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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
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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.
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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
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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
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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.
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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
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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
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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
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The theorem states that the following conditions on a square-integrable, complex valued function
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In particular, this makes the TFAE structure clearer (originally the first condition redefines
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quantifier. They leave it as an exercise to figure out which is meant. That's obnoxious.
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is square integrable (the lead at least makes this hypothesis).
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people using the word "where" in this way intend it as a
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475:{\displaystyle {\mathcal {F}}(F)(x)}
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