Some problems turn out to be easier to solve in a domain which is the dual of that where they were originally formulated. Unfortunately, my math having gone rusty for quite a while, I don't recall offhand any examples that I feel familiar enough with to discuss here

It's a little advanced, but I like Fourier's solution to the heat equation: any function satisfying certain properties can be thought of as a sum of waves (frequency being dual to position), and the heat equation is easy to solve if your initial data is such a wave. Constructing the solution from this perspective leads us to the heat kernel, which can be calculated without even breaking the function into waves in the first place!