  A new representation is proposed for functions in a Sobolev space with
dominating mixed smoothness on an $N$-dimensional hyperrectangle. In
particular, it is shown that these functions can be expressed in terms of their
highest-order mixed derivative, as well as their lower-order derivatives
evaluated along suitable boundaries of the domain. The proposed expansion is
proven to be invertible, uniquely identifying any function in the Sobolev space
with its derivatives and boundary values. Since these boundary values are
either finite-dimensional, or exist in the space of square-integrable
functions, this offers a bijective relation between the Sobolev space and
$L_{2}$. Using this bijection, it is shown how approximation of functions in
Sobolev space can be performed in the less restrictive space $L_{2}$,
reconstructing such an approximation of the function from an $L_{2}$-optimal
projection of its boundary values and highest-order derivative. This
approximation method is presented using a basis of Legendre polynomials and a
basis of step functions, and results using both bases are demonstrated to
exhibit better convergence behavior than a direct projection approach for two
numerical examples.
