Wireless ultrasound (US) systems that produce high-quality images can improve current clinical diagnosis capabilities by making the imaging process much more efficient, affordable, and accessible to users. The most common technique for generating B-mode US images is delay and sum (DAS) beamforming, where an appropriate delay is introduced to signals sampled and processed at each transducer element. However, sampling rates that are much higher than the Nyquist rate of the signal are required for high resolution DAS beamforming, leading to large amounts of data, making transmission of channel data over WIFI impractical. Moreover, the production of US images that exhibit high resolution and good image contrast requires a large set of transducers which further increases the data size. Previous works suggest methods for reduction in sampling rate and in array size. In this work, we introduce compressed Fourier domain convolutional beamforming, combining Fourier domain beamforming, sparse convolutional beamforming, and compressed sensing methods. This allows reducing both the number of array elements and the sampling rate in each element, while achieving high resolution images. Using in vivo data we demonstrate that the proposed method can generate B-mode images using 142 times less data than DAS. Our results pave the way towards wireless US and demonstrate that high resolution US images can be produced using sub-Nyquist sampling and a small number of receiving channels.