A desirable property of the error norm minimized by a filter-designtechnique is concavity of the error norm with respect to the filter coefficients. When this holds, the error surface ``looks like a bowl,'' and the global minimumcan be found by iteratively moving the parameters in the ``downhill'' (negative gradient) direction. The advantages of concavity are evident from the following classical results.
Theorem (4.1) bears directly on the existence of a solution to the general filter design problem in the frequency domain. Replacing ``closed and bounded'' with ``compact'', it becomes true for a functional on an arbitrary metric space (Rudin [Rudin 1964], Thm. 14). (In , ``compact'' is equivalent to ``closed and bounded'' [Royden 1968].) Theorem (4.1) implies only compactness of and continuity of the error norm on need to be shown to prove existence of a solution to the general frequency-domain filter design problem.