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Summary. A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m-1 fitted to n distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires order m2n operations and therefore permits efficient order m2n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires order n 3 operations.

Compressive sampling is a new paradigm for sampling, based on sparseness of signals or signal representations. It is much less restrictive than Nyquist-Shannon sampling theory and thus explains and systematises the widespread experience that methods such as the Högbom CLEAN can violate the Nyquist-Shannon sampling requirements. In this paper, a CS-based deconvolution method for extended sources is introduced. This method can reconstruct both point sources and extended sources (using the isotropic undecimated wavelet transform as a basis function for the reconstruction step). We compare this CS-based deconvolution method with two CLEANbased deconvolution methods: the Högbom CLEAN and the multiscale CLEAN. This new method shows the best performance in deconvolving extended sources for both uniform and natural weighting of the sampled visibilities. Both visual and numerical results of the comparison are provided.

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