References

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10 Index A analytic function, 46, 124 Appell polynomials, 52 asymptotic formula, 124, 158, 168 asymptotic order, 47 B Baskakov operators, 3, 96, 102, 166 Baskakov Kantorovich operators, 77 Baskakov Szász operators, 145, 165 Berens Lorentz lemma, 25, 31 Bernstein operator, 37 Bernstein polynomials, 2 Bernstein type inequalities, 31, 59 Bernstein Durrmeyer operators, 20 Binet s formula, 132 bounded variation, 122, 169 C Cauchy Schwarz inequality, 50, 58, 92, 111, 116 central moments, 4, 8, 10 Chlodovsky polynomials, 108 commutativity, 9 complete asymptotic expansion, 48 confluent hypergeometric function, 10, 138 convergence, 13 D differential operator, 31 Ditzian Totik modulus, 67 Durrmeyer variants, 6 E eigen-functions, 13 equivalence result, 28 exponential growth, 163 F Fibonacci numbers, 131, 132 G genuine operators, 159 golden ratio, 132 Grüss Voronovskaja, 169 H Hardy s inequality, 29 hybrid operators, 146, 164 hypergeometric series, 135 I interpolation, 32 inverse PKolya Eggenberger distribution, 148 inverse Pólya Eggenberger distribution, 134 inverse theorem, 32, 72 iterative combinations, 10 J Jackson-type estimate, 16 Jakimovski Leviatan operators, 52 Springer International Publishing AG 2017 V. Gupta, G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Developments in Mathematics 50, DOI /

11 186 Index K K-functional, 19, 51 Kantorovich variants, 5 L Lagrange form, 41 Laguerre polynomials, 11 Laplace transform, 18 Leibniz rule, 61 linear combination, 59 linear combinations, 5, 7, 10, 12, 18, 27, 30, 33, 84 linear functions, 84 linear interpolant, 55 linear positive operators, 2, 92, 100 link Phillips operators, 164 linking Baskakov operators, 169, 172 linking Szász Mirakjan operators, 171 Lipschitz function, 47 local and global saturation results, 31 Lupaş operators, 105 Lupaş Durrmeyer operators, 155 Lupaş Szász type operators, 165 M modified Phillips operators, 162 moduli of smoothness, 35, 43, 68 modulus of continuity, 15, 20, 90, 99 modulus of smoothness, 15, 19 moments, 118 O operators of Srivastava Gupta, 164 Q quantitative estimate, 50 R rate of convergence, 75, 169 S saturation result, 29 Schoenberg operator, 55 Schoenberg spline operator, 54 simultaneous approximation, 18, 159, 164, 166 Skeckin inequality, 35 Steklov mean, 62 Steklov means, 30 Stirling numbers, 138, 140 strong converse inequality, 58 summation integral type operators, 164 Szász Mirakjan operators, 3, 94, 103 Szász Mirakjan Baskakov operators, 22 Szász Mirakjan Durrmeyer operators, 21, 33, 64 Szász Mirakjan Laguerre operators, 13 T Taylor formula, 39 U unbounded function, 61, 144 uniform convergence, 14, 55 upper bounds, 61 P Phillips operator, 9 Phillips operators, 29, 98, 104, 165, 167 piecewise continuous, 49 piecewise linear interpolant, 55 Pochhammer symbol, 153 point-wise convergence, 37 Polya distribution, 117 polynomial growth, 99 Post Widder operators, 19 V variation-diminishing operator, 54 Voronovskaja theorem, 42, 93 Voronovskaja type, 37, 62 W weighted modulus, 92 weighted modulus of continuity, 90, 144, 145