  Adaptive approximation (or interpolation) takes into account local variations
in the behavior of the given function, adjusts the approximant depending on it,
and hence yields the smaller error of approximation. The question of
constructing optimal approximating spline for each function proved to be very
hard. In fact, no polynomial time algorithm of adaptive spline approximation
can be designed and no exact formula for the optimal error of approximation can
be given. Therefore, the next natural question would be to study the asymptotic
behavior of the error and construct asymptotically optimal sequences of
partitions. In this paper we provide sharp asymptotic estimates for the error
of interpolation by splines on block partitions in IRd. We consider various
projection operators to define the interpolant and provide the analysis of the
exact constant in the asymptotics as well as its explicit form in certain
cases.
