  We present a dimension-incremental algorithm for the nonlinear approximation
of high-dimensional functions in an arbitrary bounded orthonormal product
basis. Our goal is to detect a suitable truncation of the basis expansion of
the function, where the corresponding basis support is assumed to be unknown.
Our method is based on point evaluations of the considered function and
adaptively builds an index set of a suitable basis support such that the
approximately largest basis coefficients are still included. For this purpose,
the algorithm only needs a suitable search space that contains the desired
index set. Throughout the work, there are various minor modifications of the
algorithm discussed as well, which may yield additional benefits in several
situations. For the first time, we provide a proof of a detection guarantee for
such an index set in the function approximation case under certain assumptions
on the sub-methods used within our algorithm, which can be used as a foundation
for similar statements in various other situations as well. Some numerical
examples in different settings underline the effectiveness and accuracy of our
method.
