  Moreau's seminal paper, introducing what is now called the Moreau envelope
and the proximity operator (also known as the proximal mapping), appeared in
1965. The Moreau envelope of a given convex function provides a regularized
version which has additional desirable properties such as differentiability and
full domain. Fifty years ago, Attouch proposed using the Moreau envelope for
regularization. Since then, this branch of convex analysis has developed in
many fruitful directions. In 1967, Bregman introduced what is nowadays known as
the Bregman distance as a measure of discrepancy between two points
generalizing the square of the Euclidean distance. Proximity operators based on
the Bregman distance have become a topic of significant research as they are
useful in the algorithmic solution of optimization problems. More recently, in
2012, Kan and Song studied regularization aspects of the left Bregman-Moreau
envelope even for nonconvex functions. In this paper, we complement previous
works by analyzing the left and right Bregman-Moreau envelopes and by providing
additional asymptotic results. Several examples are provided.
