On Chebyshev Subspaces in the Space of Multivariate Differentiable Functions

In the present paper we give a characterization of Chebyshev subspaces in the space of (real or complex) continuously-differentiable functions of two variables. We also discuss various applications of the characterization theorem. Introduction. One of the central problems in approximation theory consists in determining the best approximation; that is, in a normed linear space X with a prescribed subspace U we seek for each / in X its best approximation p in U satisfying ||/ — p|| = dist(/, U). If U is finite dimensional, then every element / in X possesses a best approximant. This raises the very important and delicate question of unicity of best approximation. The finite-dimensional subspace U is called a Chebyshev subspace of X if for each / in X its best approximant in U is unique. Chebyshev subspaces have been widely investigated in different functional spaces. In case of approximation with respect to the supremum norm these investigations were initiated by the classical works of Chebyshev and Haar. Let G(if ) denote the space of real or complex continuous functions endowed with the supremum norm on the compact Hausdorff space if. The celebrated Haar-Kolmogorov theorem gives a characterization of finite-dimensional Chebyshev subspaces of C(K): the n-dimensional subspace Un C G(if ) is a Chebyshev subspace of G(if ) if and only if it satisfies the so-called Haar property, i.e. each nontrivial element of Un has at most n -1 distinct zeros at if. (This result was first proved by Haar [6] in the real case, and then by Kolmogorov [7] in the complex case.) Later Mairhuber [9] showed that in the real case C(K) possesses a Chebyshev subspace of dimension n > 1 if and only if if is homeomorphic to a subset of the circle, i.e. the study of finite-dimensional Chebyshev subspaces of C(K) is, in fact, restricted in the real case to functions of one variable. It is natural to expect that requiring unicity only with respect to a smaller subspace may lead to the extension of the family of Chebyshev subspaces. This approach is well known from the theory of Li-approximation. In case of Chebyshev approximation such investigations were initiated by Garkavi [5] who gave a characterization of Chebyshev subspaces in the space of real continuously-differentiable functions endowed with the supremum norm on [a,b]. In a series of papers [1-3] the analog of Garkavi's result was given for the real rational families. In a recent paper [8] we characterized the finite-dimensional Chebyshev subspaces in the space Cl[a, b] of real or complex continuously-differentiable functions with supremum norm on [a,b]. In order to formulate this result we shall need Received by the editors April 28, 1983 and, in revised form, May 18, 1984. 1980 Mathematics Subject Classification. Primary 41A52. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 839 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use