Title: Frequency Pooling: Shift-Equivalent and Anti-Aliasing Down Sampling. Abstract: Convolutional layer utilizes the shift-equivalent prior of images which makes it a great success for image processing. However, commonly used down sampling methods in convolutional neural networks (CNNs), such as max-pooling, average-pooling, and strided-convolution, are not shift-equivalent. This destroys the shift-equivalent property of CNNs and degrades their performance. In this paper, we propose a novel pooling method which is \emph{strict shift equivalent and anti-aliasing} in theory. This is achieved by (inverse) Discrete Fourier Transform and we call our method frequency pooling. Experiments on image classifications show that frequency pooling improves accuracy and robustness w.r.t shifts of CNNs. 