Volterra Adventures

Table of Contents

    Cover 1
    Title page 4
    Contents 8
    Preface 12
    List of Symbols 16
    Part 1 . From Volterra to Banach 18
        Chapter 1. Starting Out 20
            1.1. A vector space 20
            1.2. A linear transformation 21
            1.3. Eigenvalues 23
            1.4. Spectrum 25
            1.5. Volterra spectrum 26
            1.6. Volterra powers 28
            1.7. Why justify our “formal calculation”? 30
            1.8. Uniform convergence 31
            1.9. Geometric series 33
            Notes 36
        Chapter 2. Springing Ahead 38
            2.1. An initial-value problem 38
            2.2. Thinking differently 41
            2.3. Thinking linearly 42
            2.4. Establishing norms 43
            2.5. Convergence 45
            2.6. Mass-spring revisited 49
            2.7. Volterra-type integral equations 52
            Notes 52
        Chapter 3. Springing Higher 54
            3.1. A general class of initial-value problems 54
            3.2. Solving integral equations of Volterra type 56
            3.3. Continuity in normed vector spaces 58
            3.4. What’s the resolvent kernel? 62
            3.5. Initial-value problems redux 66
            Notes 68
        Chapter 4. Operators as Points 70
            Overview 70
            4.1. How “big” is a linear transformation? 71
            4.2. Bounded operators 73
            4.3. Integral equations done right 78
            4.4. Rendezvous with Riemann 80
            4.5. Which functions are Riemann integrable? 84
            4.6. Initial-value problems à la Riemann 86
            Notes 90
    Part 2 . Travels with Titchmarsh 96
        Chapter 5. The Titchmarsh Convolution Theorem 98
            5.1. Convolution operators 98
            5.2. Null spaces 101
            5.3. Convolution as multiplication 103
            5.4. The One-Half Lemma 106
            Notes 112
        Chapter 6. Titchmarsh Finale 114
            6.1. The Finite Laplace Transform 114
            6.2. Stalking the One-Half Lemma 116
            6.3. The complex exponential 120
            6.4. Complex integrals 122
            6.5. The (complex) Finite Laplace Transform 124
            6.6. Entire functions 125
            Notes 128
    Part 3 . Invariance Through Duality 130
        Chapter 7. Invariant Subspaces 132
            7.1. Volterra-Invariant Subspaces 132
            7.2. Why study invariant subspaces? 134
            7.3. Consequences of the VIST 140
            7.4. Deconstructing the VIST 143
            Notes 148
        Chapter 8. Digging into Duality 150
            8.1. Strategy for proving \\conjc 150
            8.2. The “separable” Hahn-Banach Theorem 153
            8.3. The “nonseparable” Hahn-Banach Theorem 161
            Notes 166
        Chapter 9. Rendezvous with Riesz 172
            9.1. Beyond Riemann 172
            9.2. From Riemann & Stieltjes to Riesz 177
            9.3. Riesz with rigor 179
            Notes 186
        Chapter 10. V-Invariance: Finale 190
            10.1. Introduction 190
            10.2. One final reduction! 191
            10.3. Toward the Proof of Conjecture U 192
            10.4. Proof of Conjecture U 195
            Notes 197
        Appendix A. Uniform Convergence 200
        Appendix B. \\CComplex Primer 202
            B.1. Complex numbers 202
            B.2. Some Complex Calculus 204
            B.3. Multiplication of complex series 205
            B.4. Complex power series 207
        Appendix C. Uniform Approximation by Polynomials 212
        Appendix D. Riemann-Stieltjes Primer 216
            Notes 228
        Bibliography 230
        Index 234
    Back Cover 240

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