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Harmonic Analysis: The Hilbert Transform
*September 30, 2010*

*Posted by Sarah in Uncategorized.*

Tags: harmonic analysis, hilbert transform

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Tags: harmonic analysis, hilbert transform

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I’ve really been enjoying my harmonic analysis class and I thought I’d write up some recent notes.

The Poisson kernel, , is defined as

This is the real part of a holomorphic function. Now, what are the properties of that holomorphic function?

We define the Herglotz kernel as

where .

As (this is why the Poisson kernel solves the Dirichlet problem in the disc.) Similarly, as , some unknown function. This is the conjugate function of f, and convolution with is known as the Hilbert transform.

Now, we can write explicitly as

similar to the Poisson kernel, except for the sign function. Summing this series,

while

This conjugate function of need not be bounded, even if is. In other words, the Hilbert transform is not bounded in the norm. It’s not bounded in the norm either. But it’s clearly bounded in the norm.

The Hilbert transform can be shown to be bounded for all , by first showing that it satisfies a weak-type inequality and then showing that all linear operators satisfying a weak-type inequality are bounded in (this is called the Marcinkiewicz Interpolation Theorem.) I might type that up another time.

If we let , then is sort of an envelope for , larger in absolute value and smoother, and having the property that it stretches and shrinks with the function.

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What is the Selberg Trace Formula?
*August 19, 2010*

*Posted by Sarah in Uncategorized.*

Tags: harmonic analysis, selberg trace formula

2 comments

Tags: harmonic analysis, selberg trace formula

2 comments

Ngo Bau Chau was one of the winners of the Fields Medal for 2010, so I thought I’d try to understand what it is he studied. The laudation gives a clear explanation of his contribution to the Langlands program. Ngo’s proof applies the Selberg trace formula, and since that’s accessible, I thought I’d try to understand that.

If is a Lie group and is a cofinite discrete group, we want to understand the operator

on .

is an integral operator, with kernel

And now we can express the trace of the operator as

The result is

where are the conjugacy classes of , is the centralizer of in , and are the automorphic representations.

When and , the Selberg Trace Formula is the Poisson Summation Formula

The Poisson Summation formula can be seen to be a trace formula as follows:

The eigenvectors of the positive Laplacian on the unit circle are for integer , and the eigenfunctions are

Consider the linear operator

with kernel

Then

and so the Poisson Summation Formula says

The Selberg Trace Formula is proven in a similar manner to the proof of the Poisson Summation Theorem. If is the regular representation of on

then we can write R as

fixing a Haar measure on . Then, by splitting the integral

.

We now take the trace of this operator in two different ways:

on one side,

We break the sum over into conjugacy classes of . Each conjugacy class contributes

On the other side, we can compute the trace alternatively by quoting a result that says decomposes into a direct sum of irreducible representations of G:

.

This completes the proof. Conjugacy classes and irreducible representations are dual — in the same way that integers (in the Fourier series) and functions are dual in the Poisson summation formula.

So what can you do with the trace formula? One thing is to let be the fundamental group of a Riemann surface, and describe the spectrum of differential operators such as the Laplace-Beltrami operator using geometric data like the lengths of geodesics. (Way back in sophomore year I was trying to learn about this from Peter Buser’s book.) The Selberg Trace Formula is also useful in analyzing the Riemann zeta function.

See these notes for a bunch of background on the Selberg Trace Formula.

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Meyer/Basarab paper
*August 18, 2010*

*Posted by Sarah in Uncategorized.*

Tags: compressed sensing, harmonic analysis, yves meyer

2 comments

Tags: compressed sensing, harmonic analysis, yves meyer

2 comments

this is what I’m reading through right now: “A variant on the compressed sensing of Emmanuel Candes,” by Yves Meyer and Matei Basarab. I’m not done with it, but it involves reconstructing a function by sub-Nyquist sampling of its Fourier coefficients, using an -minimization technique. It’s a generalization of a theorem by Terence Tao about functions on finite fields. Looks to be interesting.