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UnifConv

# UnifConv - Examples re Uniform Convergence These are the...

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Examples re. Uniform Convergence These are the examples we discussed in class on Monday, Oct. 26. On the last page you will find our discussion of the uniform convergence of H s n L on A=[-1,1], and why the convergence of H t n L is not uniform. In[20]:= Clear @ f,g,s,t,h D f @ n_ D : = x^n g @ n_ D : = n^2 * x * Exp @ - n * x D s @ n_ D : = n * Sin @ x n D t @ n_ D : = ArcTan @ n * x D h @ n_ D : = H x^2 L ^ H n H 2n - 1 LL a @ n_ D : = H x^2 + 1 n L ^ H n H 2n - 1 LL f @ n D flist = Table @ f @ n D , 8 n,15 <D ; Plot @ flist, 8 x,0,1 <D x n 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 s @ n D slist = Table @ s @ n D , 8 n,5 <D ; Plot @ slist, 8 x, - 3,3 <D nSin B x n F - 3 - 2 - 1 1 2 3 - 2 - 1 1 2

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h @ n D hlist = Table @ h @ n D , 8 n,5 <D ; Plot @ hlist, 8 x, - 1,1 <D I x 2 M n - 1 + 2n - 1.0 - 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 In class I said that the h n H x L are all differentiable functions, but that is wrong; they are not differentiable at x=0. The following modification does give a differentiable function a n H x L for each n, and has the nondifferentiable limit a n H x L fi x (although the convergence is very slow at x=0).
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