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Approximation of Non-Decaying Signals From Shift-Invariant Subspaces. (arXiv:1705.05601v1 [math.FA])
来源于:arXiv
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
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