On the Chebyshev rank of multivariate splines in L1-approximation

Weighted L1-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 L1-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.