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])?

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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.

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