Kernel projection operator's approximation error estimation in weighted mixed-norm shift-invariant subspaces

By utilizing the Strang-Fix theory, approximation of non-decaying signals from shift-invariant subspaces ($L_p$-norm) is studied by Nguyen and Unser (2019) [42]. The non-decaying function can be seemed as a kind of weighted function. In this paper, using the weighted mixed-norm Wiener amalgam space and hybrid space, we will study the approximation error bounds of the kernel projection operator in the weighted mixed-norm sense without Strang-Fix theory. Note that the condition under weighted mixed-norm hybrid space is weaker than that of Wiener amalgam space. So, in this paper, not only based on the Wiener amalgam space, we will also demonstrate that, as a comparison, the approximation results of the projection operator are also valid under the relevant hybrid space.