Suppose we define X+,X− as max and \max(-X, 0) respectively. Then, given Z = X + Y, I've been trying to figure out how to express Z^+ and Z^- in terms of X^\pm and Y^\pm, which is supposedly possible.
I know that \max(x, y) = \frac{x+y+|x-y|}2, and so Z^+ = X^+ + Y^+ + \frac{|X+Y|-|X|-|Y|}{2}, but I'm unsure what to do with this remaining term, I can't seem to figure out how to express it in terms of the other quantities. I have considered breaking he domain X, Y up into regions where X+Y\ge 0, X\ge 0 and Y\ge 0 and flip-flopping the signs, but this seemed like too many cases to be the true solution.
How exactly do you do this? I can't seem to see it.
Answer
You cannot express (X+Y)^+ alone in terms of X^\pm and Y^\pm, and likewise (X+Y)^-, but you can express the two of them together:
(X+Y)^+ - (X+Y)^- = X + Y = (X^+ - X^-) + (Y^+ - Y^-).
(Part (b) of that exercise in Rosenthal's book kind of gives it away.)
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