Previously the envelope-constrained filtering problem was formulated as designing an FIR filter such that the filter's L2 norm is minimized subject to the constraint that its response to a specified input pulse lies within a prescribed envelope. In this paper, we recast this filter design problem as a frequency-domain L infinity optimization problem with time-domain constraints. Motivations for solving this problem are given. Then recently developed infinite dimensional linear programming techniques are used for the design of the required FIR filter. For illustration, we apply the approach to a numerical example which deals with the design of an equalization filter for a digital transmission channel.