Constructive reconstruction from irregular sampling in multi-window spline-type spaces

摘要

The constructive recovery of functions in multi-window spline-type spaces from irregular samples are treated, using the concepts of frames and reproducing ker-nels. It is well known that one can describe iterative methods, which guarantee stable reconstruction, convergent in a variety of functions spaces, at least if the sampling set is dense enough. The corresponding theoretical statements either work in the context of continuous variables or discrete signals, but rarely con-sider the effect of errors that unavoidably occur if the continuous problem is implemented in a discrete (finite) setting. The analysis of the situation, using function spaces methods (in particular Wiener amalgams) and the description of implementable variants of these iterative algorithms with guaranteed rates of convergence are treated in the necessary detail here. It relies on the results of an earlier paper by the authors, demonstrating the fact that one can guarantee a good approximation of the biorthogonal family for the set of generators of the space. It is important to note, that the imperfections due to discretization do not spoil the chance of perfect reconstruction (in the limit), but only lead to a mild degradation of the rate of convergence, compared to the ideal case.