Triangular Fourier Series and Physical Reality

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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.

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

As it turns out, we indeed have both convergence results, in the $L^1$ -sense (M. Lacey and C. Thiele), and in the almost-everywhere pointwise sense (C. Muscalu, T. Tao, C. Thiele). Equation (1) is, in fact, the simplest example of triangular Fourier series, whose name is perhaps best explained by the following graph of the range of summation:

$-N \leq n_1 < n_2 \leq N$ , when $N = 5$ .

To establish the $L^1$ -theorem, we need to prove that the $L^2 \times L^2 \to L^1$ operator

$\displaystyle (f,g) \mapsto \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1) \hat{g}(n_2) e^{i n_1 x} e^{i n_2 x}$

is bounded uniformly in $N$ . Similarly, establishing the pointwise almost-everywhere convergence amounts to proving that the $L^2 \times L^2 \to L^1$ operator

$\displaystyle (f,g) \mapsto \sup_N \left| \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1) \hat{g}(n_2) e^{in_1x} e^{in_2x} \right|$

is bounded uniformly in $N$ . More generally, we can consider the multilinear operators $L^2 \times \cdots \times L^2 \to L^{2/d}$

$\begin{array}{rcl} \displaystyle (f_1,f_2,\ldots,f_d) &\mapsto& \displaystyle \sum_{-N \leq n_1 < n_2 < \cdots < n_d \leq N} \cdots \\ (f_1,f_2,\ldots,f_d) &\mapsto& \displaystyle \sup_N \left| \sum_{-N \leq n_1 < n_2 < \cdots < n_d \leq N} \cdots \right| \end{array}$ .

Part 2. The Physical Reality

Why do we care about triangular Fourier series? Let us imagine that we have $n$ particles (bodies) on the plane, which move in circular orbits. For the sake of simplicity, we assume that the trajectories of the particles are concentric at 0. We can then write $u_j(t) = c_j e^{i d_j t}$ to denote the position of the jth particle on the complex plane; we note immediately that $u_j'(t) = i d_j u_j(t)$ . Finally, let us also assume that each $d_j$ is distinct from one another.

Suppose that the particles influence each other in the following way, where $a_{kj}(t) \in \mathbb{C}$ for each $t$ :

$\displaystyle u_j'(t) = i d_1 u_1 (t) + \sum_{k \neq 1} a_{kj}(t) u_k(t)$ .

In this case, we can write $u' = iDu + Au$ , where

$\displaystyle u = \begin{pmatrix} u_1(t) \\ u_2(t) \\ \vdots \\ u_n(t) \end{pmatrix}, \mbox{ } D = \begin{pmatrix} d_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & d_n \end{pmatrix}, \mbox{ } A = \begin{pmatrix} 0 & & a_{ij} \\ & \ddots & \\ a_{ij} & & 0 \end{pmatrix}.$

We can now ask the following:

Question 1. Do we have $\|a_{ij}\|_\infty < \infty$ for all $j$ ?

To answer this question, we shall embed the system into a more general system

$u' = i \lambda D u + A u$ ,

where $\lambda \in \mathbb{R}$ . We can then ask the similar question:

Question 2. Do we have $\|a_{ij}^\lambda\|_\infty < \infty$ for all $j$ and for almost every $\lambda \in \mathbb{R}$ ?

These systems are called AKNS systems, which are connected to mathematical physics, KdV, NLS, and so on.

Let us now solved the problem posed above. An ansatz is that the solutions are of the form

$u_j(t) = e^{i \lambda d_j t} v_j(t)$

for some $v_j(t)$ . If this is the case, then we have

$v ' = Wv$

where $W$ is an n-by-n matrix such that

$W_{lm}(t) = a_{lm}(t) \cdot e^{i \lambda (d_l-d_m) t}$ .

We consider $n = 2$ first, as it is as difficult as the whole problem. In fact, if we can understand this case in full generality, then we will have obtained an understanding of the whole problem.

Since we hope to simplify the system further, we would like as many zeroes in our matrix as possible. In other words, we solve the following system:

$\begin{array}{rcl} a_{12}(t) &=& f(t) \\ a_{21}(t) &=& 0 \end{array}$ .

This yields a 2-by-2 upper triangular system

$\begin{pmatrix} 0 & f(t) e^{i\lambda(d_1-d_2)t} \\ 0 & 0 \end{pmatrix}$ ,

whence we obtain

$\begin{pmatrix}v_1'(t) \\ v_2'(t) \end{pmatrix} \begin{pmatrix} 0 & f(t) e^{i\lambda(d_1-d_2)t} \\ 0 & 0 \end{pmatrix} \begin{pmatrix} v_1(t) \\ v_2(t) \end{pmatrix}$ .

If we work in the simple subcase of $d_1-d_2 = 1$ , then we can solve this system by hand using linear algebra:

$\begin{array}{rcl} v_1'(t) &=& f(t) e^{\lambda t}v_2(t) \\ v_2'(t) &=& 0. \end{array}$

This yields, $v_2^\lambda = c_\lambda$ , which implies that

$v_1'(t) = c \cdot f(t) e^{i \lambda t}$ ,

whence

$\displaystyle v_1(t) = c \int_{-\infty}^t f(s) e^{i \lambda s} \, ds$ .

The above identity yields the bound

$\|v_1(t)\|_\infty \leq \|f\|_1$ .

We would like to have $\|v_1(t)\|_\infty < \infty$ , so that the particles will not fly away. One way to obtain this bound is to have $f$ be an $L^1$ -function, whence our problem reduces to investigating the boundedness of

$\displaystyle f \mapsto \sup_t \left| \int_{-\infty}^t f(s) e^{i \lambda s} \, ds \right|.$

We can consider another continuous analogue $C$ of the Carleson operator introduced in the first part of the talk, known as Carleson’s operator:

$\displaystyle C(g) = \sup_N \left| \int_{\xi < N} \hat{g}(\xi) e^{-i \lambda \xi} \, d\xi \right|.$

By Carleson-Hunt, we know that $C:L^p \to L^p$ is bounded for $1 < p < \infty$ . Let us study this operator further.

If $\hat{g} \in L^p$ for some $1 \leq p < 2$ , and if $p'$ is the conjugate exponent of $p$ , then Carleson-Hunt yields

$\|C(\hat{g})\|_{p'} \lesssim \|\hat{\hat{g}}\|_{p'}$ ,

whence we have

$\|\hat{\hat{g}}\|_{p'} \lesssim \|\hat{g}\|_p$

by the Hausdorff-Young inequality. Therefore, we have the following inequality:

$\displaystyle \left\| \sup_N \left| \int_{\xi < N} \hat{g}(\xi) e^{i \lambda \xi} \, d\xi \right| \right\|_{p'} \lesssim \|\hat{g}\|_p$ .

We note, however, that the $L^{p'} \to L^{p'}$ boundedness of

$\displaystyle \Xi \mapsto \sup_N \left| \int_{\xi < N} \Xi(\xi) e^{i \lambda \xi} \, d\xi \right|$

is given by Menshow-Zygmund (1930s) and so does not require Carleson-Hunt. (If $p = 2$ , then we’d need Carleson, which requires much more effort.) So then, what does Carleson by itself give? First and foremost, we have the following inequality for $1 \leq p < 2$ :

$\|C(g)\|_p \lesssim \|g\|_p$ .

If $f = \hat{g}$ and $g \in L^q$ for some $2 \leq q < 3$ , then once again Carleson gives the above estimate. Indeed, the Fourier transform of an $L^{1+\varepsilon}$ function decays very, very slowly.

The conclusion is that the question posed above has an affirmative answer if we have a potential in $L^p$ for some $1 \leq p < 2$ . Furthermore, the question also has an affirmative answer if we have a potential which is the Fourier transform of an $L^p$ -function for some $1 \leq p < 2$ .

We might ask ourselves whether this is true in general. In other words, if, for some $1 \leq p < 2$ , the functions $a_{ij}$ are either themselves $L^p$ or are Fourier transforms of $L^p$ -functions, could we say that no particles fly away? The first question has an affirmative answer, by a result of M. Christ and A. Kiselev from about ten years ago. The second question is a bit more complicated. For simplicity’s sake, we shall consider the problem in $L^2$ . The general case of our problem considers an n-by-n upper triangular matrix, which yields the operator

$\displaystyle (f_1,f_2,\ldots,f_d) \mapsto \sup_t \left| \int_{s_1 < s_2 < \cdots < s_d < t} f_1(s_1) f_2(s_2) \cdots f_d(s_d) e^{i \lambda (\#_1 s_1 + \#_2 s_2 + \cdots + \#_d s_d)} \, ds_1 \cdots s_d \right|$ .

In general, the above operator is bounded. There is also the Carleson variation of this operator—this just means we put a hat on each $f_j$ .

Mark H. Kim

Triangular Fourier Series and Physical Reality

Part 1. Triangular Fourier Series

Part 2. The Physical Reality

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Mark H. Kim

Triangular Fourier Series and Physical Reality

Part 1. Triangular Fourier Series

Part 2. The Physical Reality

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