relations 11
II Concepts from Topology 12
1.8 Sequences of rational numbers 12
1.9 A theorem for monotonic sequences 15
1.10 The concept of distance and the norm 16
1.11 Mappings from one metric space to another 17
1.12 Limits of functions 18
III Theory of Integration 20
1.13 Preliminary remarks 20
1.14 Construction of the Daniell integral 20
1.15 Identification of elements of L with numerical functions 23
1.16 Elementary properties of the integral 24
1.17 Sequence of integrals 25
1.18 Concept of a primitive. Integration by parts 26
1.19 Multiple integrals. Theorem of Fubini 27
1.20 Integrals over a noncompact interval 28
1.21 Integrals depending upon a parameter 30
1.22 The spaces L 31
Conclusions 33
xi
Xii CONTENTS
2 FOURIER TRANSFORMS OF SUMMABLE FUNCTIONS 34
I Definitions and General Properties 34
2.1 Definition and notation 34
2.2 Elementary properties 35
2.3 Riemann Lebesgue lemma 38
2.4 Continuity of the transform 39
2.5 Differentiability of the transform 39
2.6 The case in which/has an amplitude summable derivative 41
II Reciprocity of the Transform 43
2.7 Conditions for the existence of the inverse transform 43
III Convolution Theory 47
2.8 Definition 47
2.9 Some special cases 48
2.10 Remarks 49
IV Examples 50
2.11 Examples of direct and inverse transforms 50
3 FOURIER TRANSFORMS FOR
SQUARE SUMMABLE FUNCTIONS 55
I Definition and properties of the FT in L? 55
3.1 The Fourier transform for square summable functions 55
3.2 Properties of the transform 58
3.3 Return to the first definition of the FT 59
3.4 Convolution 60
II Scalar Product of Two Functions 61
3.5 Definition 61
3.6 Properties of the scalar product 62
III Example 63
3.7 Example of FT in L2. The Hermite polynomials 63
4 ELEMENTARY THEORY OF MODIFIED DISTRIBUTIONS AND
THE DEFINITION OF THEIR FOURIER TRANSFORM 68
4.1 Introduction to distributions 68
4.2 Definition of distributions 70
4.3 FT of modified distributions 72
4.4 Elementary properties of distributions 75
4.5 Properties of the FT for distributions 77
4.6 Differentiation of distributions 80
4.7 Differentiation of discontinuous functions 82
4.8 Special distributions 85
4.9 The FT of functions of S and locally summable functions 86
4.10 Example of a distribution. Systems of heavy point masses 87
4.11 Direct product of two distributions 90
4.12 Convolution product of distributions 90
4.13 Properties of the convolution product 92
4.14 Fourier series 97
4.15 Summary 102
CONTENTS Xiii
5 THE FOURIER TRANSFORM AND
DISTRIBUTIONS IN MULTIDIMENSIONAL SPACE 105
5.1 Notation 105
5.2 The FT of functions in a space of n dimensions 106
5.3 Distributions in a multidimensional space 110
5.4 Differentiation of discontinuous functions 111
5.5 Convolution 115
5.6 Direct product of two distributions 116
5.7 Radial functions or distributions 117
5.8 The nature of modified distributions 123
PART TWO EXAMPLES OF APPLICATIONS
OF THE FOURIER TRANSFORM
O DIFFRACTION AT INFINITY 127
6.1 Reduction of problems of diffration at infinity to a Fourier
transform 127
6.2 Elementary properties of the correspondence between a
diffracting pupil and a diffraction figure 128
6.3 The appearance of the distributions 131
6.4 Relation between diffraction and interference phenomena 133
6.5 The convolution product 134
6.6 Application of Fourier series 135
6.7 Imaging with coherent light 137
6.8 Interferometry in optics and radio astronomy 139
6.9 Interferometry in spectroscopy 142
7 COMPLEX IMPEDANCES AND THE FOURIER
TRANSFORM IN THE COMPLEX PLANE 1 45
7.1 Sinusoidal alternating currents and the concept of complex
impedance 145
7.2 Use of negative frequencies 147
7.3 General theorems about the complex impedance 149
7.4 Applications: Use of the FT for transient problems 159
7.5 Connection with the Laplace transform 166
7.6 The concept of an analytic signal 168
Conclusion 172
8 THE FOURIER TRANSFORM
IN MATHEMATICAL PHYSICS 173
A Detailed Study of a Partial Differential Equation 173
8.1 The wave equation in a homogeneous medium and its general
solution 173
XIV CONTENTS
8.2 Fresnel diffraction 176
8.3 The equations of convolution 190
8.4 General method for the solution of nonhomogeneous equations 191
8.5 Determination of solutions in a section of space. Limiting con¬
ditions 194
8.6 A one dimensional problem 195
8.7 A three dimensional problem 197
8.8 Green s formula 201
PART THREE LINEAR FILTERS
9 GENERAL PROPERTIES OF FUNCTIONS WHOSE FOURIER
TRANSFORM COVERS AN INTERVAL OF FINITE LENGTH 205
9.1 Linear filters in physics 205
9.2 Bernstein s theorem. Derivatives of limited spectrum functions 214
9.3 Bernstein s theorem in a two dimensional space 217
9.4 Some applications of Bernstein s theorem 218
9.5 Theorem of interpolation 224
IO RESOLVING POWER THEORY 229
10.1 Physical introduction 229
10.2 Approximation by the mean square difference 231
10.3 Insufficiency of the criterion of the mean square difference 235
10.4 Objects formed from points 235
10.5 A general method of studying the local difference between object
and image 239
10.6 Usual procedures of studying the difference 240
10.7 Mathematical results 244
10.8 The role of the various parameters 245
10.9 Localization of the approximation 246
10.10 Saturation of the approximation 248
10.11 Relations between g(x) and the filtering function 250
10.12 The Fourier procedure 253
10.13 The procedure of Fejer 254
10.14 An alternate procedure 256
10.15 General remarks 257
THE FOURIER TRANSFORM OF PROBABILITY FUNCTIONS,
mm THE AUTOCORRELATION FUNCTION,
I I AND THE SPECTRAL DISTRIBUTION OF ENERGY 260
11.1 Introduction. The problem of background noise 260
11.2 The autocorrelation function and the spectral distribution of
energy 262
11.3 Properties of the autocorrelation function and the spectral
distribution of energy 267
CONTENTS XV
11.4 Physical significance of the spectral distribution of energy 269
11.5 Random functions of the second order. 272
11.6 Discussion of a specific example. The sensitivity of the
interferometric method in radio astronomy 276
PART FOUR NUMERICAL CONSIDERATIONS
I 2 NUMERICAL METHODS 285
12.1 Problems involving the numerical calculation of Fourier
transforms 285
12.2 The formula of Poisson 287
12.3 Calculation of the value of the FT 289
12.4 Calculation of a FT in the presence of background noise 294
12.5 The numerical calculation of certain definite integrals 298
12.6 Practical calculation of a FT 301
12.7 The approximation of a function by a series of translations 303
12.8 Applications 305
12.9 General remarks 309
BIBLIOGRAPHY 311
INDEX 315
|
| any_adam_object | 1 |
| author | Arsac, Jacques 1929- |
| author_GND | (DE-588)118937030 |
| author_facet | Arsac, Jacques 1929- |
| author_role | aut |
| author_sort | Arsac, Jacques 1929- |
| author_variant | j a ja |
| building | Verbundindex |
| bvnumber | BV002046131 |
| classification_rvk | SK 450 |
| ctrlnum | (OCoLC)264339943 (DE-599)BVBBV002046131 |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T15:39:26Z |
| institution | BVB |
| language | English |
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| physical | XV, 318 S. |
| publishDate | 1966 |
| publishDateSearch | 1966 |
| publishDateSort | 1966 |
| publisher | Prentice-Hall |
| record_format | marc |
| series2 | Prentice-Hall applied mathematics series |
| spelling | Arsac, Jacques 1929- Verfasser (DE-588)118937030 aut Transformations de Fourier et théorie des distributions Fourier transforms and the theory of distributions J. Arsac Englewood Cliffs, NJ Prentice-Hall 1966 XV, 318 S. txt rdacontent n rdamedia nc rdacarrier Prentice-Hall applied mathematics series Aus d. Franz. übers. Verteilungstheorie (DE-588)4133628-8 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 gnd rswk-swf Fourier-Transformation (DE-588)4018014-1 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 s Fourier-Transformation (DE-588)4018014-1 s DE-604 Verteilungstheorie (DE-588)4133628-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001338207&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arsac, Jacques 1929- Fourier transforms and the theory of distributions Verteilungstheorie (DE-588)4133628-8 gnd Distribution Funktionalanalysis (DE-588)4070505-5 gnd Fourier-Transformation (DE-588)4018014-1 gnd |
| subject_GND | (DE-588)4133628-8 (DE-588)4070505-5 (DE-588)4018014-1 |
| title | Fourier transforms and the theory of distributions |
| title_alt | Transformations de Fourier et théorie des distributions |
| title_auth | Fourier transforms and the theory of distributions |
| title_exact_search | Fourier transforms and the theory of distributions |
| title_full | Fourier transforms and the theory of distributions J. Arsac |
| title_fullStr | Fourier transforms and the theory of distributions J. Arsac |
| title_full_unstemmed | Fourier transforms and the theory of distributions J. Arsac |
| title_short | Fourier transforms and the theory of distributions |
| title_sort | fourier transforms and the theory of distributions |
| topic | Verteilungstheorie (DE-588)4133628-8 gnd Distribution Funktionalanalysis (DE-588)4070505-5 gnd Fourier-Transformation (DE-588)4018014-1 gnd |
| topic_facet | Verteilungstheorie Distribution Funktionalanalysis Fourier-Transformation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001338207&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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