Schwartz Distribution Theory in N variables?

[lugita15]

The title says it all - is it possible to extend Schwartz's theory of distributions to N variables? I've heard that it can be easily extended to two variables using Schwartz's Kernel Theorem (does anyone know a good reference that explains this stuff?). Does this theorem carry over to N variables? Also, how does the notion of distributional derivative generalize to N variables?

Finally, a largely unrelated question: I've seen assertions that hyperfunctions are "infinite order distributions." What does that even mean? What is the order of a distribution, and how do hyperfunctions relate to distribution theory?

Any help would be greatly appreciated.

Thank You in Advance.

 

[strangerep]

does anyone know a good reference that explains [Schwartz's Kernel Theorem]

Haven't found what I'd call a "good" reference, but this one is certainly a classic reference:

Gelfand & Vilenkin (vol 4 of a series by Gelfand and several collaborators).

 

[lugita15]

Haven't found what I'd call a "good" reference, but this one is certainly a classic reference:

Gelfand & Vilenkin (vol 4 of a series by Gelfand and several collaborators).

Speaking of Gelfand Volume 4, is Schwartz's Kernel Theorem used for the distribution-valued inner product on a rigged Hilbert space?

 

[strangerep]

Speaking of Gelfand Volume 4, is Schwartz's Kernel Theorem used for the distribution-valued inner product on a rigged Hilbert space?

Not directly, afaik. But it occurs quite early in G&V (iirc), even before they start to construct nuclear spaces and their duals.

 

[lugita15]

Not directly, afaik. But it occurs quite early in G&V (iirc), even before they start to construct nuclear spaces and their duals.

But there should be some dependence, right, at least indirectly? Because the inner product is a delta function of two variables, and distributions of two variables require Schwartz's Kernel theorem.

By the way, I asked you about this in my RHS thread, but the distribution-valued inner product is a distribution on the rigged Hilbert space, so does that mean you need to define test functions on the rigged Hilbert space, and the inner product will be some kind of functional acting on those test functions, just like the plain-old Schwartz theory for ordinary spaces? Since the rigged Hilbert space (that is, the ket space) is anyway made up of distributions, we're talking about "distributions over distributions" as opposed to "distributions over numbers"!

 

[Hurkyl]

You can always apply multi-linear algebra...

If I recall correctly, the tensor product of the space of test functions in M variables with the space of test functions in N variables turns out to be the space of test functions in M+N variables, so this approach works out particularly nicely.

 

[strangerep]

But there should be some dependence, right, at least indirectly? Because the inner product is a delta function of two variables, and distributions of two variables require Schwartz's Kernel theorem.

Er,... no. A delta function $\delta(k-p)$ can be regarded as a family of ordinary distributions $\delta_p$, for each of which we have
$$
\Big(\delta_p \,,\, f\Big) ~=~ f(p)
$$
where the large parentheses denote the dual pairing operation.

By the way, I asked you about this in my RHS thread, but the distribution-valued inner product is a distribution on the rigged Hilbert space, so does that mean you need to define test functions on the rigged Hilbert space, and the inner product will be some kind of functional acting on those test functions, just like the plain-old Schwartz theory for ordinary spaces? Since the rigged Hilbert space (that is, the ket space) is anyway made up of distributions, we're talking about "distributions over distributions" as opposed to "distributions over numbers"!

Hmmm. Almost everything you said in that paragraph is wrong, or distorted. I don't know where to start...

Let's put distribution-valued inner products aside for a while.

Tell me which texts on rigged Hilbert space you've actually studied in detail, and then tell me what you think a "linear functional" is, what a "rigged Hilbert space" is, and what a "bilinear functional" is.

 

[strangerep]

You can always apply multi-linear algebra...

If I recall correctly, the tensor product of the space of test functions in M variables with the space of test functions in N variables turns out to be the space of test functions in M+N variables, so this approach works out particularly nicely.

Yes, but the step implied in the above, i.e.,
$$
(V \otimes V)^* ~=~ V^* \otimes V^*
$$
is straightforward in the finite-dim case but tricky in the inf-dim case -- which is what the SKT covers, iiuc.

 

[lugita15]

Er,... no. A delta function $\delta(k-p)$ can be regarded as a family of ordinary distributions $\delta_p$, for each of which we have
$$
\Big(\delta_p \,,\, f\Big) ~=~ f(p)
$$
where the large parentheses denote the dual pairing operation.

Where can I find more information about constructing a two-dimensional delta function as a family of delta functions? Would it be there in Gelfand?

Tell me which texts on rigged Hilbert space you've actually studied in detail

Just various bits and pieces gathered from Gelfand vol 4, Rafael de la Madrid's various papers, and PhysicsForums. Nothing in real detail.

and then tell me what you think a "linear functional" is, what a "rigged Hilbert space" is, and what a "bilinear functional" is.

A linear functional is a linear mapping from a vector space to the real line, and a bilinear functional is a linear mapping from the Cartesian product of a vector space and itself to the real line. A rigged Hilbert space arises from the Gelfand triplet consisting of a Hilbert space, a nuclear subspace on which unbounded Hilbert space operators are defined, and a space of anti-linear functionals, essentially Schwartz distributions except for the anti-linear part, using the nuclear subspace as the set of test functions on which the functionals act. But you'll have to excuse me, because I often slip up and just refer to the ket space, i.e. the space of anti-linerar functionals, out of this triplet as the "Rigged Hilbert Space".

 

[strangerep]

A linear functional is a linear mapping from a vector space to the real line, and a bilinear functional is a linear mapping from the Cartesian product of a vector space and itself to the real line.

OK, that's reasonably close. The actual definitions are a little more general, cf.
http://en.wikipedia.org/wiki/Bilinear_functional
Also, for QM we usually take the range to be the complex plane rather than just the real line.

A rigged Hilbert space arises from the Gelfand triplet consisting of a Hilbert space, a nuclear subspace on which unbounded Hilbert space operators are defined, and a space of anti-linear functionals, essentially Schwartz distributions except for the anti-linear part, using the nuclear subspace as the set of test functions on which the functionals act.

The case of Schwartz test functions and tempered distributions is a special case. But ok, that's enough to go on with...

[...] I often [...] just refer to the ket space, i.e. the space of anti-linear functionals, out of this triplet as the "Rigged Hilbert Space".

IMHO, that's actually a reasonable complaint. There's no overarching term for the whole thing, afaik. Maybe it deserves to be called a "Dirac space", since it was his approach to QM that motivated much of the rigorous development. :-)

Where can I find more information about constructing a two-dimensional delta function as a family of delta functions? Would it be there in Gelfand?

It depends what you mean by "two-dimensional delta function". I interpret it to mean a
bilinear functional $\delta_{pq},~ p,q \in \mathbb{C}$, acting on some vector space $\mathbb{F}$ of functions over $\mathbb{C}^2$, and defined via
$$
\def\Cz{\mathbb{C}}
\def\Fz{\mathbb{F}}
\Big( \delta_{pq} \,,\, f \Big) ~=~ f(p,q)
$$
Normally, we'd like to write the above as
$$
\int dx \delta(p-x) \int dy \, \delta(q-y) \; f(x,y) ~=~ f(p,q) ~.
$$

But now consider an arbitrary bilinear functional K satisfying
$$
\Big( K \,,\, f \Big) ~=~ z_f
$$
where $z_f$ is a family of complex numbers dependent on the function f.

The Schwartz kernel theorem (iiuc!) states that one can always express $K$
and its action in the form
$$
\int dx \int dy \, K(x,y) \; f(x,y) ~=~ z_f ~.
$$
That's why it's called the "kernel" theorem. The quantity $K(x,y)$ is the "integral kernel" defining the bilinear functional.

Depending on the details of K, it may or may not be possible to express it as a simple product $A(x)Y(y)$ as we did for the 2D delta distribution.

So maybe you can now re-study the sections in G+V on the kernel theorem, and you (or anyone else -- please!) can tell me whether I've got it right... :-)