Inspired by the graph Laplacian and the point integral method, we introduce a novel weighted graph Laplacian method to compute a smooth interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image inpainting show that the weighted graph Laplacian is a reliable and efficient interpolation method. In addition, it is easy to implement and faster than graph Laplacian.

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A Sobolev space in several variables in an$L$^{1}-type norm is not complemented in its second dual. Hence it is not isomorphic as a Banach space to any complemented subspace of a Banach lattice.