T. Hangan ELASTIC STRIPS AND DIFFERENTIAL GEOMETRY

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188 L. Rempulska – Z. Walczak 1.2. In [1], [2] and [7] (also [3], [4]) were examined approximation properties of the Szász- Mirakyan operators ∑ ∞ S n ( f ; x) := e −nx (nx) k ( k f k! n) and the Baskakov operators V n ( f ; x) := k=0 k=0 ∞∑ ( ) ( n − 1 + k k x k k (1 + x) −n−k f , n) x ∈ [0,∞], n = 1, 2,... , for functions f continuous on the interval [0,∞]. The results given in [2] show that for every r-th times (r ≥ 2) differentiable function f we have |S n ( f ; x) − f (x)| = O x (n −1) , |V n ( f ; x) − f (x)| = O x (n −1) , for n ∈ N and every x ≥ 0, i.e. the order of approximation of f by S n ( f ) and V n ( f ) is independent on differential properties of functions f if r ≥ 2. 1.3. In this paper we shall introduce the certain class of linear operators of the Szász- Mirakyan and Baskakov type L n,r ( f ; A; x) = ∞∑ a nk (x) k=0 r∑ j=0 f ( j) ( k n ) in the space of r-th times differentiable functions f . For these operators we shall define the strong differences k=0 j=0 j! ( x − k n) j , ⎧ ⎨ ∞∑ Hn,r( q r∑ f f ; A; x) = a nk (x) ( j) ( ) k ( n ⎩ x − k j q − f (x) ∣ j! n) ⎫ ⎬ ∣ ⎭ with q > 0 and we shall prove that H q n,r( f ; A; x) = o x (n −r/2) as n → ∞, at every x ≥ 0 and q > 0. We can verify that the formula (6) of L n,0 ( f ) contains the Szász-Mirakyan and Baskakov operators S n ( f ) and V n ( f ). 1/q
The strong approximation 189 From results given in Sections 2 and 3 we can deduce that introduced operators L n,r , r ≥ 2, have better approximation properties than classical Szász-Mirakyan and Baskakov operators S n ( f ) and V n ( f ). The order of approximation of r-th times differentiable function f by L n,r ( f ) improves if r grows. Moreover we shall observe that the order of approximation is independent on q > 0 and if q = 1, then the result on strong approximation implies identical result for ordinary approximation of f by L n,r ( f ). 2. Definitions and preliminary properties 2.1. Let C B be the space of all real-valued functions f uniformly continuous and bounded on R 0 = [0,∞) with the norm (1) ‖ f ‖ ≡ ‖ f (·) ‖ := sup x∈R 0 | f (x)|. For f ∈ C B we shall consider the modulus of continuity (2) ω( f ; t) := sup ‖ h f (·)‖, t ≥ 0, 0≤h≤t where h f (x) := f (x + h) − f (x). It is known ([3]) that ω( f ; λt) ≤ (1 + λ)ω( f ; t) for λ, t ≥ 0 and lim t→0+ ω( f ; t) = 0 for every f ∈ C B . Let r ∈ N 0 and let C r B be the class of all functions f ∈ C B having the derivatives f ′ ,..., f (r) ∈ C B . The norm in C r B is given by (1) (C0 B ≡ C B). 2.2. Denote by the set of all infinite matrices A = [a nk (x)] n∈N, k∈N0 , N ∈ {1, 2,...}, of functions a nk ∈ C B having the following properties: (i) a nk (x) ≥ 0 for x ∈ R 0 , n ∈ N, k ∈ N 0 , (ii) ∑ ∞ k=0 a nk (x) = 1 for x ∈ R 0 , n ∈ N, (iii) for every p ∈ N the series ∑ ∞ k=0 k p a nk (x) is uniformly convergent on R 0 and its sum p (·; A) is function depending on p and A such that (1 + x p ) −1 p (x; A) belongs to the space C B . (iv) for every p ∈ N there exists a positive constant M 1 (p, A) depending on p and A such that the function T n,2p (x; A) := ∞∑ ( ) k 2p a nk (x) n − x , x ∈ R 0 , n ∈ N, k=0
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T. Hangan ELASTIC STRIPS AND DIFFERENTIAL GEOMETRY

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