Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen.

每个最大单调算子都可以与一个凸函数族相关联，称为Fitzpatrick 族或代表函数族. 令人惊讶的是，在 2017 年，Burachik 和 Martínez-Legaz 表明，众所周知的 Bregman 距离是一般距离家族的一个特例，每个距离都由特定的最大单调算子及其代表函数之一的特定选择引起。对于广义 Bregman 距离族，最近已经提供了凸性、矫顽力和超矫顽力的充分条件。受这些进步的启发，我们在本文中介绍了广义的左右包络和邻近算子，并提供了参数的渐近结果。某些结果很容易从更具体的 Bregman 上下文中扩展，而其他结果仅扩展到某些一般情况。为了说明，我们从 Bregman 泛化案例构建示例，