Functional transforms make it possible to investigate problems from a different perspective and sometimes simplify their investigation. In convex analysis, the most suitable notion of a transform is the Legendre transform, which maps a function to its Fenchel conjugate. This transform is studied in detail in this chapter. In particular, it is shown that the conjugate of an infimal convolution is the sum of the conjugates. The key result of this chapter is the Fenchel–Moreau theorem, which states that the proper convex lower semicontinuous functions are precisely those functions that coincide with their biconjugates.
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