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.

翻译：我们提出了$N$维超矩形上Sobolev空间函数的一种新表示，通过将此类函数表示为沿定义域下边界求值的可容许导数形式。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$L_{2}$中——不受Sobolev空间固有连续性约束的限制。此外，我们证明了从该边界值空间到Sobolev空间的映射由具有多项式核的积分算子给出，并证明了该映射是可逆的。基于此结果，我们提出了一种Sobolev空间函数的多项式逼近方法，通过从边界值的多项式投影重构此类逼近。我们证明了该逼近关于离散-连续Sobolev范数是最优的，并通过数值算例表明其比函数的直接投影具有更优的收敛性。最后，我们展示了该方法还可通过引入阶梯函数基，构建不受吉布斯现象等影响的精确分段多项式逼近。