# Approximation of Non-Decaying Signals From Shift-Invariant Subspaces

## TL;DR

This paper extends the Strang-Fix theory to show that non-decaying signals in weighted-$L_p$ spaces can be approximated with error decreasing as $O(h^L)$, using shift-invariant spaces and specific kernel conditions.

## Contribution

It introduces a generalized approximation error decay rate for non-decaying signals, extending classical theory to weighted spaces and providing conditions for both projection and interpolation schemes.

## Key findings

- Error decays as $O(h^L)$ with decreasing sampling step h.
- Approximation is stable under specific kernel and signal smoothness conditions.
- Both projection and interpolation methods achieve the decay, with interpolation requiring more signal smoothness.

## Abstract

In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted-$L_p$ spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang-Fix theory to show that, for $d$-dimensional signals whose derivatives up to order $L$ are all in some weighted-$L_p$ space, the weighted norm of the approximation error can be made to go down as $O(h^L)$ when the sampling step $h$ tends to $0$. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang-Fix conditions of order $L$. We show that the $O(h^L)$ behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of $d/p+\varepsilon$, for arbitrary $\varepsilon >0$. This extra amount of derivatives is used to make sure that the direct sampling is stable.