A multisensor deconvolution problem (Q818503)

The authors consider the multisensor deconvolution problem which, mathematically, is as follows. Given a collection of compactly supported distributions \(\mu_i\in{\mathcal E(\mathbb R})\), \(i=1,2,\ldots l\), often called convolvers, find a collection of compactly supported distributions \(\{\nu_i\}_{i=1}^l\), often called deconvolvers, such that \(\sum_{i=1}^l \mu_i *\nu_i=\delta\), where \(\delta\) is the Dirac distribution. From a signal processing point of view, the multisensor deconvolution problem is as follows. Suppose a signal or image \(f\) is remotely sensed through a bank of \(l\) linear, translation invariant sensors with impulse responses \(\{\mu_i\}_{i=1}^l\), then \(f\) is rendered into a vector of data \({\mathbf s}=(s_1,s_2,\ldots s_l)\) where \(s_i=f*\mu_i\). The problem is to recover \(f\) exactly from \(\mathbf s\). The authors find conditions equivalent to Hörmander's condition for a ring \(A_p(\Omega)\) of analytic functions defined on \(\Omega\) to be generated by a given set of its elements. Weaker conditions, also equivalent to Hörmander's conditions, are found so that the representation theorem for convolution operators on the space of Gevrey functions hold. An explicit formula is given by means of the nonperiodic sampling method.