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.

翻译：针对定义在$N$维超矩形上具有主导混合光滑性的Sobolev空间，本文提出了一种新的函数表示方法。具体而言，研究证明此类函数可借助其最高阶混合导数及其在区域适当边界上求值的低阶导数进行表达。该展开式被证明具有可逆性，能够通过函数的导数与边界值唯一确定Sobolev空间中的任意函数。由于这些边界值要么是有限维的，要么存在于平方可积函数空间中，因此这一表示在Sobolev空间与$L_{2}$空间之间建立了双射关系。利用该双射，本文展示了如何在限制性较弱的$L_{2}$空间中对Sobolev空间中的函数进行逼近——通过从边界值与最高阶导数的$L_{2}$最优投影中重构函数的近似形式。该逼近方法分别采用Legendre多项式基与阶梯函数基进行实现，两个数值算例的结果表明，两种基下的逼近均比直接投影方法展现出更优的收敛特性。