Error estimates for two-dimensional cross approximation

In this paper we deal with the approximation of a given function ff on [0,1]2[0,1]2 by special bilinear forms ∑i=1kgi⊗hi via the so-called cross approximation. In particular we are interested in estimating the error function f−∑i=1kgi⊗hi of the corresponding algorithm in the maximum norm. There is a large amount of publications available that successfully deal with similar matrix algorithms in applied situations, for example in connection with HH-matrices (see Boerm and Grasedyck (2003) [9] or Hackbusch (2007) [16] for many references). But as they do not give satisfactory error estimates, we concentrate on the theoretical issues of the problem in the language of functions. We connect it with related results from other areas of analysis in a historical survey and give a lot of references. Our main result is the connection of the error of our algorithm with the error of best approximation by arbitrary bilinear forms. This will be compared with the different approach in Bebendorf (2008) [6].