Tkachenko M. The questions of uniqueness of elements of the best approximation in integral metric

The object of the research is approximations and non-symmetric approximations of continuous functions in integral metric. Characterizations of unicity subspaces and unicity subspaces for the best non-symmetric approximation for continuous functions in metric of space L1 are the aim of the research. Modern methods of theory of functions, mathematical and functional analysis, approximation theory have been used in this paper. The characterization of subspaces of uniqueness of element of the best L1-approximation and of the best non-symmetric approximation for continuous functions on a metric compact with values in Banach space in the term of classes of "test" functions is obtained. The existence of element of the best approximation in the week finite dimensional subspace for continuous functions on a metric compact with values in separable Banach space is proved. The dual theorems for the best non-symmetric and one-sided approximations of elements of KB-space are proved. The new characterization of unicity subspaces for real-valued continuous functions by the set of linear combinations of finite set of basis functions with coefficieht constraints is obtained. The results are new. They have been published in leading scientific editions and they can be used in further investigations of problems of uniqueness of the best L1-approximant.