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What is the reason maximum modulus principle will fail for non-holomorphic functions in the closed disk? I am reading the part in Rudin on Poisson summation formula and I was wondering what a good counterexample would be like. I am assuming $f$ to be continuous and is bounded on the unit disk, so functions like $\log[x],\frac{1}{x}$ are excluded.

Rudin proved this holds for all polynomial functions, and it is not too difficult to extend it to all holomorphic functions on the unit disk via classical arguments. So i am thinking about given a function on the boundary of the circle and run a heat equation on it, or given a heat distribution in the middle circle of the disk instead. So in the extreme case I would have a function has value 1 on $0$, and 0 on the circle. This contradicts the maximum modulus principle - but why can't we approximate this simple function via holomorphic functions and use dominated convergence theorem? In other words, what is the reason that $C_{0}(D^{2})$ is not a completion of $H(D^{2})$?

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A continuous function can almost behave in whatever way you want, and there is absolutely no reason to think that continous functions satisfy a maximum modulus principle. (The example you give yourself, shows this.)

For the second question, it's not difficult to show that if $f_j$ are holomorphic and converge locally uniformly to a function $f$, then $f$ is also holomorphic. (This follows, for example from Morera's theorem.) Even if you just assume that the $f_j$:s converge pointwise, there is a theorem by Osgood showing that limit function is holomorphic on an open, dense set, so we are very far from being able to approximate arbitrary continuous functions by holomorphic ones.

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Do you know where can I look up the result on Osgood? For me pointwise convergence (or stronger, convergence in measure) is suffice, since I could use DCT. –  Bombyx mori Dec 30 '12 at 13:37
    
Osgood's theorem is not very well known, and missing from most textbooks. Some googling turned up this reference, mathdl.maa.org/images/upload_library/22/Chauvenet/… (see p.131). Another one is math.wustl.edu/~sk/limits.pdf (which deals with limits of holomorphic functions, and could be of interest to you) –  mrf Dec 30 '12 at 13:42

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