Talk:Titchmarsh theorem

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Where[edit]

This article says:

F(x) = ƒ(x) − ig(x) where ƒ is a real-valued square-integrable function and g is the Hilbert transform of ƒ.

I don't have a problem with using the word "where" to explain notation, e.g. "where 2 is the smallest integer larger than 1" etc. But in this case I'm guessing it means "for some real-valued square-integrable function ƒ". Is that what it means? If so, it shouldn't be expressed by using "where". In effect, "where" becomes an existential quantifier. But sometimes people using the word "where" in this way intend it as a universal quantifier. They leave it as an exercise to figure out which is meant. That's obnoxious. Michael Hardy (talk) 04:45, 14 May 2008 (UTC)

I don't typically mind using where to have an implied existential quantifier, but usually because the primary subject of the sentence is also just being introduced. In this case though, it really does read poorly. I replaced "where" with "for some" as you suggested. The functions f and g are both real valued, so they are just F's real and imaginary parts, and the theorem is saying they must be in the relation that −g is the hilbert transform of f. This could probably be phrased more clearly, but it would take a more awake editor than myself. JackSchmidt (talk) 05:13, 14 May 2008 (UTC)
I'm sorry if you found the phrasing awkward. It looked alright to me at the time (in fact, it still does). Anyway, I have changed JackSchmidt's version because I feel it obscured the punchline somehow (i.e., that g is the Hilbert transform of f). Let me know if this is better. Thanks, silly rabbit (talk) 12:06, 14 May 2008 (UTC)
This is mostly what I had in mind, but there is a problem with the theorem right now. Basically the second condition as stated simply evaluates to false, so something needs to be done to fix its quantifiers. I suggest something like:

The theorem states that the following conditions on a square-integrable, complex valued function F on the real line are equivalent:

  • F(x) is the limit as z → x of a holomorphic function F(z) in the upper half-plane such that
  • −Im(F) is the Hilbert transform of Re(F), where Re(F) and Im(F) are real-valued functions with F = Re(F) + i Im(F).
  • The Fourier transform vanishes for x < 0.
In particular, this makes the TFAE structure clearer (originally the first condition redefines F and the second condition quantifies over all F, but actually one intends a specific F to remain fixed), and emphasizes the hilbert transform even more. One also does not need to mention the real and imaginary parts are square integrable, once one has been more clear about the overarching hypothesis that F is square integrable (the lead at least makes this hypothesis). JackSchmidt (talk) 13:45, 14 May 2008 (UTC)
Ok. Done (I hope). silly rabbit (talk) 14:11, 14 May 2008 (UTC)
Looks good to me. I like R→C better than "on the real line" too. If one was looking for a brevity award, "complex valued" is now redundant, but especially on a wiki prone to vandalism, a little redundancy never hurt. JackSchmidt (talk) 14:15, 14 May 2008 (UTC)