On Optimal Interpolation by Linear Functions on n-Dimensional Cube

Let , and let be the unit cube . By we denote the space of continuous functions with the norm by - the set of polynomials of variables of degree (or linear functions). Let be the vertices of an -dimnsional nondegenerate simplex . The interpolation projector corresponding to the simplex is defined by the equalities The norm of as an operator from to can be calculated by the formula Here are the basic Lagrange polynomials with respect to is the set of vertices of . Let us denote by the minimal possible value of Earlier the first author proved various relations and estimates for values and , in particular, having geometric character. The equivalence takes place. For example, the appropriate according to dimension inequalities can be written in the form If the nodes of a projector coincide with vertices of an arbitrary simplex with maximum possible volume, then we have When an Hadamard matrix of order exists, holds In the present paper, we give more precise upper bounds of for . These estimates were obtained with application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements Also we systematize and comment the best nowaday upper and low estimates of for concrete n.