  Group symmetry is inherent in a wide variety of data distributions. Data
processing that preserves symmetry is described as an equivariant map and often
effective in achieving high performance. Convolutional neural networks (CNNs)
have been known as models with equivariance and shown to approximate
equivariant maps for some specific groups. However, universal approximation
theorems for CNNs have been separately derived with individual techniques
according to each group and setting. This paper provides a unified method to
obtain universal approximation theorems for equivariant maps by CNNs in various
settings. As its significant advantage, we can handle non-linear equivariant
maps between infinite-dimensional spaces for non-compact groups.
