The direct and inverse problems on the degree of best approximation in Banach spaces

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Let X be a Banach space and B = {P_j : j = 0, ±1, ±2,…} a total, fundamental sequence of mutually orthogonal bounded linear projection operators of X into itself. For each nonnegative integer n, M_n strands for the linear span of {P_j(X) : |j| 【less than or equal】 n}. Let T^*_n be a family of bounded linear projection operators of X onto M_n and S a bounded linear operator of X into itself. Let S_n = Σ^n_<j=-n>P_j be the nth partial sum operator of the Fourier series Σ^∞_<j=-∞>P_j(f) (F ∈ X) with respect to B. Then I proved that S_n is an operator of best approximation to S from T^*_n, under certain suitable conditions. And I estimated the degree of approximation by convex sums of convolution type operators associated with a periodic type, strongly continuous group T of bounded linear operators of X into itself by means of the modulus of continuity with respect to T and established the direct and inverse theorems for approximation by the generalized Rogosinski operators. Furthermore, I … More applied these results to the best approximation of multiplier operators induced by B as well as to homogeneous Banach spaces which include the classical function spaces, as special cases.I introduced the integral operators in the space of X-valued bounded continuous functions on a metric space, and established the approximation theorem and the Korovkin-type convergence theorem for them. Moreover, I applied these results to interpolation type operators as well as convolution type operators. Several concrete approximate kernels are the Gauss-Weierstrass, Picad, Bui-Federov-Cervakov, Landau, Mamedov, de la Vallee-Poussin kernels, and so on.In the Banach lattice of all real-valued bounded continuous functions on a metric space, I established the Korovkin-type approximation theorem for nets of positive linear operators, and gave a quantitative version of this result by means of the modulus of continuity and higher order moments induced by systems of test functions. Furthermore, I applied these results to the multi-dimensional Bernstein, Szasz, Baskakov-type, Meyer-Konig and Zeller operators. Less