Mathematics (or something like that)

This blog on PF is a copy of the posts on my blogspot.com blog. There are pages and content not compatible with PF blogs, so please check out the original! The URL is http://letepsilonbegreaterthanzero.blogspot.com

I will very quickly summarize the main points of the blog here

For the summer of 2012 I am reading through two books, Fourier Analysis by Stein and Shakarchi, and The Road to Reality by Penrose. I am blogging about thoughts I have from these books. I think it would be pretty cool to have feedback here =)

In fall 2012, I will be starting a PhD program with plans of continuing the blog over my course work. If I pass quals in two years (fingers crossed) this blog may change focus.

Anyways, every time I have a new post I will post a link here. I won't have any of the pages or other info on this blog, but at least the bulk of the post will be the same.
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fourier analysis: orthonormal bases

Posted Jun3-12 at 10:45 PM by theorem4.5.9 (Mathematics (or something like that))
View full entry at my blogspot.com blog: fourier analysis: orthonormal bases

One of the reasons Fourier analysis works so well is that $\left\{ e_n := e^{inx}\right\}_{n=-\infty}^\infty$ forms an orthogonormal subspace of $L^2[-\pi,\pi]$ functions. Of course the inner product is the usual one for complex functions on the circle, $<f,g> := \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) \overline{g(x)} dx$

Hence it is not surprising that Hilbert space geometry plays a crucial...
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road to reality: chapters 6-7

Posted Jun3-12 at 10:44 PM by theorem4.5.9 (Mathematics (or something like that))
View full entry at my blogspot.com blog: road to reality: chapters 6-7

Levels of smoothness:

I really enjoyed how Penrose spent so much time talking about what a mathematician means by smooth. The hierarchy he has is

one continuous derivative $\rightarrow$ derivatives of all order $\rightarrow$ analytic (has a power series expansion)

I like mentioning this because before I had not considered analyticy as the next level of "smoothness",...
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fourier analysis: infinite series

Posted Jun3-12 at 10:43 PM by theorem4.5.9 (Mathematics (or something like that))
View full entry at my blogspot.com blog: fourier analysis: infinite series

There are some clever tricks for getting information from divergent series that I have not seen before. I will probably forget this in the future, so I would like to take some time and write some notes on Cesaro and Abel summability techniques (which just so happen to be useful in Fourier analysis!) Note: this post covers chapter 2, section 5 of the text.

I am going to make up some terminology...
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fourier analysis: nonzero solution of the laplacian

Posted Jun3-12 at 10:42 PM by theorem4.5.9 (Mathematics (or something like that))
View full entry at my blogspot.com blog: fourier analysis: nonzero solution of the laplacian

In the last remark of chapter 2, the authors make mention that if $u$ is harmonic on the disc (i.e. $\triangle u = 0$) and converges uniformly to $0$ as $r \rightarrow 1$, then $u$ must be identically $0$. However, if we relax the boundary condition to pointwise convergence, the conclusion no longer holds. Here is some matlab code I wrote up to investigate this problem (the solution is provided...
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fourier analysis: convolutions

Posted Jun3-12 at 10:40 PM by theorem4.5.9 (Mathematics (or something like that))
View full entry at my blogspot.com blog: fourier analysis: convolutions

I did not understand convolutions well so I tried to find an analogous operation for finite dimensional vector spaces. This turned out unsatisfactory but rewarding. What I focused on was the result
$$S_N(f)(x) := \sum_{n=-N}^{N}\hat{f}(n)e^{inx} = (f *D_N) $$

where $D_N = \sum_{n=-N}^{N}e^{inx}$, the so called Dirichlet kernel. Basically, convolution replaces the pointwise product in many...
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