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.

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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])?

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