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By Erkus E., Duman O.

During this paper, utilizing the concept that ofA-statistical convergence that is a regular(non-matrix) summability technique, we receive a common Korovkin kind approximation theorem which matters the matter of approximating a functionality f via a series {Lnf } of confident linear operators.

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X, are independent, then D r + X,] i = 1Xi = r li J i D[Xi]. If these X, have the same covariance matrix C, then D L T Xi] = nC, i 1 and the sample mean vector X = - Xi has covariance matrix D[%] = ni 1 ( : y n Z = Z, a familiar formula in the univariate case. -[: 0 -1 -:I. 2: Let X,, X,, . . , X, be independent random variables, all with variance (1'. Define = XI+ . + X, for k = 1,. . , n. Find DCY] for Y' = (Y,, . . , Also find R[Y], the correlation matrix. x). 's X and Y such that X and Y have covariance 0, but are not independent.

Suppose that k p(yl V ) = 1 p(ylxj) for every vector y E R Prove that xl,. . ,x, are mutually j- 1 orthogonal. Hint: Consider the vector y = x i for each i. Define the f(x)y(x) dx. Such an inner product space, with the correct definition of the integral, and a more subtle property called completeness, is called a Hilbert space after the great German mathematician, David Hilbert, of the late nineteenth and early twentieth centuries. X is not finite dimensional, but our projection theory still applies because we will be interested in projections on finite dimensional subspaces.

Let V = Vo @ VR@ 4. Then p(y1 I/) = 4, + gR + Ec, where j 0 = p(yl Vo) = I.. - y . ) R i , and 9c = p(yI vf) = C ( j . j- j.. 4: The null space of an m x n matrix A is the collection of vectors x E R , such that Ax = 0. We denote this null space by N(A). The column (or range) space of A is C(A) = f x l x = Ab for some b}. 8: Let A be an rn x n matrix. 1) Proof: w E N ( A ) o w 1 (row space of A) o w I(column space of A') o w E C(A')'. 1) follows by taking complements on -. : I both sides. -. 9: Let X be an n x k matrix.

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


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