Hilbert Space Interpretation of Fourier Transform

[thegreenlaser]

I've been taught (in the context of Sturm-Liouville problems) that Fourier series can be explained using inner products and the idea of projection onto eigenfunctions in a Hilbert space. In those cases, the eigenvalues are infinite, but discrete. I'm now taking a quantum mechanics course, and reading the 3rd chapter of Griffiths' Quantum book piqued my interest about the idea of continuous sets of eigenvalues and their corresponding eigenfunctions which may not be normalizable in the usual sense. Being in electrical engineering, the Fourier Transform is something that's relatively familiar to me, and it seems very much related to this idea. So my question is, is it possible to interpret the Fourier transform in terms of projection onto a set of eigenfunctions corresponding to a continuous set of eigenvalues (similar to Fourier series)? If so, can anyone point me to some resources explaining how this is done which would be understandable to a student who isn't majoring in math (but is willing to learn)?

As an idea of what I've done so far, I've been trying to read up on Rigged Hilbert Spaces and distribution theory, but I've mostly been getting bogged down in unfamiliar mathematics and I haven't been able to make a solid connection between those ideas and the Physics/Electrical Engineering I'm trying to understand better.

Thanks for any help.

 

[Tosh5457]

Sorry I don't know the answer, but can you explain what linear operator are you talking about when you say its eigenvalues are infinite?

I've been taught (in the context of Sturm-Liouville problems) that Fourier series can be explained using inner products and the idea of projection onto eigenfunctions in a Hilbert space. In those cases, the eigenvalues are infinite, but discrete

 

[thegreenlaser]

Sorry I don't know the answer, but can you explain what linear operator are you talking about when you say its eigenvalues are infinite?

Sorry, I meant to say there are (or at least could be) an infinite number of eigenvalues.

 

[Bacle2]

I don't know if this is what you were asking, but the Fourier series is the orthogonal projection of a function ( the function f to be represented by the F. series) into the space spanned by the eigenfunctions, and so, by properties of Hilbert spaces ( the orthogonal projection from a point into a subspace minimizes the distance of the point to the subspace) it --the F. series--is the best possible rep. of f, i.e., the Fourier coefficients a_i minimize the difference:

|| f- Ʃa_ie_i||


Where the e_i are the eigenfunctions.

 

[thegreenlaser]

I don't know if this is what you were asking, but the Fourier series is the orthogonal projection of a function ( the function f to be represented by the F. series) into the space spanned by the eigenfunctions, and so, by properties of Hilbert spaces ( the orthogonal projection from a point into a subspace minimizes the distance of the point to the subspace) it --the F. series--is the best possible rep. of f, i.e., the Fourier coefficients a_i minimize the difference:

|| f- Ʃa_ie_i||


Where the e_i are the eigenfunctions.

That's the sort of thing I'm, talking about, but I already understand it for Fourier series. What I'm asking is whether (and specifically how) those ideas can be extended to the Fourier transform, where the idea of an "orthonormal basis" seems to be a little bit fuzzy.

 

[jbunniii]

That's the sort of thing I'm, talking about, but I already understand it for Fourier series. What I'm asking is whether (and specifically how) those ideas can be extended to the Fourier transform, where the idea of an "orthonormal basis" seems to be a little bit fuzzy.

The ideas are analogous but the technical details are different. All $L^1(\mathbb{R})$ (absolutely integrable) functions have a Fourier transform, because
$$\left| \int_{-\infty}^{\infty} f(x) e^{i u x} dx \right| \leq \int_{-\infty}^{\infty} |f(x) e^{i u x}| dx = \int_{-\infty}^{\infty} |f(x)| dx < \infty$$
But the apparent "basis" functions are not in $L^1$:
$$\int_{-\infty}^{\infty} |e^{i u x}| dx = \int_{-\infty}^{\infty} (1) dx = \infty$$
and the inner product between any pair of them diverges, so you cannot say that they are orthogonal:
$$\langle e^{iux}, e^{ivx} \rangle = \int_{-\infty}^{\infty} e^{i u x} e^{-i v x} dx = \int_{-\infty}^{\infty} e^{i(u-v)x} dx \space \text{ diverges }$$

 

[Tosh5457]

Sorry, I meant to say there are (or at least could be) an infinite number of eigenvalues.

But what linear application are you talking about? Since eigenvalues are associated to a linear application.

 

[thegreenlaser]

But what linear application are you talking about? Since eigenvalues are associated to a linear application.

I'm not really sure what you're asking for to be honest. I don't mean this to be tied to a specific case. I'm wondering about how the theory of operators on a Hilbert space with discrete sets of eigenvalues (in particular ideas like projecting a vector onto a basis of eigenfunctions) can be extended to operators with continuous sets of eigenvalues (the Fourier transform being a particular example I'm interested in).

The ideas are analogous but the technical details are different. All $L^1(\mathbb{R})$ (absolutely integrable) functions have a Fourier transform, because
$$\left| \int_{-\infty}^{\infty} f(x) e^{i u x} dx \right| \leq \int_{-\infty}^{\infty} |f(x) e^{i u x}| dx = \int_{-\infty}^{\infty} |f(x)| dx < \infty$$
But the apparent "basis" functions are not in $L^1$:
$$\int_{-\infty}^{\infty} |e^{i u x}| dx = \int_{-\infty}^{\infty} (1) dx = \infty$$
and the inner product between any pair of them diverges, so you cannot say that they are orthogonal:
$$\langle e^{iux}, e^{ivx} \rangle = \int_{-\infty}^{\infty} e^{i u x} e^{-i v x} dx = \int_{-\infty}^{\infty} e^{i(u-v)x} dx \space \text{ diverges }$$

This is exactly the kind of thing I'm looking for! Could you point me to any resources that go more in-depth on this? For example, I'm curious to know how the Hilbert space theory underlying Fourier Series has to be altered/extended to deal with things like "basis vectors" which aren't actually members of the space they span so that it can be applied to Fourier Transforms (or other expansions corresponding to continuous sets of eigenvalues).

 

[strangerep]

Wow. A thread in the math forums I can usefully respond to...

As an idea of what I've done so far, I've been trying to read up on Rigged Hilbert Spaces and distribution theory, but I've mostly been getting bogged down in unfamiliar mathematics and I haven't been able to make a solid connection between those ideas and the Physics/Electrical Engineering I'm trying to understand better.

[...] I'm curious to know how the Hilbert space theory underlying Fourier Series has to be altered/extended to deal with things like "basis vectors" which aren't actually members of the space they span so that it can be applied to Fourier Transforms (or other expansions corresponding to continuous sets of eigenvalues).

OK, you do want Rigged Hilbert Spaces and distribution theory. Which books have you been reading? The standard QM text I recommend for getting an overview of RHS is Ballentine's "QM -- A Modern Development", chapter 1. I'd guess that's probably the best starting point to see the connection between the maths and the physics -- assuming your current math skills allow you to study and understand that chapter successfully. Then probably revisit the math textbooks for more detail and rigor.

 

[Tosh5457]

I'm not really sure what you're asking for to be honest. I don't mean this to be tied to a specific case. I'm wondering about how the theory of operators on a Hilbert space with discrete sets of eigenvalues (in particular ideas like projecting a vector onto a basis of eigenfunctions) can be extended to operators with continuous sets of eigenvalues (the Fourier transform being a particular example I'm interested in).

I'm just asking this question out of ignorance, since I never thought of an orthogonal base of a Hilbert space as eigenfunctions. I know the base can be the eigenfunctions of a normal matrix, associated to a normal application. But how is that related to the Fourier Series? Are the coefficients of the Fourier series the eigenvalues? And what is that linear application?

 

[thegreenlaser]

I'm just asking this question out of ignorance, since I never thought of an orthogonal base of a Hilbert space as eigenfunctions. I know the base can be the eigenfunctions of a normal matrix, associated to a normal application. But how is that related to the Fourier Series? Are the coefficients of the Fourier series the eigenvalues? And what is that linear application?

Are you familiar with the idea of a treating functions as vectors in a Hilbert space? Correct me if I'm wrong, but when you're saying things like "normal matrix" and "normal application" I'm getting the sense you've taken some linear algebra but haven't really seen those ideas applied to functions. If that's the case I would suggest reading up on function spaces/inner product spaces and then maybe look at boundary-value problems as an application. I don't mean to ignore your questions, but if you're not familiar with these ideas I think it's too much to introduce in a single post, and it wouldn't help this thread very much.

 

[thegreenlaser]

Wow. A thread in the math forums I can usefully respond to...

OK, you do want Rigged Hilbert Spaces and distribution theory. Which books have you been reading? The standard QM text I recommend for getting an overview of RHS is Ballentine's "QM -- A Modern Development", chapter 1. I'd guess that's probably the best starting point to see the connection between the maths and the physics -- assuming your current math skills allow you to study and understand that chapter successfully. Then probably revisit the math textbooks for more detail and rigor.

That's actually one of the books I've been reading in the last couple days. So far it's been the most helpful, although I wish he went on to discuss it more. For example, near the end of that section he says that the "generalized spectral theorem" states that all self-adjoint operators in the Hilbert space have a complete set of eigenfunctions in the extended space (where the delta function lives). But in what sense are they "complete"? If elements of the extended space are functionals and not functions, then what exactly does it mean to write a function in a Hilbert space as a linear combination of them?

 

[micromass]

That's actually one of the books I've been reading in the last couple days. So far it's been the most helpful, although I wish he went on to discuss it more. For example, near the end of that section he says that the "generalized spectral theorem" states that all self-adjoint operators in the Hilbert space have a complete set of eigenfunctions in the extended space (where the delta function lives). But in what sense are they "complete"? If elements of the extended space are functionals and not functions, then what exactly does it mean to write a function in a Hilbert space as a linear combination of them?

Talking about a "function in Hilbert space" is fine, but I consider it to be ambiguous. I would prefer to talk about elements of the Hilbert space. The thing you should realize is that elements of the Hilbert space can be identified with functionals on the Hilbert space. In QM notation, this is obvious: given the element $|x>$, we can associate $<x|$. And $<x|$ is a linear functional on the Hilbert space. If you want to talk about rigged hilbert spaces, then you are going to take element $|x>$ and associate it with a linear functional again. But the domain of the functional is not longer the Hilbert space, but a different space. The association element -> functional can be seen as an identification. So we don't distringuish between the element and the functional. So if we write an element as a linear combination of functionals, then that means that we write the associated functional as a linear combination. The key idea is that there are a lot more functionals than there are elements in the Hilbert space. Because there are more functionals, this allows us to find a complete eigenbasis.

An extremely good book is the following: https://www.amazon.com/dp/0821847902/?tag=pfamazon01-20
I think that book contains the answer to your OP. It deals with Fourier transforms in the setting you want. It also deals with distribution theory: so he explains what a I wrote above in more detail. In particular, Folland explains how to see an element of a space as a functional. He does not go into rigged Hilbert spaces though.
There are also not a lot of prerequisites, so I think you can start reading the book immediately.

 

[Tosh5457]

Are you familiar with the idea of a treating functions as vectors in a Hilbert space? Correct me if I'm wrong, but when you're saying things like "normal matrix" and "normal application" I'm getting the sense you've taken some linear algebra but haven't really seen those ideas applied to functions. If that's the case I would suggest reading up on function spaces/inner product spaces and then maybe look at boundary-value problems as an application. I don't mean to ignore your questions, but if you're not familiar with these ideas I think it's too much to introduce in a single post, and it wouldn't help this thread very much.

Yes I'm familiar with that. It's just that when you talk of eigenfunctions and eigenvalues, they have to be associated with a linear application, and I'm not seeing what that linear application is. And the only reason the base is made of the eigenfunctions of an application it's because it's a normal application.
In my course we only saw that the complex exponentials are the basis of the Hilbert space of functions in the Fourier Theorem conditions (which are solutions of some PDEs with certain boundary conditions as you said), and because that basis is countable and infinite, we get a series, with the coefficients being the inner product between the function and each basis function. But we never talked about them being the eigenfunctions of some application, like you're suggesting...

EDIT: Ah I think I understand now. Basically you write the PDE with certain boundary conditions in the form as an hermitian operator (a particular case of a normal application), like in Quantum Mechanics. So that's why you refer to the basis of the space as eigenfunctions, because the eigenfunctions of that PDE operator are the basis of the space.

 

[thegreenlaser]

Talking about a "function in Hilbert space" is fine, but I consider it to be ambiguous. I would prefer to talk about elements of the Hilbert space. The thing you should realize is that elements of the Hilbert space can be identified with functionals on the Hilbert space. In QM notation, this is obvious: given the element $|x>$, we can associate $<x|$. And $<x|$ is a linear functional on the Hilbert space. If you want to talk about rigged hilbert spaces, then you are going to take element $|x>$ and associate it with a linear functional again. But the domain of the functional is not longer the Hilbert space, but a different space. The association element -> functional can be seen as an identification. So we don't distringuish between the element and the functional. So if we write an element as a linear combination of functionals, then that means that we write the associated functional as a linear combination. The key idea is that there are a lot more functionals than there are elements in the Hilbert space. Because there are more functionals, this allows us to find a complete eigenbasis.

An extremely good book is the following: https://www.amazon.com/dp/0821847902/?tag=pfamazon01-20
I think that book contains the answer to your OP. It deals with Fourier transforms in the setting you want. It also deals with distribution theory: so he explains what a I wrote above in more detail. In particular, Folland explains how to see an element of a space as a functional. He does not go into rigged Hilbert spaces though.
There are also not a lot of prerequisites, so I think you can start reading the book immediately.

Thanks for this! I got that book out from the library and I'm slowly working through it. Hopefully between that and the Ballentine book I'll be able to figure everything out.

 

[strangerep]

[...] I wish [Ballentine] went on to discuss it more. [...]

For a longer discussion in a physics context, you could also try this paper: http://arxiv.org/abs/quant-ph/0502053

 

[strangerep]

An extremely good book is the following: https://www.amazon.com/dp/0821847902/?tag=pfamazon01-20

I took a closer look through that book and it is indeed more helpful than various other Fourier Transform books in the context of this thread. So I purchased a copy.