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Thoughts on least squared-error optimal windows

Recently, a simple and versatile method for the design of linear phaser FIR filters with spline transition bands and optimal in a least-squared sense was introduced. The following question is raised: Given an arbitrary window, say, for example, a Hamming window, does there exist a transition function (like the spline function above) such that the Hamming window is least-squares optimal? A related question is the following: Given a transition function, does there exist a window sequence w(n) such that the least squared optimal FIR filter is given by g(n)w(n)? This correspondence shows that all windows have associated transition functions that make them least-squared optimal. For every window, there exists a transition function that makes it superoptimal.