2/ I've n't yet done the 2nd part of the post as promised, but this will happen this week. Instead, I spent some time improving on the 1st part. I've made it as complete possible, w/o introducing any superfluous material. The most important ad-on is Danskin-Bertsekas Theorem.
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Just tweeted the second part of the posthttps://twitter.com/dohmatobelvis/status/1194348149336207360 …
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Eagerly awaiting the next installment!
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I should be able to do the second blog-post tomorrow evening, Paris time :). Will be announced on in a tweet.
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I've never learned why convex conjugation is said to be related to the Fourier transform. I'd be curious to hear more about that. The convex conjugate views a function as a supremum of affine functions, whereas FT decomposes into pure frequencies. I don't see a clear connection.
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1/ The connection is algebraic, not analytically. You won't find them in "frequencies" :) So, in harmonic analysis, there is a notion of convolution and of course, the Fourier transform. In convex analysis, you have inf-convolution and of course, Fenchel conjugates.
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The dual norm definition seems to be missing the absolute value (or my phone is not rendering it).
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You don't need absolute value. It's a definition. Even if you put absolute value, it wouldn't change anything. Indeed, |<x*,x>| = max(<x*,x>, <-x*,x>). Since for every x* in X*, -x* is also in X* (a vector space), sup_{x* in X*} <x*,x> is equal to sup_{x* in X*} |<x*,x>|.
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Ooops, that's a typo. The sup should be over C, not X^* :). Good catch :). Fixed (should take < 1 minute to go live).
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Čini se da učitavanje traje već neko vrijeme.
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