We describe a general and exact method to considerably speed up linear object detection systems operating in a sliding, multi-scale window fashion, such as the individual part detectors of part-based models. The main bottleneck of many of those systems is the computational cost of the convolutions between the multiple rescalings of the image to process, and the linear filters. We make use of properties of the Fourier transform and of clever implementation strategies to obtain a speedup factor proportional to the filters' sizes. The gain in performance is demonstrated on the well known Pascal VOC benchmark, where we accelerate the speed of said convolutions by an order of magnitude.