In this paper we extend a number of important results of the classical
Chebyshev approximation theory to the case of simultaneous approximation of two
or more functions. The need for this extension is application driven, since
such kind of problems appears in the area of curve (signal) clustering. In this
paper we propose a new efficient algorithm for signal clustering and develop a
procedure that allows one to reuse the results obtained at the previous
iteration without recomputing the cluster centres from scratch. This approach
is based on the extension of the classical de la Vallee-Poussin's procedure
originally developed for polynomial approximation. In this paper, we also
develop necessary and sufficient optimality conditions for two curve Chebyshev
approximation, that is our core tool for curve clustering. These results are
based on application of nonsmooth convex analysis