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Bergmann Space

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data

    If G = { z in C: 0<|Z|<1} show that every f in L_{a}^{2}(G) has a removable singularity at z = 0


    We must show that lim z->0 z*f(z) = 0 for all f in L_{a}^{2}(G)

    By a corollary 1.12, if f in L_{a}^{2}(G), a in G and 0<r<dist(a,bdr G), thne
    |f(a)| <= 1/(r\sqrt(\pi))||f||_2,

    for |z| < 1/2 we have that

    |f(z)| <= 1/(|z|\sqrt(pi)/sqrt(2)) ||f||_2 = 2/(|Z|sqrt(pi) ||f||_2

    so that |zf(z)| = |z||f(z)| <= 2/sqrt(pi) ||f||_2.

    how do I proceed from there

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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