This book provides a systematic survey of classical and recent
results on hyperbolic cross approximation. Motivated by numerous
applications, the last two decades have seen great success in
studying multivariate approximation. Multivariate problems have
proven to be considerably more difficult than their univariate
counterparts, and recent findings have established that
multivariate mixed smoothness classes play a fundamental role in
high-dimensional approximation. The book presents essential
findings on and discussions of linear and nonlinear approximations
of the mixed smoothness classes. Many of the important open
problems explored here will provide both students and professionals
with inspirations for further research.
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