Smithies F., Hall P. (ed.) — INTEGRAL EQUATIONS (No. 49) :: Электронная библиотека попечительского совета мехмата МГУ

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Smithies F., Hall P. (ed.) — INTEGRAL EQUATIONS (No. 49)

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Название: INTEGRAL EQUATIONS (No. 49)

Авторы: Smithies F., Hall P. (ed.)

Аннотация:

This tract is devoted to the theory of linear equations, mainly of the second kind, associated with the names of Volterra, Fredholm, Hilbert and Schmidt. The treatment has been modernised by the systematic use of the Lebesgue integral, which considerably widens the range of applicability of the theory. Special attention is paid to the singular functions of non-symmetric kernels and to obtaining as strong results as possible for the convergence of the expansions in infinite series. References are given to work on numerical methods of solution. Individual chapters deal with the resolvent kernel and the Neumann series, the Fredholm theorems, orthonormal systems of functions, the classical Fredholm theory, the Fred-holm formulae for ß2 kernels, Hermitian kernels, singular functions and singular values.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1958

Количество страниц: 172

Добавлена в каталог: 09.12.2009

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Предметный указатель

$(u_{n}, v_{n}; \mu_{n})$       145
$(x_{n}; \lambda_{n})$       114
$a\otimes b$       36
$H_{\lambda}$       17
$K\begin{pmatrix}u_{1},u_{2},...,u_{n}\\v_{1},v_{2},...,v_{n}\end{pmatrix}$       70
$K^{*}, \tilde{K}$       20
$K^{n}$       11
$x \sim (\xi_{n})$ , $x \sim \underset{n}{\sum}\xi_{n}e_{n}$       56
$x(t) = ^{\circ}y(t)$ , $x(t) \leqslant ^{\circ}y(t)$       6
$x_{n}\rightarrow x (\mathfrak{L}^{2})$ , $\sum\limits^{\infty}_{n=1} x_{n} = ^{\circ}s(\mathfrak{L}^{2})$       57
$\lambda^{+}_{n}$ , $\lambda^{-}_{n}$ , $\kappa^{+}_{n}$ , $\kappa^{-}_{n}$ , $x^{+}_{n}$ , $x^{-}_{n}$       131
$\mathfrak{L}^{2}$ , convergence      57
$\mathfrak{L}^{2}$ , function      6
$\mathfrak{L}^{2}$ , kernel      11 14
$\rho(K)$       31
$\sigma_{n}$       82 94
$\tau(K)$       79
$\| K \|$       10 13
$\| K \|_{2}$       13
$\| K \|_{c}$       10
$\| x \|_{c}$ $\| x \|$ , $\| x \|_{2}$       7
(X, y)      8
Abspaltungsverfahren      36
Adjoint equation      20 43 48 50 103
Adjoint kernel      20 103
Adjugate matrix      39
Almost normal kernels      151 160
Antilinear      9
Approximation in mean square      40 54 148
Associated homogeneous equation      2
Bessel's inequality      55 62
Birkhoff, G.      122
Bocher, M.      35
Bueckner, H.      52 75 131
Burkill, J.C.      8 12 58
Canonical kernels of finite rank      80 f.
Canonical kernels of finite rank, Fredholm formulae      84 ff.
Carleman, T.      99
Cauchy sequence of $\mathfrak{L}^{2}$ functions      58
Cauchy's inequality      7
Characteristic functions      21
Characteristic functions full orthonormal system      103 113 152
Characteristic functions full system      49
Characteristic functions orthogonality properties      104 112
Characteristic subspace      21
Characteristic system      114 152
Characteristic values      21
Characteristic values enumerability of set      51 112
Characteristic values existence for Hermitian kernels      107
Characteristic values extremal properties      130 ff. 159
Characteristic values minimax properties      133 ff.
Characteristic values reality for Hermitian kernels      112
Characteristic values relation with regular values      22 47 102
Characteristic values relation with zeros of Fredholm determinant      75 100
Characteristic values, rank      49 102
Complete orthonormal systems      53 60 118 123 127 139 141 147
Continuous kernels      9 ff. 18f. 22 30 35 40 52 65 124 139 141 147 165
Courant, R.      6 40 106 107 131 133
Definite kernel      127
Determinant, Fredholm      71 ff.
Determinant, modified Fredholm      97 ff.
Determinant-free theorems      52 102ff.
Differential equations, connexion with integral equations      3 ff.
Dini's theorem      124
Dini, U.      124
Dissection, $\omega-$       41
Eigenfunction      22
Eigenvalue      22
Equivalent functions      6
Equivalent, linearly      61
Fan, Ky      134
Finite rank, approximation by      40 f. 148
Finite rank, canonical      80 ff.
Finite rank, Fredholm formulae      84 ff.
Finite rank, kernels of      36
First kind, integral equations      2 164
Fischer, E.      58
Fourier coefficients      56
Fredholm alternative      52
Fredholm determinant      71 ff.
Fredholm determinant, modified      97 ff.
Fredholm formulae for $\mathfrak{L}^{2}$ kernels      94 ff.
Fredholm formulae for canonical kernels of finite rank      84 ff.
Fredholm formulae for continuous kernels      70 ff.
Fredholm integral equation      3
Fredholm minor, first      71
Fredholm minor, modified      97
Fredholm, I.      65 68
Fubini's theorem      12
Full orthonormal system of characteristic functions      103 113 152
Full system of characteristic functions      49
Full system of pairs of singular functions      145
Functional equation of resolvent      31
Goldfain, I.A.      161
Goursat, E.      77 106
Gram, J.P.      62
Hadamard's inequality      68
Hadamard, J.      68
Hardy, G.H.      7 69
Hellinger, E.      127
Hermitian adjoint      20
Hermitian kernels      20 106 150
Hilbert formula      119
Hilbert, D.      6 40 68 99 106 107 131 133
HK      10
Hobson, E.W.      40
Homogeneous equations      2 47 75
Homogeneous equations with Hermitian kernels      107 ff.
Homogeneous equations, associated      2
Identity operator      16
inf      131
Inner product      8
Integral equation      1
Integral equation of first kind      2 164
Integral equation of second kind      2
Integral equation, connexion with differential equations      3 ff.
Integral equation, Fredholm      3
Integral equation, homogeneous      2
Integral equation, linear      1
Integral equation, Volterra      2 31
Iterated kernels (iterates)      11
Iterated kernels of Hermitian kernels      107 f. 121
Iterated kernels of Volterra kernels      32

Iterated kernels, characteristic values      111 121
Iterated kernels, traces      82 ff. 109
Kaczmarz, S.      53
kernels      2
Kernels of finite rank      36
Kernels, $\mathfrak{L}^{2}$       11
Kernels, $\mathfrak{L}^{2}$ in the wide sense      14
Kernels, adjoint      20
Kernels, canonical      80 f.
Kernels, continuous      9 ff.
Kernels, definite      127
Kernels, Hermitian      20 106 150
Kernels, iterated      11
Kernels, norm      10 13
Kernels, normal      20 150
Kernels, rank      37
Kernels, resolvent      17
Kernels, symmetric      20
Kernels, transposed      20
Kernels, Volterra      32
Kneser, A.      107
Kx      10
Le Roux, J.      32
Lewis, D.C.      143
Linear, integral equation      1
Linear, operator      10
Linearly equivalent      61
Liouville — Neumann series      30
Liouville, J.      30
Littlewood, J.E.      7 69
Lovitt, W.V.      6 77 106
Martin, R.S.      98
Mean square, approximation      54
Mean square, convergence      56 f.
Mean square, distance      56
Mean square, limit      57
Mercer's theorem      127 f.
Mercer, J.      128
Metric      56
Michal, A.D.      98
Minimax properties of characteristic values      133 ff.
Minkowski's inequality      8
Mollerup, J.      143
Moore, E.H.      23
Negative definitive      127
Neumann series      26 30
Neumann series for Volterra kernels      32 ff.
Neumann, C.      30
Non-negative definite      127
Non-positive definite      127
Norm of a continuous kernel      10
Norm of a function      7
Norm of an $\mathfrak{L}^{2}$ kernel      13
Normal kernels      20 150
Normal, almost      151
Null function      6
Operator multiplication      10
Operator, identity      16
Operator, linear      10
Orthogonal functions      8
Orthogonalization process      61 f.
Orthonormal systems      53
Orthonormal systems of characteristic functions      103 113 152
Orthonormal systems of functions of two variables      62 ff.
Parseval formulae      61
Picard, E.      164
Plemelj, J.      98
Positive definite      127
Radon, J.      36
Rank of characteristic value      49
Rank of kernel      37
Rank, finite      36
Regular values      17
Regular values, relation with characteristic values      22 47 102
Regular values, relation with Fredholm determinant      71 97
Regular values, structure of set      30
Relatively uniform absolute convergence      23 ff.
Relatively uniform convergence      23 ff.
Relatively uniform convergence, relation with mean square convergence      58
Representable functions      116
Resolvent equation      17
Resolvent kernel      17
Resolvent kernel for canonical kernel      87
Resolvent kernel for Hermitian kernel      137 ff.
Resolvent kernel for kernel of finite rank      39
Resolvent kernel for normal kernel      159
Resolvent kernel for Volterra kernel      34
Resolvent kernel, Fredholm formulae      71 97
Resolvent kernel, functional equation      31
Resolvent kernel, meromorphic character      44 71 139
Riesz — Fischer theorem      58 59 62
Riesz, F.      32 58
Ruston, A.F.      98
Salem, R.      127
Schmidt, E.      36 62 106 107 142
Schwarz's inequality      7
Second kind, integral equation      2
Singular functions      142 ff.
Singular functions for Hermitian kernel      150
Singular system      145
Singular system for Hermitian kernel      150
Singular system for normal kernel      153
Singular values      142 ff.
Singular values for Hermitian kernel      150
Singular values for normal kernel      153 ff.
Smithies, F.      32 98 106 142
Steinhaus, H.      53
Strong convergence      57
Sup      7
Symmetric kernels      20 106
Titchmarsh, E.C.      71 90
Toeplitz, O.      127
Tonelli — Hobson extenstion of Fubini's theorem      12
Trace      79 f.
Trace of iterated kernels      82 ff. 109
Transposed kernel      20
Vergerio, A.      143
Volterra integral equations      2 31
Volterra kernel      32
Volterra, V.      32
Weyl, H.      58
Zaanen, A.C.      98
[u, v]      143
{f(x):P(x)}      131

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