| Description |
1 online resource (xxv, 534 pages) : illustrations |
| Series |
Applied and numerical harmonic analysis |
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Applied and numerical harmonic analysis.
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| Contents |
Machine generated contents note: pt. I Primer on Functional Analysis -- 1. Banach Spaces and Operator Theory -- 1.1. Definition and Examples of Banach Spaces -- 1.2. Holder's and Minkowski's Inequalities -- 1.3. Basic Properties of Banach Spaces -- 1.4. Linear Combinations, Sequences, Series, and Complete Sets -- 1.5. Hilbert Spaces -- 1.6. Orthogonal Sequences in Hilbert Spaces -- 1.7. Operators -- 1.8. Bounded Linear Functional and the Dual Space -- 2. Functional Analysis -- 2.1. Hahn-Banach Theorem and Its Implications -- 2.2. Reflexivity -- 2.3. Adjoints of Operators on Banach Spaces -- 2.4. Adjoints of Operators on Hilbert Spaces -- 2.5. Baire Category Theorem -- 2.6. Uniform Boundedness Principle -- 2.7. Open Mapping Theorem -- 2.8. Topological Isomorphisms -- 2.9. Closed Graph Theorem -- 2.10. Weak Convergence -- pt. II Bases and Frames -- 3 |
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Note continued: 5.7. Perturbations of Bases -- 6. Unconditional Bases in Banach Spaces -- 6.1. Basic Properties and the Unconditional Basis Constant -- 6.2. Characterizations of Unconditional Bases -- 6.3. Conditionality of the Schauder System in C[0,1] -- 6.4. Conditionality of the Haar System in L1[0, 1] -- 7. Bessel Sequences and Bases in Hilbert Spaces -- 7.1. Bessel Sequences in Hilbert Spaces -- 7.2. Unconditional Bases and Riesz Bases in Hilbert Spaces -- 8. Frames in Hilbert Spaces -- 8.1. Definition and Motivation -- 8.2. Frame Expansions and the Frame Operator -- 8.3. Overcompleteness -- 8.4. Frames and Bases -- 8.5. Characterizations of Frames -- 8.6. Convergence of Frame Series -- 8.7. Excess -- pt. III Bases and Frames in Applied Harmonic Analysis -- 9. Fourier Transform on the Real Line -- 9.1. Summary: Main Properties of the Fourier Transform on the Real Line -- 9.2. Motivation: The Trigonometric System -- 9.3. Fourier Transform on L1(R) -- 9.4. Fourier Transform on L2(R) -- 9.5. Absolute Continuity -- 10. Sampling, Weighted Exponentials, and Translations -- 10.1. Bandlimited Functions -- 10.2. Sampling Theorem -- 10.3. Frames of Weighted Exponentials -- 10.4. Frames of Translates -- 11. Gabor Bases and Frames -- 11.1. Time-Frequency Shifts -- 11.2. Painless Nonorthogonal Expansions -- 11.3. Nyquist Density and Necessary Conditions for Frame Bounds -- 11.4. Wiener Amalgam Spaces -- 11.5. Walnut Representation -- 11.6. Zak Transform -- 11.7. Gabor Systems at the Critical Density -- 11.8. Balian-Low Theorem -- 11.9. HRT Conjecture -- 12. Wavelet Bases and Frames -- 12.1. Some Basic Facts -- 12.2. Wavelet Frames and Wavelet Sets -- 12.3. Frame Bounds and the Admissibility Condition -- 12.4. Multiresolution Analysis -- 12.5. All About the Scaling Function, I: Refinability |
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Note continued: 12.6. All About the Scaling Function, II: Existence -- 12.7. All About the Wavelet -- 12.8. Examples -- pt. IV Fourier Series -- 13. Fourier Series -- 13.1. Notation and Terminology -- 13.2. Fourier Coefficients and Fourier Series -- 13.3. Convolution -- 13.4. Approximate Identities -- 13.5. Partial Sums and the Dirichlet Kernel -- 13.6. Cesaro Summability and the Fejer Kernel -- 13.7. Inversion Formula for L1(T) -- 14. Basis Properties of Fourier Series -- 14.1. Partial Sum Operators -- 14.2. Conjugate Function -- 14.3. Pointwise Almost Everywhere Convergence -- pt. V Appendices -- A. Lebesgue Measure and Integration -- A.1. Exterior Lebesgue Measure -- A.2. Lebesgue Measure -- A.3. Measurable Functions -- A.4. Lebesgue Integral -- A.5. Lp Spaces and Convergence -- A.6. Repeated Integration -- B. Compact and Hilbert-Schmidt Operators -- B.1. Compact Sets -- B.2. Compact Operators -- B.3. Hilbert-Schmidt Operators -- B.4. Finite-Rank Operators and Tensor Products -- B.5. Hilbert-Schmidt Kernel Theorem |
| Bibliography |
Includes bibliographical references (pages 515-526) and index |
| Notes |
Print version record |
| Subject |
Function spaces.
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Espacios funcionales
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Function spaces
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| Form |
Electronic book
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| ISBN |
9780817646875 |
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0817646876 |
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9780817646868 |
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0817646868 |
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