Onmi contains lately accomplished domestic/international tasks, that are categorised by way of topic. It presents the perception into the tendencies of global structure via numerous drawings, skeches and pictures.
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Then Lf − f + ∞ ≤ f ∞ · L1 − 1 1 ω1 f , D1 2 ∞ + f ∞ (M − m) + D1 + (L (t − ·)) (·) D12 . 3. 12) D12 , 4 (M − m) where D1 = (L (|t − A|)) (A) . 5 (, p. 421) Let L be a positive linear operator from C ([m, M]) into itself. Here D2 := Let f ∈ C 2 ([m, M]). Then L (t − ·)2 (·) 1 2 ∞ < +∞. 2 Main Results 53 Lf − f ∞ ≤ f ∞ f + + · L1 − 1 ∞ ∞ + f L (t − ·)2 (·) 2 (L (t − ·)) (·) ∞ ∞ ∞ D2 D22 (M − m) + + . 5. Then (L f ) (A) − f (A) ≤ f (A) + + (L1) (A) − 1 H + f (A) f (A) 2 1 ω1 f , D2 D2 2 L (t − A)2 (A) D2 D22 (M − m) + + , 3 2 4 (M − m) where L (t − A)2 D2 = (L (t − A)) (A) (A) 1 2 .
25 Next we assume that the spectrum of A is [0, 1]. For example, it could be A f = x f (x) on L 2 ([0, 1]) which is a self adjoint operator and it has spectrum [0, 1]. ) (2n)! 66) (a − k), a ∈ R, b ∈ N, n ∈ N, x ∈ [0, 1] . 27 (, p. 75) Let f ∈ C ([0, 1]), n ∈ N. 67) ∀ x ∈ [0, 1], where c1 > 0 is an absolute constant independent of n, f and x. 28 Let f ∈ C ([0, 1]), n ∈ N. Then Sn − Mn−1,−1 (A) = Sn − Mn−1,−1 ∞ ≤ c1 ω4 f, 4 3 . 68) References 1. G. Anastassiou, Self Adjoint Operator Korovkin Type and Polynomial Direct Approximations with Rates, (2016, submitted) 2.
102) z (1 − z) , ∀ z ∈ [0, 1]. N Here we consider f ∈ C ([a, b]), and the general Bernstein positive linear polynomial operators from C ([a, b]) into itself, defined by (see , p. 80) 22 1 Self Adjoint Operator Korovkin Type Quantitative Approximation Theory N (B N f ) (s) = i=0 N i s−a b−a (b − a) N a+i f i b−s b−a N −i , ∀ s ∈ [a, b]. 103) By , p. e. B N f → f , uniformly, as N → ∞, ∀ f ∈ C ([a, b]), the convergence is given with rates. e. B N (1) = 1. 106) = 1 − y. N i a+ f i (b − a) N y i (1 − y) N −i .