A straightforward generalization of Diliberto and Straus' algorithm does not work

An algorithm for best approximating in the sup-norm a function f C0, 1]² by functions from tensor-product spaces of the form π_k C0, 1] + C0, 1] π_l, is considered. For the case k = L = 0 the Diliberto and Straus algorithm is known to converge. A straightforward generalization of this algorithm to general k, l is formulated, and an example is constructed demonstrating that this algorithm does not, in general, converge for k² + l² > 0.