A Sampling Theory for Non-Decaying Signals

The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted $L _{ p }$ spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its d∕p + ε derivatives, for arbitrarily small ε > 0, grow no faster than a polynomial in the $L _{ p }$ sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.