This paper proposes minimax design of digital linear phase FIR filters that have adjustable bandwidth(s) whereas the phase response is fixed. For this purpose, a structure is employed in which the overall transfer function is a weighted linear combination of fixed subfilters and where the weights are directly determined by the bandwidth(s). These filters can be designed using linear programming which generates globally optimal overall filters in the minimax (Chebyshev) sense. The paper also introduces a new structure for bandpass and bandstop filters with individual adjustable upper and lower band edges, and with a substantially lower arithmetic complexity compared to structures that make use of two separate adjustable-bandwidth lowpass and highpass filters in cascade or in parallel.