$$K_{a}$$ K a -convergence and Korovkin type approximation
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K a -convergence and Korovkin type approximation Sevda Orhan1 · Kamil Demirci1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract In the present paper, we study a Korovkin type approximation theorem in the setting of K a -convergence that contains the classical result. We also study the rate of K a convergence and afterwards, we give some concluding remarks. Keywords K a -convergence · Positive linear operator · Korovkin theorem · Statistical convergence · Almost convergence Mathematics Subject Classification 40A35 · 41A25 · 41A36
1 Introduction The notion of statistical convergence was first introduced by Fast [3] and Steinhaus [12], independently and studied by many authors [4,5,14]. Let E be a subset of N, the set of natural numbers, then the natural density of E, denoted by δ(E), is given by: 1 δ(E) := lim |{k ≤ n : k ∈ E }| n n whenever the limit exists, where |B| denotes the cardinality of the set B [11]. Then a sequence x = (xk ) of numbers is statistically convergent to l provided that, for every ε > 0, δ ({k ∈ N : |xk − l| ≥ ε}) = 0. In this case we write st − lim xk = l [3,12]. We note that every convergent sequence is statisk
tically convergent, but not conversely. Also, as it is well known, statistical convergence and almost convergence [9,10] overlap. These convergence methods provide interesting results in summability theory and play a vital role not only in pure mathematics but also in other
B 1
Sevda Orhan [email protected] Department of Mathematics, Sinop University, Sinop, Turkey
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S. Orhan, K. Demirci
branches of science involving mathematics. Another interesting convergence method is K a convergence [8]. If the sequence x = (xk ) is convergent and ∞ k=1 |ak | < ∞, then it is K a -convergent, too. It is now natural to ask whether these three convergence methods contain each other. Our answer is no, i.e. these methods are overlap, neither contains the other. In the present paper, we study a Korovkin type approximation theorem via K a -convergence and the rate of K a -convergence and afterwards, we give some concluding remarks. We now turn our attention to the K a -convergence: The idea of K a -convergence was defined by Lazic and Jovovic in 1993 [8], which is obviously associated to the matrix ⎞ ⎛ a1 0 0 0 . . . ⎟ ⎜ a2 a1 0 0 ⎟ ⎜ ⎟. ⎜ A = ⎜ a3 a2 a1 0 ⎟ ⎠ ⎝ . . Let a = (ak ) and x = (xk ) be number sequences, set K a (x) = y, where y = (yk ) and k yk = i=1 ak−i+1 xi (k = 1, 2, 3, . . .), then we say that y = (yk ) is the K a -transformation of the x = (xk ). Definition 1.1 [8] The sequence x = (xk ) of real numbers is said to be K a -convergent to the number l if, its K a -transformation y = (yk ) converges to the number l, i.e. lim yk = l. k
This limit is denoted by K a − lim xk = l. k
Proposition 1.2 [8] Let a = (ak ) be a number sequence and the series convergent, i.e. ∞ |ak | < ∞.
ak be absolutely (1.1)
k=1
(i) If x = (xk ) is convergent, lim xk = l and the condition (1.1) is satisfied then, k
K a − lim xk = l k
∞ ak . k=1
(ii) A convergence method K a is regul
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