On the unicity of bestL 1-approximation by polynomials of several variables

Abstract

In the present paper we shall consider certain problems of best mean approximation for functions of several variables. It is wellknown that there is an essential difficulty in the extension of the theory of best Chebyshev approximation to the multivariate case. The basic problem here is the absence of Haar subspaces of functions of more than one variable. However, this fact seems not to be an insuperable obstacle in developing the theory of best Lt-approximation of functions of several variables. Some first steps in this direction have been made in [1]. In particular, it was shown in [1] that the L~-approximation of continuous functions of two variables by the tensor product of a two-dimensional Haar subspaces and an arbitrary Haar subspace is unique. This result gives some hope that the general problem of unicity of best Ll-approximation of continuous functions of several variables by tensor products ofHaar subspaces can be solved positively. Nevertheless, this problem is still unsolved even in the case of ordinary polynomials. In the present paper we continue the study of uniqueness of multivariate L~-approximation presenting another approach to this problem. We shall consider approximation by arbitrary multivariate polynomials but restrict ourselves only to approximating rational functions. The main advantage of this approach is that it includes the study of the unicity of polynomials of several variables with fixed leading coefficients which have minimal Ll-deviation from zero. In case of Chebyshev approximation this problem was considered in [2], [6] and [7]. The results of our paper imply the unicity of the polynomials of least Ll-deviation having fixed leading coefficients. Thus we show that the solution of the Korkin--Zolotarjov type extremal problem is unique in the multivariate case, too. For rectangular regions this unique solution is given by the product of Chebyshev polynomials of second kind.