On the Chebyshev rank of multivariate splines in L1-approximation

Weighted L-1-approximation of continuous multivariate real-valued functions by finite-dimensional subspaces of multivariate splines of degree m >= 1 is studied. Lower and upper bounds for the Chebyshev rank, i.e., for the largest dimension of the sets of best L-1-approximations, are established. The exact value of the Chebyshev rank of linear splines is determined. Our investigations extend known results for bivariate splines to the multivariate case. (C) 2012 Elsevier Inc. All rights reserved.