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

Abstract

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.