  Hyperbolic cross approximation is a special type of multivariate
approximation. Recently, driven by applications in engineering, biology,
medicine and other areas of science new challenging problems have appeared. The
common feature of these problems is high dimensions. We present here a survey
on classical methods developed in multivariate approximation theory, which are
known to work very well for moderate dimensions and which have potential for
applications in really high dimensions. The theory of hyperbolic cross
approximation and related theory of functions with mixed smoothness are under
detailed study for more than 50 years. It is now well understood that this
theory is important both for theoretical study and for practical applications.
It is also understood that both theoretical analysis and construction of
practical algorithms are very difficult problems. This explains why many
fundamental problems in this area are still unsolved. Only a few survey papers
and monographs on the topic are published. This and recently discovered deep
connections between the hyperbolic cross approximation (and related sparse
grids) and other areas of mathematics such as probability, discrepancy, and
numerical integration motivated us to write this survey. We try to put emphases
on the development of ideas and methods rather than list all the known results
in the area. We formulate many problems, which, to our knowledge, are open
problems. We also include some very recent results on the topic, which
sometimes highlight new interesting directions of research. We hope that this
survey will stimulate further active research in this fascinating and
challenging area of approximation theory and numerical analysis.
