Author(s): Debnath L., Mikusinski P.
Language: English
Pages: 600
Contents......Page 9
Preface to the Third Edition......Page 13
Preface to the Second Edition......Page 15
Preface to the First Edition......Page 17
Introduction......Page 21
Vector Spaces......Page 22
Normed Spaces......Page 28
Banach Spaces......Page 39
Linear Mappings......Page 45
Banach Fixed Point Theorem......Page 52
Exercises......Page 54
Introduction......Page 59
Step Functions......Page 60
Lebesgue Integrable Functions......Page 65
The Absolute Value of an Integrable Function......Page 68
Series of Integrable Functions......Page 70
Norm in L1(R)......Page 72
Convergence Almost Everywhere......Page 75
Fundamental Convergence Theorems......Page 78
Locally Integrable Functions......Page 82
The Lebesgue Integral and the Riemann Integral......Page 84
Lebesgue Measure on R......Page 87
Complex-Valued Lebesgue Integrable Functions......Page 91
The Spaces Lp(R)......Page 94
Lebesgue Integrable Functions on RN......Page 98
Convolution......Page 102
Exercises......Page 104
Introduction......Page 113
Inner Product Spaces......Page 114
Hilbert Spaces......Page 119
Orthogonal and Orthonormal Systems......Page 125
Trigonometric Fourier Series......Page 142
Orthogonal Complements and Projections......Page 147
Riesz Representation Theorem......Page 152
Exercises......Page 155
Introduction......Page 165
Examples of Operators......Page 166
Bilinear Functionals and Quadratic Forms......Page 171
Adjoint and Self-Adjoint Operators......Page 178
Normal, Isometric, and Unitary Operators......Page 183
Positive Operators......Page 188
Projection Operators......Page 195
Compact Operators......Page 200
Eigenvalues and Eigenvectors......Page 206
Spectral Decomposition......Page 216
Unbounded Operators......Page 221
Exercises......Page 231
Introduction......Page 237
Basic Existence Theorems......Page 238
Fredholm Integral Equations......Page 244
Method of Successive Approximations......Page 246
Volterra Integral Equations......Page 248
Method of Solution for a Separable Kernel......Page 253
Abel's Integral Equation......Page 256
Ordinary Differential Equations......Page 259
Sturm-Liouville Systems......Page 267
Inverse Differential Operators......Page 273
The Fourier Transform......Page 278
Applications of the Fourier Transform......Page 291
Exercises......Page 299
Introduction......Page 307
Distributions......Page 308
Sobolev Spaces......Page 320
Fundamental Solutions......Page 323
Elliptic Boundary Value Problems......Page 343
Applications of the Fourier Transform......Page 349
Exercises......Page 363
Introduction......Page 371
Basic Concepts and Equations......Page 372
Postulates of Quantum Mechanics......Page 383
The Heisenberg Uncertainty Principle......Page 397
The Schrödinger Equation of Motion......Page 399
The Schrödinger Picture......Page 415
The Heisenberg Picture......Page 421
The Interaction Picture......Page 425
The Linear Harmonic Oscillator......Page 427
Angular Momentum Operators......Page 432
The Dirac Relativistic Wave Equation......Page 440
Exercises......Page 443
Brief Historical Remarks......Page 453
Continuous Wavelet Transforms......Page 456
The Discrete Wavelet Transform......Page 464
Multiresolution Analysis......Page 472
Examples of Orthonormal Wavelets......Page 482
Exercises......Page 493
Introduction......Page 497
The Gateaux and Fréchet Differentials......Page 498
Optimization Problems......Page 510
Minimization of Quadratic Functionals......Page 525
Variational Inequalities......Page 527
Optimal Control Problems......Page 530
Approximation Theory......Page 537
The Shannon Sampling Theorem......Page 542
Linear and Nonlinear Stability......Page 546
Bifurcation Theory......Page 550
Exercises......Page 555
1.7 Exercises......Page 567
2.16 Exercises......Page 568
3.8 Exercises......Page 570
5.13 Exercises......Page 571
6.7 Exercises......Page 573
7.12 Exercises......Page 576
8.6 Exercises......Page 580
9.11 Exercises......Page 581
Bibliography......Page 585
Index......Page 591
