Statistical approximation by positive linear operators. (English) Zbl 1049.41016

The sequences of some classical approximation operators tend to converge to the values of the function they approximate, except perhaps at points of discontinuity, where in several cases such sequences do not converge to any value. Statistical convergence, which is a regular non-matrix summability method, has revealed effective to correct the lack of convergence. Using \(A\)-statistical convergence, the authors prove a Korovkin type approximation which concerns the problem of approximating a function \(f\) by means of a sequence \((T_n)\) of positive linear operators acting from a weighted space \(C_{\rho_1}\) into a weighted space \(B_{\rho_2}\). Let \(A = (a_{jn})\) be an infinite summability matrix. For a given sequence \(x = (x_n)\), the \(A\)-transform \(Ax := ((Ax)_j)\) of \(x\) is given by \((Ax)_j = \sum_{n=1}^\infty a_{jn}x_n\), provided the series converges for each \(j\). It is said that \(A\) is regular if \(\lim_j (Ax)_j = L\) whenever \(\lim_j x_j = L\). If \(A\) is a non-negative regular summabilitiy matrix and \(K\) is a subset of N \(:= \{1,2,...\}\), then the \(A\)-density of \(K\) is defined as \(\delta_A(K) = \lim_j \sum_{n=1}^\infty a_{jn} \chi_K (n)\) provided the limit exists, where \(\chi_K\) is the characteristic function of \(K\). A sequence \(x = (x_n)\) is said to be \(A\)-statistically convergent to a number \(L\) (st\(_A\)-\(\lim x = L\)) if, for every \(\varepsilon > 0\), \(\delta_A (\{n \in\) N\(:\, | x_n - L| \geq \varepsilon\}) = 0\). The concept of \(A\)-statistical convergence can be given for sequences \((x_n)\) in a normed space. Let R denote the set of real numbers. A weight function is a continuous function \(\rho :\)R \(\to [1,+\infty )\) tending to \(\infty\) as \(| x| \to \infty\). In such a case, the weighted space \(B_\rho\) is the space of real-valued functions \(f\) defined on R satisfying \(| f(x)| \leq M_f \rho (x)\) for all \(x \in\)R, where \(M_f\) is a constant depending on \(f\). The weighted subspace \(C_\rho\) is \(C_\rho := \{f \in B_\rho: \, f\) is continuous on R\(\}\). The spaces \(B_\rho\) and \(C_\rho\) are Banach spaces with the norm \(\| f\| _\rho = \sup | f| /\rho\), where the supremum is taken over R. A classical Korovkin type approximation theorem asserts the following: Assume that \(\rho_1, \, \rho_2\) are weight functions satisfying \((*)\) \(\lim_{| x| \to \infty} \rho_1(x)/\rho_2(x) = 0\) and that \(T_n: C_{\rho_1} \to B_{\rho_2}\) (\(n \geq 1\)) is a sequence of positive linear operators. Then \(\lim_n \| T_nf - f\| _{\rho_2} = 0\) for all \(f \in C_{\rho_1}\) if and only if \(\lim_n \| T_n F_\nu - F_\nu\| _{\rho_1} = 0\) for \(\nu = 0,1,2\), where \(F_\nu (x) := x^\nu \rho_1(x)/(1 + x^2)\). The authors prove an extension of the last theorem with the ordinary limit operator replaced by an \(A\)-statistical limit operator. Specifically, they demonstrate the following. Let \(A = (a_{jn})\) be a non-negative regular summability matrix and let \((T_n)\) as before, where the weight functions \(\rho_1, \, \rho_2\) satisfy \((*)\). Then st\(_A\)-\(\lim_n \| T_nf - f\| _{\rho_2} = 0\) for all \(f \in C_{\rho_1}\) if and only if st\(_A\)-\(\lim_n \| T_n F_\nu - F_\nu\| _{\rho_1} = 0\) for \(\nu = 0,1,2\). If, in addition, \(\varphi\) is a continuous increasing function on R and \(\rho_1(x) = 1 + \varphi^2 (x)\), then the last property is equivalent to st\(_A\)-\(\lim_n \| T_n \varphi^\nu - \varphi^\nu\| _{\rho_1} = 0\) for \(\nu = 0,1,2\).


41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
47B38 Linear operators on function spaces (general)
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