The authors suggest three independent methods for computing upper and lower bounds of the frequency response gains. The first method is derived with the lifting approach and does not have an approximation parameter. Although this excludes the possibility of adjusting the accuracy, the method quite often gives almost coincident upper and lower bounds in a short computational time. This is particularly useful for the combined use with the bisection algorithm, enabling one to arrive at an exact gain to any degree of accuracy with only numerically reliable computations. The second and the third methods are derived with the FR-operator approach and have an approximation parameter that can be used to improve accuracy at the expense of the computational load. They give not only the lower bound but also the upper bound of the frequency response gain. It has been also clarified that the arguments discussed here have close relationships to the sampled-data \(H_2\) and \(H_\infty\) problems.