# Weak, Strong and Linear Convergence of the CQ-Method Via the Regularity of Landweber Operators

## TL;DR

This paper investigates the convergence properties of the CQ-method for split convex feasibility problems, introducing the Landweber transform which preserves key operator properties and ensures convergence.

## Contribution

It introduces the Landweber transform for operators, extending the CQ-method's applicability and establishing its property preservation for broader classes of operators.

## Key findings

- Landweber transform preserves nonexpansiveness.
- Landweber transform maintains regularity properties.
- Ensures convergence of CQ-type methods under broader conditions.

## Abstract

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne (2002), we define an abstract andweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves many interesting properties. For example, the Landweber transform of a (quasi/firmly) nonexpansive mapping is again (quasi/firmly) nonexpansive. Moreover, the Landweber transform of a (weakly/linearly) regular mapping is again (weakly/linearly) regular. The preservation of regularity is important because it leads to (weak/linear) convergence of many CQ-type methods.