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Fractional integral transforms : theory and applications / Ahmed I. Zayed
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Publisher: Boca Raton : CRC Press, 2024Edition: First editionDescription: 1 Online-RessourceISBN:- 9781040003718
- 1040003710
- 9781003089353
- 1003089356
- 9781040003664
- 519.2/3 23/eng/20231031
- QA432
Contents:
Summary: "Fractional Integral Transforms: Theory and Applications presents over twenty-five integral transforms, many of which have never before been collected in one single volume. Some transforms are classic, such as Laplace, Fourier, etc, and some are relatively new, such as the Fractional Fourier, Gyrator, Linear Canonical, Special Affine Fourier Transforms, as well as, continuous Wavelet, Ridgelet, and Shearlet transforms. This book provides an overview of the theory of fractional integral transforms with examples of such transforms, before delving deeper into the study of important fractional transforms, including the fractional Fourier transform. Applications of fractional integral transforms in signal processing and optics are highlighted. The book's format has been designed to make it easy for readers to extract the essential information they need to learn about the fundamental properties of each transform. Supporting proof and explanations are given throughout"--PPN: PPN: 1907959629Package identifier: Produktsigel: ZDB-4-NLEBK | BSZ-4-NLEBK-KAUB
1.3.1. Non-orthogonal Bases and Frames -- 1.3.2. Reproducing-Kernel Hilbert Spaces -- 1.4. SHIFT-INVARIANT SPACES -- 1.5. GENERALIZED FUNCTIONS AND DISTRIBUTIONS -- 1.5.1. Testing-Function Spaces and Their Duals -- 1.5.2. Spaces of Generalized Functions -- 1.5.3. A Special Type of Generalized Functions -- 1.6. SAMPLING AND THE PALEY-WIENER SPACE -- 1.7. POISSON SUMMATION FORMULA -- 1.8. UNCRTAINTY PRINCIPLE -- CHAPTER 2: Integral Transformations -- 2.1. INTRODUCTION AND BRIEF HISTORY -- 2.2. WHAT IS AN INTEGRAL TRANSFORM? -- 2.3. EXAMPLES OF INTEGRAL TRANSFORMS
2.3.1. One-Dimensional Integral Transforms -- 2.3.2. Higher Dimensional Transforms -- 2.3.3. Special Cases of Higher Dimensional Transforms -- 2.4. GENERAL PROPERTIES OF INTEGRAL TRANSFORMATIONS -- 2.5. WHY INTEGRAL TRANSFORMS? -- CHAPTER 3: Fractional Integral Transforms -- 3.1. INTRODUCTION -- 3.2. PRELUDE TO FRACTIONAL INTEGRAL TRANSFORMS -- 3.2.1. The Fractional Fourier Transform -- 3.2.2. The Fractional Hankel Transform -- 3.3. GENERAL CONSTRUCTION OF FRACTIONAL INTEGRAL TRANSFORMS -- 3.3.1. Examples of the General Construction
3.3.2. Fractional Integral Transforms Associated With the Jacobi Polynomials -- 3.4. FRACTIONAL DERIVATIVES AND INTEGRALS VERSUS FRACTIONAL INTEGRAL TRANSFORMS -- 3.5. OTHER FRACTIONAL INTEGRAL TRANSFORMS -- CHAPTER 4: The Fractional Fourier Transform (FrFT) -- 4.1. HISTORICAL OVERVIEW -- 4.2. PRELIMINARIES -- 4.3. OPERATIONAL CALCULUS -- 4.3.1. Convolution Theorem -- 4.3.2. Poisson Summation Formula for the Fractional Fourier Transform -- 4.3.3. Sampling Theorem for the Fractional Fourier Transform -- 4.3.4. The Wigner Distribution -- 4.4. THE FRACTIONAL HILBERT TRANSFORM
4.5. FRACTIONAL TIME-FREQUENCY REPRESENTATIONS -- 4.5.1. Fractional Wigner Distributions -- 4.5.2. Fractional Time and Frequency Shifts -- 4.5.3. The Fractional Cross-Ambiguity Function -- 4.5.4. Fractional Windowed (Sliding-Window)-Fourier Transform -- 4.6. UNCERTAINTY PRINCIPLE FOR THE FRACTIONAL FOURIER TRANSFORM -- 4.7. FRACTIONAL FOURIER TRANSFORM OF GENERALIZED FUNCTIONS -- 4.7.1. The Embedding Method -- 4.7.2. The Space of Boehmians -- 4.7.3. The Algebraic Method -- 4.8. APPLICATIONS OF THE FRACTIONAL FOURIER TRANSFORM
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