Erkus E., Duman O.'s A -Statistical extension of the Korovkin type approximation PDF

By Erkus E., Duman O.

During this paper, utilizing the idea that ofA-statistical convergence that's a regular(non-matrix) summability procedure, we receive a basic Korovkin style approximation theorem which matters the matter of approximating a functionality f via a chain {Lnf } of confident linear operators.

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G 1 need not be continuous in general. Fortunately, we have at our disposal a powerful theorem by Ellis [65, 66] that settles the matter in most cases of interest to us. 4 (Ellis’s Theorem). Let G be a locally compact space that is algebraically a group. Suppose further that multiplication in G is separately continuous. Then G is a topological group. 4 in its general form. 5. Let S be a locally compact topological semigroup. If S is algebraically a group, then it is a topological group. 2. 2. Let S be a locally compact topological semigroup.

Let M; N 2 G, a compact group of nonnegative matrices. Let P be of rank k. 14 and its proof, MN 1 is the identity permutation. 22. The only compact subgroup of

The following matrix semigroups are abelian: 9 80 1 ˇ = < 1 a b ˇ ˇ (i) S D @0 1 aA ˇ a 0; b 2 < , ˇ ; : 0 0 1ˇ ) ( à ˇ a b ˇ (ii) S D ˇ a; b 2 < , b a ˇ 9 80 1 ˇ = < a b c ˇ ˇ (iii) S D @c a b A ˇ a; b; c 2 < . ˇ ; : b c a (i) does not admit a completely simple minimal ideal: S is not even simple, since the entry a at (1,2) and (2,3) is increasing; if a > 0, the corresponding element generates an ideal properly contained in S . <; C/. The minimal ideal in (ii) is f0g. S has divisors of zero, for example,  a a a a Ã à  à a a 0 0 D : a a 0 0 Idempotents of rank 1 are  and  à 1=2 1=2 1=2 1=2 1=2 1=2 à 1=2 : 1=2 Subsemigroups of matrices of rank Ä 1 is not 0-simple, since  1=2 1=2 à  à 1=2 1=2 1=2 …S : 1=2 1=2 1=2 (iii) S has of course a zero.

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A -Statistical extension of the Korovkin type approximation theorem by Erkus E., Duman O.

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