Part One. Linear Operators Chapter 1. Differentiation ...................................... 1 1.1 Normed Spaces .............................................. 1 1.1.1 Norm ................................................ 1 1.1.2 Banach Spaces ....................................... 4 1.1.3 Linear Operators .................................... 4 1.1.4 Operator Algebra .................................... 6 1.1.5 Small Mappings ...................................... 8 1.2 Definition of Differential ................................ 10 1.2.1 Increment of a Function ............................ 10 1.2.2 Differential ....................................... 11 1.2.3 Derivative ......................................... 14 1.2.4 Partial Differentials .............................. 16 1.2.5 Partial Derivatives ................................ 17 1.3 Rules for Differentiation ................................. 19 1.3.1 Differentiation of a Linear Combination ............ 19 1.3.2 Differentiation of a Product ....................... 19 1.3.3 The Chain Rule ..................................... 21 1.3.4 Lagrange's Theorem ................................. 23 1.3.5 Total Differentials ................................ 27 1.4 Solution of Functional Equations .......................... 32 1.4.1 Illustrative Examples and Definitions .............. 32 1.4.2 The Implicit Function Theorem ...................... 36 1.4.3 Theorem of Inverse Function ........................ 37 1.4.4 The Fixed Point Theorem ............................ 39 1.4.5 Sum of Operator Progression ........................ 40 1.5 Taylor's Formula .......................................... 41 1.5.1 Sequential Derivatives ............................. 42 1.5.2 Maclaurin's Formula ................................ 44 1.5.3 Taylor's Formula ................................... 45 1.5.4 Symmetry of Derivatives ............................ 46 1.5.5 Particular Case .................................... 47 1.6 Local Minima .............................................. 49 1.6.1 Definitions ........................................ 49 1.6.2 Unconditional Local Minima ......................... 50 1.6.3 Conditional Local Minima ........................... 51 1.6.4 The Lagrange Method ................................ 53 1.6.5 Smooth Surfaces .................................... 54 Chapter 2. Integration ......................................... 60 2.1 Measures .................................................. 60 2.1.1 Simple Sets and Simple Functions ................... 61 2.1.2 Measures on Simple Sets ............................ 62 2.2 Integral Sums ............................................. 63 2.2.1 Definition ......................................... 63 2.2.2 Order Properties ................................... 65 2.3 Convergence Almost Everywhere ............................. 66 2.3.1 The Concept of "Almost Everywhere" ................. 66 2.3.2 Continuity and Countable Additivity of Measures .... 67 2.3.3 Convergence of Monotonie Sequences ................. 68 2.4 Integrals ................................................. 70 2.4.1 Definition of Measurable Functions ................. 71 2.4.2 Approximating Sequences ............................ 72 2.4.3 Integrable Functions ............................... 74 2.4.4 Definition of Integral ............................ 76 2.4.5 Order Properties of Integrals ...................... 79 2.5 Limit Theorems ............................................ 80 2.5.1 Levy's Theorem ..................................... 80 2.5.2 Fatou's Theorem .................................... 81 2.5.3 Integrability Criterion ............................ 84 2.5.4 Lebesgue's Theorem ................................. 86 2.5.5 Fourier Transform .................................. 87 2.6 Measurable Functions ...................................... 88 2.6.1 Measurability and Integrability .................... 88 2.6.2 Sequences of Measurable Functions .................. 88 2.6.3 Measurable and Integrable Sets ..................... 89 2.7 Fubini and Tonelli's Theorems ............................. 90 2.7.1 Product of Measures ................................ 91 2.7.2 Double and Iterated Integrals ...................... 92 2.7.3 Fubini's Theorem ................................... 93 2.7.4 Tonelli's Theorem .................................. 97 2.7.5 Rules for Operations Under the Integral Sign ....... 99 2.8 Indefinite Integrals ..................................... 108 2.8.1 The Radon-Nikodym Theorem ......................... 108 2.8.2 Theorem on the Change of Variables in Integration ....................................... 109 Chapter 3. Linear Operators ................................... 112 3.1 Hilbert Spaces ........................................... 112 3.1.1 Euclidean Spaces .................................. 112 3.1.2 Geometry of Euclidean Spaces ...................... 115 3.1.3 Orthogonal Projection ............................. 116 3.1.4 Linear Functionals ................................ 119 3.1.5 Hilbert Space 2 ................................. 122 3.2 Fourier Series ........................................... 127 3.2.1 Orthonormal Bases ................................. 127 3.2.2 The Bessel Inequality ............................. 128 3.2.3 Fourier Series Expansion .......................... 130 3.2.4 The Parseval Equation ............................. 132 3.2.5 The Riesz-Fisher Theorem .......................... 132 3.3 Spaces of Functions ...................................... 133 3.3.1 Metric Spaces ..................................... 133 3.3.2 Smooth Functions .................................. 138 3.3.3 Lebesgue Spaces ................................... 143 3.3.4 Distributions ..................................... 147 3.3.5 Sobolev Spaces .................................... 155 3.4 Fourier Transforms ....................................... 157 3.4.1 Fourier Transforms of Functions of Rapid Decrease .......................................... 157 3.4.2 Transformations of Tempered Distributions ......... 159 3.4.3 The Fourier-Planeherel Transform .................. 160 3.4.4 The Fourier-Stieltjes Transform ................... 161 3.4.5 The Radon Transform ............................... 162 3.5 Bounded Linear Operators ................................. 163 3.5.1 Extension of Functionals .......................... 163 3.5.2 Uniform Boundedness of Operators .................. 164 3.5.3 Inverting Operators ............................... 166 3.5.4 Closedness of Operator Graphs ..................... 168 3.5.5 Weak Compactness .................................. 169 3.6 Compact Linear Operators ................................. 170 3.6.1 Examples of Compact Operators ..................... 170 3.6.2 Properties of Compact Operators ................... 172 3.6.3 Adjoint Operators ................................. 173 3.6.4 Fredholm Operators ................................ 174 3.6.5 Fredholm's Theorems ............................... 177 3.7 Self-Adjoint Operators ................................... 179 3.7.1 Banach Space Adjoints ............................. 179 3.7.2 Hilbert Space Adjoints ............................ 180 3.7.3 Hermitian and Normal Operators .................... 182 3.7.4 Unitary Operators ................................. 183 3.7.5 Positive Operators ................................ 183 3.8 Operator Spectra ......................................... 184 3.8.1 Classification of Spectra ......................... 184 3.8.2 Spectrum of a Closed Operator ..................... 189 3.8.3 Spectrum of a Bounded Operator .................... 191 3.8.4 Spectrum of a Compact Operator .................... 192 3.8.5 Spectrum of a Self-Adjoint Operator ............... 193 3.9 The Spectral Theorem ..................................... 200 3.9.1 Projection-Valued Measures ........................ 200 3.9.2 Integrals of Bounded Functions .................... 205 3.9.3 Integrals of Unbounded Functions .................. 212 3.9.4 The Spectral Theorem .............................. 215 3.9.5 Operator-Valued Functions ......................... 219 3.10 Operator-Valued Exponential .............................. 220 3.10.1 Problem Formulation ............................... 221 3.10.2 Operator Semigroup ................................ 222 3.10.3 The Laplace Transform ............................. 223 3.10.4 Stone's Theorem ................................... 224 3.10.5 Evolution Equations ............................... 225 Part Two. Ill-Posed Problems Chapter 4. Classical Problems ................................. 229 4.1 Mathematical Description of Physical Laws ................ 229 4.2 Classification of Differential Equations of the Second Order .................................................... 234 4.3 Elliptic Equations ....................................... 236 4.4 Hyperbolic and Parabolic Equations ....................... 241 4.5 Concept of Well-Posedness ................................ 243 Chapter 5. Ill-Posed Problems ................................. 245 5.1 Ill-Posed Cauchy Problems ................................ 245 5.2 Analytic Continuation and Interior Problems .............. 248 5.3 Physical Problems Leading to Ill-Posed Formulations ...... 250 5.4 Reduction of Ill-Posed Problems to Integral Equations .... 251 5.5 Well-Posedness of Integral Equations ..................... 254 5.6 Two Ill-Posed Problems ................................... 255 5.7 Tomography ............................................... 256 Chapter 6. Operator and Integral Equations .................... 260 6.1 Fredholm Equations of the Second Kind .................... 260 6.2 Equations of the First Kind .............................. 264 6.3 Well-Posedness in the Sense of Tychonoff ................. 266 6.4 Regularization ........................................... 269 6.5 Equations with Bounded Operators ......................... 274 6.6 Equations with Weak Singularities ........................ 276 6.7 Volterra Equations for Functions of One Variable ......... 277 6.8 Operator Volterra Equations .............................. 280 Chapter 7. Evolution Equations ................................ 283 7.1 Cauchy Problems and Operator Semigroups .................. 283 7.2 Equations in Hilbert Spaces .............................. 285 7.3 Equations with Variable Operator ......................... 288 7.4 Equations of the Second Order ............................ 290 7.5 Well-Posed and Ill-Posed Cauchy Problems ................. 292 Chapter 8. Problems of Integral Geometry ...................... 293 8.1 Determining a Function from Its Spherical Averages ....... 293 8.2 General Problems of Plane Integral Geometry .............. 298 8.3 General Problems of Integral Geometry in Space ........... 306 8.4 Volterra-Type Problems of Integral Geometry with Motion-Group Invariant Manifolds ......................... 319 Supplement. Inversion Formulas in Inverse Problems (A.L. Bukhgeim) Introduction .................................. 323 Chapter I. Tomography and the Theory of A-Analytic Functions ..................................................... 324 1.1 Radon's Problem: a Complex Function Approach ............. 324 1.2 A-Analytic Functions ..................................... 332 1.2.1 The Cauchy Formula ................................ 332 1.2.2 Limit Values of Cauchy-Type Integrals ............. 337 1.2.3 Illustrative Examples ............................. 341 1.3 A-Harmonic Functions ..................................... 347 1.3.1 Green's Formulas .................................. 347 1.3.2 The Dirichlet Problem ............................. 356 1.4 The Radon Problem on a Ring with Weight .................. 360 1.5 Problems ................................................. 364 Chapter 2. Reconstruction of Phonon Spectra from the Heat Capacity Data ................................................. 367 2.1 Problem Statement and Theoretical Background ............. 367 2.2 Inversion Formulas ....................................... 370 2.3 Problems ................................................. 374 Reference Guide ............................................... 376 References .................................................... 377 Subject Index ................................................. 379