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

设X是Banach空间，B={P_j：J=0，±1，±2，…}X到其自身的相互正交的有界线性投影算子的全基本序列。对于每个非负整数n，M_n链{P_j(X)：|j|[小于或等于]n}的线性跨度。设T^n是X到M_n上的有界线性投影算子族，S是X到M_n上的有界线性投影算子。设S_n=Σ^n_&lt；j=-n&gt；P_j是关于B的傅立叶级数Σ^∞_&lt；j=-∞&gt；P_j(F)(F∈X)的第n个部分和算子，在一定条件下证明了S_n是由T^*_n逼近S的算子。利用关于T的连续模估计了与周期型强连续的有界线性算子群T相关的卷积型算子的凸和对自身的逼近程度，并建立了广义Rogosinski算子逼近的正逆定理。此外，I…进一步将这些结果应用于B诱导的乘子算子的最佳逼近以及包括经典函数空间在内的齐次Banach空间，作为特例，我在度量空间上的X值有界连续函数空间中引入了积分算子，并建立了它们的逼近定理和Korovkin型收敛定理。此外，我将这些结果应用于插值型算子和卷积型算子。在度量空间上所有实值有界连续函数的Banach格中，建立了正线性算子网的Korovkin型逼近定理，并利用测试函数系诱导出的连续模和高阶矩给出了这一结果的量化形式.此外，我将这些结果应用于多维Bernstein、Szasz、Baskakov型、Meyer-Konig和Zeller算子。较少

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