Time domain interpretations of frequency domain inequalities on (semi)finite ranges

Many of the significant results in systems and control literature rely on characterizations of system properties in terms of frequency domain inequalities (FDIs) and/or time domain inequalities (TDIs). Classical FDIs, required to hold on the entire frequency range, have been interpreted by equivalent TDIs so that satisfaction of one implies that of the other. Recent developments have addressed FDIs within (semi)finite frequency ranges to increase flexibility in the system analysis and synthesis. This paper provides necessary and sufficient conditions for a general FDI to hold within a restricted frequency range in terms of TDIs to be satisfied by every inputs within a certain class specified by a matrix-valued integral quadratic constraint.