Rice J.R. — Linear Theory. Volume 1. The approximation of functions :: Электронная библиотека попечительского совета мехмата МГУ

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Rice J.R. — Linear Theory. Volume 1. The approximation of functions

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Название: Linear Theory. Volume 1. The approximation of functions

Автор: Rice J.R.

Аннотация:

The central problem and theme of this book is the approximation of a real continuous function by an approximating function depending on a fixed finite number of parameters. Thus we consider a real and continuous function f(x) and an approximating function F(A, x) which may depend on the parameters A in a linear or nonlinear way. The problem is to determine those parameters A* which make F(A*, x) closest to f(x) by some measure of closeness. This book gives an exposition of the many facets of this theme, such as existence, uniqueness, and characteristic properties of the solutions to this problem.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1964

Количество страниц: 213

Добавлена в каталог: 12.11.2012

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Предметный указатель

$L_{1}$ -approximation      94
$L_{1}$ -approximation, canonical points      106
$L_{1}$ -approximation, characterization      102
$L_{1}$ -approximation, characterization on finite sets      114
$L_{1}$ -approximation, characterization, special case      105
$L_{1}$ -approximation, descent for finite point sets      165
$L_{1}$ -approximation, existence      94
$L_{1}$ -approximation, finite point sets      112
$L_{1}$ -approximation, linear programming      182
$L_{1}$ -approximation, polynomials      110
$L_{1}$ -approximation, tangent planes      96
$L_{1}$ -approximation, trigonometric sums      110
$L_{1}$ -approximation, uniqueness      104
$L_{1}$ -approximation, uniqueness and Tchebycheff sets      106
$L_{1}$ -approximation, weight functions      118
$l_{1}$ -norm      20 94
$L_{2}$ -approximation      30
$L_{2}$ -norm      20
$L_{p}$ -approximation      164
$L_{p}$ -approximation, nonlinear programming      186
$L_{p}$ -distance function      4 29
$L_{p}$ -norms      4
$L_{p}$ -norms, finite point sets      7
$L_{p}$ -norms, weighted      21
$\epsilon_{m}$ -algorithm      152 186
$\epsilon_{m}$ -algorithm, Pade table      153
Abel transformation      150
Abel — Poisson kernel      130
Aitken's $\delta^{2}$ -process      153
Alternation      54
Approximating functions      15
Approximation problem in $E_{m}$       11
Approximation problem, definition      2
Approximation problem, general approach      3
Approximation problem, Tchebycheff      52
Arithmetic means      128
Axial set      83
Bernstein kernel      130 131
Bernstein polynomials      131 145
Bessel's inequality      141
Best approximation      2
Canonical points      106
Canonical points, polynomials      112
Canonical points, trigonometric sums      111
Cesaro sums      128
Cesaro transformation      150 152 186
Christoffel — Darboux formula      37
Closure in C[0,1]      123
Computational methods      144
Condition E      27 75
Continued fractions      146
Continued fractions, conversion to rational functions      147
Continued fractions, telescoping procedures      156
Contraction mappings      169
Contraction mappings, descent mappings      189
Convex      13 95
de la Vallee Poussin algorithm      66
de la Vallee Poussin algorithm, rate of convergence      69
Degree of convergence      119 133
Degree of convergence with Lipshitz condition      137
Degree of convergence, continuous periodic functions      134
Degree of convergence, Jackson's theorem      140
Degree of convergence, remarks      141
Descent mappings      168 187
Descent mappings, convergence      170
Descent mappings, matrix iteration      187
Descent mappings, one-for-one exchange      187
Descent mappings, one-to-many mappings      187
Descent mappings, steepest descent      171
Deviation      52
Deviation, estimation      60
Dirichlet kernel      125 130
Distance function      2
Distance function, $L_{p}$       4 29
Dual program      182
Equicontinuous      44
Error curve      5 53
Euler's transformation      149
Exchange, duality with Zuhovickii method      185
Exchange, one-for-one      173
Exchange, Remes      176
Existence theorems      23
Existence theorems with Condition E      27
Existence theorems, Tchebycheff approximation      23
Extremal points      54 66
Fejer kernel      129 130 142 150
Fourier series      124 145
Gegenbauer polynomials      36
Gradient      159
Gram polynomials      41
Gram — Schmidt orthogonalization      46
Haar problem      87
Hermite polynomials      36 49
Hessian matrix      161
Hilbert matrix      47
Hoelder condition      67
Ill-conditioned      46
Independence assumption      3
Jackson kernel      130 137
Jacobi polynomials      36
Kernel functions      125 131
Kernel functions, peaks      132
Krawtchouk polynomials      41
Kummer's transformation      148
Laguerre polynomials      36 48
Landau kernel      130
Laurent series      145
Least squares      30
Least squares, approximations      49
Least squares, characterization of best approximation      33
Least squares, continuous dependence      51
Least squares, finite point sets      40
Least squares, normal equations      31
Legendre polynomials      36 39 47
Linear programming      180
Linear programming, constraints      181
Linear programming, dual program      182
Linear programming, objective function      181
Linear programming, simplex method      185
Linear space      10
Linear space, normed      29
Lipschitz condition      67
Lipschitz condition, second order      143
Lipschitz — Dini condition      142
Markov set      117
Measure      98
Measure of approximation      2
Method of ascent      171

Method of ascent, convergence      187
Method of ascent, descent mappings      173
Method of ascent, duality      183
Method of ascent, one-for-one exchange      173
Method of descent      158
Method of descent, $L_{p}$ -approximation      164
Method of descent, descent mappings      171
Method of descent, method of gradients      162
Method of descent, polytopes      164
Method of descent, rational functions      189
Method of descent, steepest descent      162
Method of descent, steepest descent for poly topes      165
Method of descent, Zuhovickii      166
Metric space      11
Modulus of continuity      141
Muntz's theorem      124
Nonlinear approximating functions      70
Normal equations      31
Norms      2
Norms, $L_{p}$       4
Norms, strict      14
Norms, Tchebycheff      8
Objective function      181
One-for-one exchange      173 187 188
Orthogonal functions      30 34
Orthogonal functions, finite point sets      40
Orthogonal polynomials      36
Orthogonal polynomials, Christoffel — Darboux formula      37
Orthogonal polynomials, classical      36
Orthogonal polynomials, classical, characterization of      50
Orthogonal polynomials, finite points sets      41
Orthogonal polynomials, recurrence relation      37
Orthogonal polynomials, zeros      39
Orthogonal system      33
Orthogonalization      45
Orthonormal systems      33
Orthonormal systems, equivalent      45
p-algorithm      153
Pade table      147 153 186
Parameter space      2 26
Parameter space, topology      26
Planes of support      96
Planes of support, uniqueness      100
Polya Algorithm      8
Polynomial approximation, $L_{1}$ -norm      110
Polynomials      15 120 188
Polynomials, evaluation      145
Polynomials, Gram      41
Polynomials, Krawtchouk      41
Polynomials, orthogonal      36
Polynomials, piece wise      16
Polynomials, Tchebycheff      41
Polynomials, Tchebycheff set      55
Polytope      113
Polytopes, method of descent      164
Property Z      71 82 87
Property Z, rational functions      78
Quotient-difference (QD) algorithm      152
Rational functions      16 74
Rational functions, approximation examples      93
Rational functions, Condition E      75
Rational functions, continued fractions      146
Rational functions, degree      78
Rational functions, existence of Tchebycheff approximations      77
Rational functions, method of descent      189
Rational functions, nonexistence of Tchebycheff approximations      76
Rational functions, Property Z      78
Rational functions, solvence      79
Rational functions, telescoping procedures      157
Recurrence relation      37 41
Remes algorithm      176 187
Remes algorithm, descent      177
Remes algorithm, Murnaghan and Wrench      189
Remes algorithm, unisolvent functions      178
Rhombus algorithms      151
Sequence transformations      148
Sigma, factors transformation      150
Simplex method      185
Simplex method, Zuhovickii method      186
Solvent      71 79
Spline functions      17
Taylor's series      144 180
Tchebycheff approximation      52
Tchebycheff approximation, characterization of best approximations      55
Tchebycheff approximation, continuous dependence      62
Tchebycheff approximation, descent for finite point sets      165
Tchebycheff approximation, finite point sets      65 66
Tchebycheff approximation, linear programming      181
Tchebycheff approximation, method of ascent      171
Tchebycheff approximation, polynomials      92 188
Tchebycheff approximation, uniqueness      60
Tchebycheff polynomials      41 48 93
Tchebycheff polynomials, polynomial evaluation      145
Tchebycheff sets      55 61 86 88 89 106
Tchebycheff sets, $L_{1}$ -approximation      94
Tchebycheff sets, Property Z      71
Tchebycheff sets, zeros      57 91
Tchebycheff type theory, existence      80
Tchebycheff type theory, existence and characterization      82
Tchebycheff type theory, uniqueness      87
Tchebycheff type theory, uniqueness, example of nonuniqueness      89
Telescoping procedures      154
Telescoping procedures with Tchebycheff polynomials      156
Telescoping procedures, continued fractions      156
Telescoping procedures, linear approximating functions      155
Telescoping procedures, rational functions      157
Transformations, linear      148
Transformations, sequence      147
Trigonometric sums      15
Trigonometric sums, orthogonality      35
Trigonometric sums, Tchebycheff set      55
Unisolvent functions      70 91 93
Unisolvent functions, examples      74
Unisolvent functions, Remes algorithm      178
Unisolvent functions, Tchebycheff approximation      73
Unit sphere      12
Weierstrass kernel      130
Weierstrass theorem      119
Weierstrass theorem, polynomials      121
Weierstrass theorem, trigonometric sums      122
Weight function      6 118
Zeros, double      57
Zeros, multiple      60
Zeros, orthogonal polynomials      39
Zeros, Property Z      71
Zuhovickii's method of descent      166 188
Zuhovickii's method of descent, duality with exchange      185
Zuhovickii's method of descent, simplex method      186

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