Smooth orthonormal basis for L2 (R2) using smooth projections on L2(R2) associated with rectangles in R2

For L-2(R-2), one can easily construct an orthonormal basis from an orthonormal basis of L-2(R) using the tensor product; and that orthonormal basis is infact separable orthonormal basis. The separable basis have a number of disadvantages, as they have very little design freedom. Furthermore, the separability imposes an unnecessary product structure on R-2, that is artificial for natural images. In this paper, we construct a compactly supported nonseparable smooth orthonormal basis for L-2(R-2) by constructing smooth projection P-IxM associated with a rectangle I x M = [a, b] x [c, d] subset of (R-2) on L-2(R-2) without using tensor product.