The principle of operation of intense radiation devices such as microwave tubes, free-electron lasers, and masers, is based on a distributed interaction between an electron beam and electromagnetic radiation. Some of the effects emerging during the interaction involve a continuum of frequencies in their broadband spectrum. We developed a three-dimensional, space-frequency theory for the analysis and simulation of radiation excitation and propagation in electron devices and free-electron lasers operating in an ultrawide range of frequencies. The total electromagnetic field (radiation and space-charge waves) is presented in the frequency domain as an expansion in terms of transverse eigenmodes of the (cold) cavity, in which the field is excited and propagates. The mutual interaction between the electron beam and the electromagnetic field is fully described by coupled equations, expressing the evolution of mode amplitudes and electron beam dynamics. The approach is applied in a numerical particle code WB3D, simulating wideband interactions in free-electron lasers operating in the linear and nonlinear regimes.