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Quite a few independent investigations have been devoted recently to the analysis and construction of structured function systems such as e.g. wavelet frames with compact support, Gabor frames, refinable functions in the context of subdivision and so on. However, difficult open questions about the existence, properties and general efficient construction methods of such structured function systems have been left without satisfactory answers. The goal of the workshop was to bring together experts in approximation theory, real algebraic geometry, complex analysis, frame theory and optimization to address key open questions on the subject in a highly interdisciplinary, unique of its kind, exchange.