Chebyshev approximations by least squares method

We consider the problem of linear approximation in the form of the minimization problem of the weighted Chebyshev norm, and that in the form of the minimization problem of the weighted Euclidean norm of the residual vector. We give an algorithm for the unambiguous calculation in all cases of the Chebyshev approximation that does not require the Haar condition. The theorem obtained indicates that any approximation by the method of least squares (for any set of positive weight coefficients in the minimized Euclidean norm) can be represented as the Chebyshev approximation based on the choice of weight coefficients in the Chebyshev norm. As an example we consider the approximation of the reduced fuel supply costs of a settlement based on an energy plantation as a quadratic dependence on volumes of reserved funds.