  String-averaging is an algorithmic structure used when handling a family of
operators in situations where the algorithm at hand requires to employ the
operators in a specific order. Sequential orderings are well-known and a
simultaneous order means that all operators are used simultaneously (in
parallel). String-averaging allows to use strings of indices, constructed by
subsets of the index set of all operators, to apply the operators along these
strings and then to combine their end-points in some agreed manner to yield the
next iterate of the algorithm. String-averaging methods were discussed and used
for solving the common fixed point problem or its important special case of the
convex feasibility problem. In this paper we propose and investigate
string-averaging methods for the problem of best approximation to the common
fixed point set of a family of operators. This problem involves finding a point
in the common fixed point set of a family of operators that is closest to a
given point, called an anchor point. We construct string-averaging methods for
solving the best approximation problem to the common fixed points set of either
finite or infinite families of firmly nonexpansive operators in a real Hilbert
space. We show that the simultaneous Halpern-Lions-Wittman-Bauschke algorithm,
the Halpern-Wittman algorithm and the Combettes algorithm, which were not
labeled as string-averaging methods, are actually special cases of these
methods. Some of our string-averaging methods are labeled as static because
they use a fixed pre-determined set of strings. Others are labeled as
quasi-dynamic because they allow the choices of strings to vary, between
iterations, in a specific manner and belong to a finite fixed pre-determined
set of applicable strings.
