Methods of the Theory of Generalized Functions. (2014)

• Record Type: Book
• Further Information: Note: V.S. Vladimirov.
• Authors: Vladimirov, V. S
• Contents: Cover; Half title; Analytical Methods and Special Functions; Title; Copyright; CONTENTS; Preface; Symbols and Definitions; CHAPTER 1 Generalized Functions and their Properties; 1. Test and Generalized Functions; 1.1. Introduction; 1.2. The space oftest functions 1>(0; 1.3. The space of generalized functions 1>'(0; 1.4. The completeness ofthe space of generalized functions 1>'(0; 1.5. The support of a generalized function; 1.6. Regular generalized functions.; 1.7. Measures; 1.8. Sochozki formulae; 1.9. Change of variables in generalized functions 1.10. Multiplication of generalized functions2. Differentiation of Generalized Functions; 2.1. Derivatives of generalized functions; 2.2. The antiderivative (primitive) of a generalized function; 2.3. Examples.; 2.4. The local structure of generalized functions; 2.5. Generalized functions with compact support; 2.6. Generalized functions with point support; 2.7. Generalized functions P(rr.., lzl0 -- 1; 3. Direct Product of Generalized Functions; 3.1. The definition of a direct product; 3.2. The properties of a direct product; 3.3. Some applications 3.4. Generalized functions that are smooth with respect to some of the variables4. The Convolution of Generalized Functions; 4.1. The definition of convolution; 4.2. The properties of a conwlution; 4.3. The existence of a convolution; 4.4. Cones in 1Rn; 4.5. Convolution algebras V'(r+) and l>'(r; 4.6. Mean functions of generalized functions; 4.7. Multiplication of generalized functions; 4.8. Convolution as a continuous linear translationinvariant operator; 4.9. Some applications; 5. Tempered Generalized Functions; 5.1. The spaceS of test (rapidly decreasing) functions 5.2. The spaceS' of tempered generalized functions5.3. Examples oftempered generalized functions and elementary operations in S; 5.4. The structure of tempered generalized fUnctions; 5.5. The direct product of tempered generalized functions; 5.6. The convolution of tempered generalized fUnctions; 5.7 . Homogeneous generalized fUnctions; Chapter 2. Integral Transformations of Generalized Fuaetioas; 6. The Fourier Transform of Tempered Generalized Functions; 6.1. The Fourier transform of test functions in S; 6.2. The Fourier transform of tempered generalized functions 6.3. Properties of the Fourier transform6.4. The Fourier transform of generalized functions with compact support; 6.5. The Fourier transform of a convolution; 6.6. Examples; 6.7. The Mellin transform; 7. Fourier Series of Periodic Generalized Functions; 7.1. The definition and elementary properties of periodic generalized functions; 7.2. Fourier series of periodic generalized functions; 7.3. The convolution algebra 1; 7 .4. Examples; 8. Positive Definite Generalized Functions; 8.1. The definition and elementary properties of positive definite generalized fUnctions
• Publisher Details: Boca Raton, FL : CRC Press
• Publication Date: 2014
• Extent: 1 online resource
• Subjects: 515.782; Mathematical analysis; Mathematics; Mathematical analysis; Mathematics; Electronic books
• Languages: English
• ISBNs: 9781482288162; 1482288168
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.