In this paper the following question is answered for the line and for the circle. When is a trapezoid t the graph of a function whose Fourier norm is smallest among functions whose graphs coincide with t on its intervals of constancy? When such functions are viewed as frequency responses of low-pass filters, they are optimally stable. In the periodic case this means that their Fourier transforms have the property that when implemented as digital filters, the least upper bound of all ratios of norms of output sequences to norms of corresponding input sequences is as small as possible.