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: convolutions

Posted Jun3-12 at 10:40 PM by theorem4.5.9

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 instances of complex analysis. In finding an analogy in $\mathbb{R}^n$, this is exactly the case. Take for example $\mathbb{R}^2$, with the usual basis. Then denote the convolution operator as $(\cdot , \cdot )_e$ where the subscript denotes our basis. We can define this operator as
$$\left(\begin{matrix} a_1 \\ a_2 \end{matrix} * \begin{matrix}b_1\\b_2\end{matrix}\right)_e = \begin{pmatrix}a_1 b_1\\a_2 b_2 \end{pmatrix}$$

From what I can tell, this function has all of the important properties of a convolution on $\mathbb{R}^n$ (continuity in finite dimensional vector spaces doesn't say much). I made explicit the basis because this "convolution" is dependent upon it. This exploration ended with the conclusion that the convolution is very specially equipped to handle function-multiplication of complex functions.
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