I am aware that in the theory of classical infinite sums , one can not generally interchange the order of a double sum or do other infinite sum manipulations. However, these infinite sum manipulations can be valid if absolute convergence is available.
I am working on a problem , where I would like to manipulate a double infinite sum but absolute convergence need be satisfied in my context. I am under the impression that these classically illegal infinite sum manipulations can hold under less strict conditions in the context of non classical summation methods. I would be glad if someone can write an answer that discusses when interchanging the order of a double sum is allowed in the non classical setting and direct me to a place where I should read about this. Thank you a lot.