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| Ïîèñê ïî óêàçàòåëÿì |
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| Stakgold I. — Boundary value problems of mathematical physics |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel's formula for the Wronskian 60
Absolutely continuous functions 145
Action of distribution 28 29 31
Adjoint see also "Self-adjoint operators" "Symmetry"
Adjoint boundary conditions 70
Adjoint for differential system 84
Adjoint for integral equation 81 212
Adjoint for linear equations 80
Adjoint Green's function 71
Adjoint of integral operator 212
Adjoint of matrix 80 152—154
Adjoint of transformation on Euclidean space 152—154
Adjoint of transformation on Hilbert space 170—180
Adjoint, formal 40 68
Adjoint, null space of 171
Admissible pair 170
Alternative Theorems 79—86
Alternative theorems for completely continuous operators 187
Alternative theorems for differential equations 82—86
Alternative theorems for equations in Euclidean space 153—154
Alternative theorems for integral equations 81 221
Alternative theorems for linear equations 79—81 153
Alternative theorems for operators with closed range 172
Banach spaces 106
Basis for Hilbert space 119
Basis for vector space 97
Basis of eigenvectors 160
Basis, dual 138
Basis, orthonormal 123—124 129
Basis, orthonormal, characterization of 127—128
Bessel equation 329—332
Bessel equation as singular problem 301 305—308 313—315 317 321
Bessel equation, eigenvalues of 282
Bessel equation, Green's function for 75 77
Bessel functions 329
Bessel functions, modified 331
Bessel's inequality 125
Bi-orthogonal set 256
Bilinear concomitant 69
Bilinear expansion for Green's function 261 273 290
Bilinear expansion for kernel 238—240
Bilinear series see "Bilinear expansion"
boundary conditions 64—69 see
Boundary conditions, adjoint 70
Boundary conditions, unmixed 64 66
Boundary value problem regular 269 270
Boundary value problem regular for second order equation 64—91
Boundary value problem regular, singular 269 283—322
Cauchy principal value 49
Cauchy sequences 100
Closable operator 142
Closed operator 142 168
Closed set. 114
Compact operator see "Completely continuous transformation"
Compact set 114 164
Compatibility conditions see "Consistency conditions"
Complete orthonormal set 127 see
Completely continuous transformation 185
Completely continuous transformation, alternative theorem for 187
Completely continuous transformation, integral operator as 192
Completeness of metric space 101—105
Completeness of orthonormal set 127
Concentrated source see "Dirac delta function"
Conjunct 69
Consistency conditions 80 83 85 154 201 221 265
Convergence in the mean 117—118 131
Convergence of distributions 42
Convergence of test functions 30
Convergence, Cauchy 100
Convergence, uniform 131
Convergence, weak 133
Courant minimax principle 225
Deficiency 181
Degenerate eigenvalue 213
Delta function see "Dirac delta function"
Delta sequence 24—27
Dense set 114
Dependence of set of functions 59—60 118
Dependence of set of vectors 97 118
Dependence, criterion for 60
Dimension of vector space 97
Dipole 25—26 35
Dirac delta function 6—7 18—27 33
Dirac delta function, expansion of 273
Dirac delta function, sifting property of 22—24
Distance function see "Metric"
Distributional solutions of differential equations 51—58
Distributions 28—58
Distributions as linear functionals 28 31
Distributions as solutions of differential equations 53
Distributions, action on test functions 28—29 31
Distributions, convergence of 42
Distributions, delta 33
Distributions, differentiation of 37 41
Distributions, dipole 35
Distributions, divergent integrals and 47—50
Distributions, Fourier series of 45—47
Distributions, fundamental solution and 54 56
Distributions, operations on 35—36
Distributions, order of 40
Distributions, pseudofunctions and 48—51
Distributions, regular 33
Distributions, singular 33
Distributions, values of 37
Domain of differential operator 76
Domain of function 93 95
Domain of transformation 95
Eigenfunctions of integral operator 213
Eigenfunctions of self-adjoint system 220 270
Eigenfunctions, completeness of 219 220
Eigenfunctions, normalization of 274 278
Eigenfunctions, orthogonality of 213 270
Eigenmanifold 156
Eigenvalues of integral operator 212—213
Eigenvalues of transformation on Euclidean space 154 156
Eigenvalues of transformation on Hilbert space 181
Eigenvalues, approximations for 228—236
Eigenvalues, degenerate 213
Eigenvalues, existence of 214
Eigenvalues, extended problem of 248—249
Eigenvalues, extremal principles for 162 187—190 214—217
Eigenvalues, multiplicity of 156—157 214
Eigenvalues, simple 213
Equal almost everywhere 33 105
Essentially regular transformation 166
Existence and uniqueness for inhomogeneous integral equation 221
Existence and uniqueness for initial value problems 58 59 61
Existence and uniqueness of Green's function 66
Expansion theorem 218
Extremal principles for eigenvalues 162 187—190 214—214 223—236
Extremal principles for inhomogeneous equation 245—250
Finite part of divergent integrals 48
Fourier coefficients 125
Fourier cosine transform 293
Fourier series of distributions 44—46 50 51
Fourier sine transform 287
Fourier sum 125
Fourier transform 47 294
Fourier — Bessel expansion 308
Fredholm integral equations of first kind 195 201
Fredholm integral equations of second kind 195
Functionals 134 245 see
Fundamental solution 54 62 see
Fundamental solution, causal 56
Fundamental solution, pole of 62
GALERKIN equations 244 245 248
Galerkin's method 244 245
Generalized function 34 see
Generalized solutions of differential equations 53
Gram — Schmidt procedure 122
| Green's formula 70
Green's function 1—18 54—56 65—69 71—78 86—91 219—220 261—268 271—282 284—295 303—321
Green's function as integral operator 12 199 219
Green's function for Bessel's equation 75 77
Green's function for initial value problems 56 66
Green's function for regular boundary value problems 66 271
Green's function for singular problems 303—304
Green's function, adjoint 70
Green's function, existence and uniqueness of 66
Green's function, modified 86—91 316—319
Green's function, singularities of 274 304
Hankel functions 329
Hankel transforms 315—316
Heat conduction equation 327—328
Heaviside function 20 27 34 38
Heaviside sequence 27
Hermite equation 302 320—321
Hermite polynominals 321
Hermitian see "Symmetric"
Hilbert spaces, definition of 110
Hilbert spaces, functionals on 135—139
Hilbert spaces, properties of 116—135
Hilbert spaces, separable 116
Hilbert spaces, transformations on 140—146 165—190
Hilbert — Schmidt operators see "Integra] operators"
Impulse response 78 see
Impulsive source see "Dirac delta function"
Indefinite operator 165 224
Independence of set of functions 59—60 118
Independence of set of vectors 97 118
Independence, criterion for 60
Influence function see "Green's function"
Initial conditions 64 67
Initial value problem 59
Initial value problem, Green's function for 56 66
Inner product on complex linear space 109
Inner product on real linear space 107
Inner product spaces 107—114 see
Inner product spaces, complex 109
Inner product spaces, real 107
Integral equations 191—258
Integral equations, approximate methods for 241—250
Integral equations, differential equations and 199—206
Integral equations, eigenvalue problem for 195 212—220
Integral equations, Fredholm 195
Integral equations, Volterra 196
Integral operators of Hilbert — Schmidt type 193
Integral operators, adjoint of 212
Integral operators, complete continuity of 193
Integral operators, eigenvalues and eigenfunctions of 195—196
Integral operators, Green's functions and 199—206
Integral operators, indefinite 224
Integral operators, kernel of 192
Integral operators, negative and nonnegative 224
Integral operators, nonsymmetric 250—258
Integral operators, positive and nonpositive 224
Integral operators, real 225
Integral operators, spectrum of 212—220
Inverse operator 12 140 151 165—180
Kantorovich — Lebedev transform 317
Kernel of integral operator 144
Kernel, bilinear series for 238—240
Kernel, Hilbert — Schmidt 193
Kernel, iterated 206 239
Kernel, left iterate of 251
Kernel, nonsymmetric 250—258
Kernel, right iterate of 251
Kernel, separable 192 198
kernel, symmetric 212
Kohn — Kato method 235 236
Lagrange's identity 70
Least squares approximation 124 224
Lebesgue convergence theorem 105
Lebesgue integral 104—105
Legendre equation 302 319 320
Legendre polynomials 123 130 319
Limit circle case for singular problem 297 301 310—313
Limit point case for singular problem 297 301 310
Linear functionals on Euclidean space 135—139
Linear functionals on Hilbert space 135—139
Linear functionals on space of test functions 31
Linear functionals, boundedness of 135 137
Linear functionals, continuity of 135
Linear functionals, extension of 137
Linear functionals, norm of 135
Linear functionals, Riesz representation theorem for 136
Linear manifolds 120 121
Linear spaces 96—99 105—190 see
Linear transformations see "Transformations"
Locally integrable 31
MacDonald functions 331
Mapping see "Operator" "Transformations"
Matrix 79 146
Mellin cosine transform 316
Mellin sine transform 316 317
Mellin transform 308 309
Mercer's theorem 240
Metric 100 117
Metric spaces 99—105 114—116
Metric spaces, Cauchy sequences in 100
Metric spaces, closed sets in 114
Metric spaces, compact sets in 114
Metric spaces, completeness of 101
Metric spaces, convergence in 100
Metric spaces, dense sets in 114
Metric spaces, triangle inequality for 100
Metric, generated by norm 106
Minimax principle 225
Modified Green's function 86—91 317—319
Modified Green's function, symmetric 88
Multiplicity, algebraic 157
Multiplicity, geometric 156
Negative operator 165 224
Neumann functions 329
Neumann series 206—211
Neumann series for Volterra equation 209
Nonnegative operator 165 224
Nonpositive operator 165 224
Norm 105 117
Norm of functional 135
Norm of s type 296
Norm of transformation 140
Norm, generated by inner product 108
Norm, metric generated by 106
Normal form of differential equation 279
Normalization of eigenfunctions 274 278
Normed linear spaces 105—107
Nullity of transformation 153
One-to-one transformation 93
Operator see also "Transformations"
Operator, bounded below 277
Operator, closable 142
Operator, closed 142 168
Operator, continuous spectrum of 181
Operator, differentiation 174
Operator, extremal principles for 162 214 226 245
Operator, point spectrum of 181
Operator, projection 149
Operator, regular value of 181
Operator, residual spectrum of 181
Operator, resolvent set of 181
Operator, shifts 169
Operator, spectrum of 180
Order of distribution 40
Orthogonal complement 121
Orthogonal sets 108 120
Orthogonalization by Gram — Schmidt procedure 122
Orthonormal basis 123—124 127—128 129
Parallelogram law 113
Parseval's identity 127
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