Constructive Representation of Functions in $N$-Dimensional Sobolev Space

We propose a new representation of functions in Sobolev spaces on an $N$-dimensional hyper-rectangle, expressing such functions in terms of their admissible derivatives, evaluated along lower-boundaries of the domain. These boundary values are either finite-dimensional or exist in the space $L_{2}$ of square-integrable functions -- free of the continuity constraints inherent to Sobolev space. Moreover, we show that the map from this space of boundary values to the Sobolev space is given by an integral operator with polynomial kernel, and we prove that this map is invertible. Using this result, we propose a method for polynomial approximation of functions in Sobolev space, reconstructing such an approximation from polynomial projections of the boundary values. We prove that this approximation is optimal with respect to a discrete-continuous Sobolev norm, and show through numerical examples that it exhibits better convergence behavior than direct projection of the function. Finally, we show that this approach may also be adapted to use a basis of step functions, to construct accurate piecewise polynomial approximations that do not suffer from e.g. Gibbs phenomenon.