This paper studies the problem of time series forecasting (TSF) from the perspective of compressed sensing. First of all, we convert TSF into a more inclusive problem called tensor completion with arbitrary sampling (TCAS), which is to restore a tensor from a subset of its entries sampled in an arbitrary manner. While it is known that, in the framework of Tucker low-rankness, it is theoretically impossible to identify the target tensor based on some arbitrarily selected entries, in this work we shall show that TCAS is indeed tackleable in the light of a new concept called convolutional low-rankness, which is a generalization of the well-known Fourier sparsity. Then we introduce a convex program termed Convolution Nuclear Norm Minimization (CNNM), and we prove that CNNM succeeds in solving TCAS as long as a sampling condition—which depends on the convolution rank of the target tensor—is obeyed. This theory provides a meaningful answer to the fundamental question of what is the minimum sampling size needed for making a given number of forecasts. Experiments on univariate time series, images and videos show encouraging results.