Mathematics | Mark H. Kim

Some Learning Oppurtinities in (Harmonic) Analysis

ca.classical-analysis, MK Blathers No Responses »

Apr 012012

My poor blog! I left it abandoned and barren for too long. I will try to post more regularly, though, as usual, I can’t promise anything.

I have been traveling in the past two weeks, visiting graduate schools and trying to decide where I want to spend my life for the next five years. I think it would be impolite to relay my impressions of the schools I visited on a blog, so I will resist the urge. Instead, I would like to share some information I have gathered at the visits that are of interest to fellow apprentice analysts.

On Friday, I learned that Professor Brett Wick at Georgia Tech is running the Internet Analysis Seminar, which, I believe, is on its third year. The idea is to collaborate on studying the basic material via the internet, and then to gather at a week-long conference to discuss the more advanced material. In a sense, it is fairly similar to Professor Christoph Thiele‘s Summer Schools in Analysis, except that Thiele’s summer school does not have the centralized coverage of the basic material that Wick’s seminar has. Indeed, Wick writes up lecture notes for the seminar, which can be found here and there.

Also, the University of Wisconsin at Madison has been awarded the NSF Research Training Groups grant in analysis. UW’s summary of the grant is as follows:

UW Math Professors Alexander Kiselev, Andreas Seeger and Leslie Smith have received an RTG grant from the NSF. The title of their grant is “Analysis and Applications”, and its approved budget is $1.8 million for 5 years. The funds are mostly for support of graduate students, postdocs, and undergraduate research. The proposal ranked first among RTG in Analysis submitted this year to the NSF. The grant involves faculty members in both analysis and applied mathematics groups.

I expect that UW will host many seminars, conferences, and workshops on harmonic analysis, partial differential equations, fluid dynamics, and mathematical biology.

Triangular Fourier Series and Physical Reality

ca.classical-analysis, ds.dynamical-systems, la.linear-algebra No Responses »

Oct 192011

This is a transcription of the September 15 talk by Prof. Camil Muscalu at the Courant Institute. Any errors in this post are due to my interpretation of the talk.

* * * * *

Part 1. Triangular Fourier Series

What is a triangular Fourier series? Let us suppose that $f$ is a $2\pi$ -periodic function on $\mathbb{R}$ . If $f \in L^p([0,2\pi])$ for some $1 < p < \infty$ , we know that the classical Fourier series

$\displaystyle \sum_{n=-\infty}^{\infty} \hat{f}(n) e^{i n x}$

converges to $f(x)$ , both in the $L^p$ -sense (M. Riesz) and in the pointwise almost-everywhere sense (L. Carleson and R. Hunt). These results can be phrased in terms of boundedness of certain operators. For one, the $L^p$ -convergence happens if and only if the operator

$\displaystyle f \mapsto \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx}$

is bounded on $L^p$ for all $N$ . Similarly, the almost-everywhere convergence happens if and only if the operator

$\displaystyle f \mapsto \sup_N \left| \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \right|$

is bounded on $L^p$ .

So then, we have a clear notion of the convergence

$\displaystyle \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \xrightarrow{N \to \infty} f(x)$ .

Why not take the square, and obtain:

$\displaystyle \left( \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \right)^2 \xrightarrow{N \to \infty} f(x)^2$ .

We expand the left-hand side as follows:

$\begin{array}{rcl} \displaystyle \left( \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \right)^2 &=& \displaystyle \sum_{-N \leq n_1,n_2 \leq N} \hat{f}(n_1) \hat{f}(n_2) e^{in_1x} e^{in_2x} \\ &=& \displaystyle \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1)\hat{f}(n_2) e^{in_1x}e^{in_2x} \\ & & + \displaystyle \sum_{-N \leq n_2 < n_1 \leq N} \hat{f}(n_1)\hat{f}(n_2) e^{in_1x}e^{in_2x} \\ & & + \displaystyle \sum_{-N \leq n_1 = n_2 \leq N} \hat{f}(n_1)\hat{f}(n_2) e^{in_1x}e^{in_2x}\end{array}$ .

The last term is the convolution $(f*f)(2x)$ .

We can now ask ourselves the following questions: (1) Does the following convergence happen?

$\displaystyle \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1) \hat{f}(n_2) e^{in_1x} e^{in_2x} \to \frac{1}{2} (f(x)^2 - (f*f)(2x));$

(2) Similarly, does the following convergence happen?

$\textbf{(1) }\displaystyle \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1) \hat{g}(n_2) e^{in_1x} e^{in_2x} \to \frac{1}{2}(f(x)g(x) - (f*g)(2x)),$

where $f,g \in L^2([0,2\pi])$ ?

Connectedness and Discrete-Valued Maps

gn.general-topology 1 Response »

Aug 032011

I found a neat characterization of connectedness in Bredon’s Topology and Geometry: so nice, in fact, that I am compelled to write up a quick note about it.

Let us recall that a topological space $X$ is disconnected if there exists a disjoint pair of open subsets $U$ and $V$ of $X$ whose union is $X$ , and that $X$ is connected otherwise. A discrete-valued map is a continuous map $d:X \to D$ from a topological space $X$ to a discrete space $D$ , which is a topological space in which every subset is open. An alternate characterization of connectedness is as follows:

Proposition 1 (Alternate characterization of connectedness). A topological space $X$ is connected if and only if every discrete-valued map $d:X \to D$ on $X$ is constant.

The proof is quite simple. If $X$ is connected, then the preimage $d^{-1}(y)$ of an element $y$ in the image of $d$ is nonempty, open, and closed: therefore, $d^{-1}(y)$ must be the whole space. Conversely, if $X$ is not connected, then we can find a disjoint pair of open subsets $U$ and $V$ of $X$ whose union is $X$ , whence the map $d:X \to \{0,1\}$ which is 0 on $U$ and 1 on $V$ is a nonconstant discrete-valued map on $X$ .

Why would anyone think of such an alternate definition? Glen Wilson offered the following perspective, which I very much like. First off, following the “categorical” way of thinking, we submit that describing a collection of object in terms of maps between them—as opposed to the objects themselves—is a good thing. But why constant maps? We can consider “labeling” each connected component in a topological space $X$ by quotienting $X$ out by its connected components. The resulting space $Y$ is a discrete space, and the quotient map $\pi:X \to Y$ is a surjective continuous map that is constant if and only if $Y$ is a singleton. Of course, this happens precisely when $X$ has one connected component, i.e., if $X$ is connected.

This alternate characterization leads us to swift proofs of the key properties of connected space. Let us first consider the following

Proposition 2. The continuous image of a connected space is connected.

Here is a one-liner for the proof: If $X$ is a connected space, $f:X \to Y$ a continuous map, and $d:f(X) \to D$ any discrete-valued map, then the composition $d \circ f:X \to D$ must be constant, and so $f(X)$ is connected. The following triumph of intuition also admits a devastatingly simple proof:

Proposition 3. If a collection of connected sets share a point, then the union is connected.

Here, any discrete-valued map $d$ must be constant on each connected set, and the value of $d$ must be the same because they all share a point: it follows that $d$ is constant on the union. Another illustrative example is as follows:

Proposition 4. If $A$ is a connected subset of a topological space $X$ , and if $B$ is a subset of $X$ such that $A \subseteq B \subseteq \bar{A}$ , then $B$ is connected.

Again, the proof is very short. Any discrete-valued map $d$ on $X$ is a continuous, hence sequentially continuous, map that is constant on $A$ , whence we conclude that the value of $d$ on any limit point of $A$ is the same as the value $d$ takes on $A$ . It follows that $d$ is constant on $\bar{A}$ , hence on $B$ .