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Chebfun versus Parks-McClellan

There is a Matlab-based package called Chebfun that (among other things) computes polynomial approximations of functions. See here: http://www2.maths.ox.ac.uk/chebfun/. I don't know anything about ...
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What is the difference between Remez exchange and Parks-McClellan filter design?

A question recently came up regarding Parks-McClellan and some of the comments point out that the wikipedia article on Parks-McClellan states, ...the Parks-McClellan algorithm is a variation of ...
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How can I design Nyquist interpolation filters with the Parks-McClellan algorithm?

We can easily design interpolation filters that obey certain frequency-domain constraints using the Parks-McClellan algorithm. However, it's not immediately clear how to enforce time-domain ...