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Subject Index Alternant 16, 21 Alternant theorem 20, 144 - for rational approximation 1 61 , 162 Alternants, results concerned with 101 Anomalies in rational approximation

163, 165 Approximation on small intervals 86,

167 - by polynomials in several variablcs

71, 103, 104 - in the space Lq 129 Approximation results for the Fejer

operator 48 Approximation theorems in the

complex plane 10 Approximation theorems of ACHIESER

9, 10 of KOROVKIN 6 of MUNTZ 9 of STONE 9 of WALSH 11, 12 of WEIERSTRASS 7, 8

Asymptotic results for polynomial approximations 90

--- for rational approximations 167, 168

Asymptotic convexity 136, 138

Basic polynomials 73 BERNSTEIN, the theorems of 57 Bernstein operator 7, 68 Bessel function 96 Best approximation 1

Convergence theorem for the Remez algorithm 108

- of VElD INGER 111

Dense systems 5 Direct methods 122 Discretization 124

Error integral 175 Exchange algorithms 106, 107 Existence theorem of RICE for

exponential approximation 179

Existence theorem of RICE for linear approximation 1

- for rational approximation 156, 158

Exponential approximation 176 Exponential type, entire functions of

67 Extremal point 14

Fejer operator 47 Fixed point theorem of BROUWER 27,

158 - of SCHAUDER 27

Gram determinants 4

Haar condition 16 -,local 142 HAHN-BANACH, theorem of 5 Heat equation 151 Hilbert space 2

Inequalities of BERNSTEIN 57, 62, 67 - of A. A. MARKOFF 67 - of W. A. MARKOFF 67 Initial approximations for exponential

approximation 181 - for linear approximation 116 Interpolation formula of LAGRANGE

73 Invariants 26, 154, 157 Iterative methods of REMEZ 105-116

KOLMOGOROFF, the theorem of 15 - Generalization of the theorem of

133, 134, 136 KOROVKIN, operators of 50

Linear functionals 4 Linear programming 125, 126 Lipschitz-condition 48 Lower bounds for En (I) 82

Maehly's method of rational approximation 174

Maximal linear functionals 4

198 Subject Index

Maximum-minimum methods 128 Mean square error method, with

bounded absolute value 130 Modulus of continuity 51 Moment problem 182

Newton iteration method 113, 114, 149 Non-linear approximation 131 Normal equations 20 Numerical methods of linear

Tchebycheff approximation 105 - of rational Tchebycheff approxi­

mation 170

Operator of BERNSTEIN and ROGOSINSKI 66

- of DE LA VALLEE-POUSSIN 66

Pre-iteration 121 Polynomial approximation 45, 46, 72 Polynomial operators, linear 45 Projection operator 45

Rational approximation 154 Reality theorem of WERNER 172 Reference sets 124 Reference deviation 125 Reference functions 125 Regularity ellipse 90 REMEZ, bound of 83 RICE, the investigations of 148

Simplex method 126 Simultaneous exchange method 107 Stiefel method 72, 124 Strict convexity 2 Strictly normed 2

Tchebycheff norm 13 Tchebycheff polynomials 31 Tchebycheff systems 28 Telescoping method 31, 122 Trigonometric approximation 45

Uniqueness theorem of COLLATZ 24 - for exponential approximation

178 - of Haar 16, 146 - for rational approximation 161 - of RIVLIN and SHAPIRO 24 Unisolvency 148 Upper bounds for En (I) 77

DE LA VALLEE-POUSSIN, bound of 82 Varisolvency 149 Vector valued functions 28

Walsh approximation theorem for rational approximation 158

Walsh table 162

Zolotareff polynomials 42, 44 ZYGMUND, the theorems of 57

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