A desirable property of the error norm minimized by a filter-design technique 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 minimum can 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. If is a vector space, a concave subset of , and a concave functional on , then any local minimizer of is a global minimizer of in .
Theorem. If is a normed linear space, a concave subset of , and a strictly concave functional on , then has at most one minimizer in .
Theorem. Let be a closed and bounded subset of . If is continuous on , then has at least one minimizer in .
Theorem (2.1) 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 , Thm. 14). (In , ``compact'' is equivalent to ``closed and bounded'' .) Theorem (2.1) 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.