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Fourier transform conjucture/theorem

  1. Apr 20, 2008 #1
    fourier transform conjecture/theorem

    Is this a theorem?

    Let [itex]f\in L^2(\mathbb{R},\mathbb{R})[/itex] be such function that f is continuous on some sets [itex]]x_0-\delta, x_0[[/itex] and [itex]]x_0, x_0+\delta[[/itex] with [itex]\delta >0[/itex], and

    [tex]
    \lim_{x\to x_0^-} f(x) \neq \lim_{x\to x_0^+}f(x).
    [/tex]

    (So we cannot choose a continuous g such that [g]=[f] in [itex]L^2[/itex] sense)

    There does not exist such [itex]M,C,\epsilon>0[/itex], that

    [tex]
    |(\mathcal{F} f)(k)| < C \frac{1}{|k|^{1+\epsilon}},\quad \forall |k|> M.
    [/tex]
     
    Last edited: Apr 20, 2008
  2. jcsd
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