From: baez@math.ucr.edu (john baez)
Newsgroups: sci.math.research
Subject: Re: Rigorous theory of digital signal processing?
Date: 29 Jun 1998 14:45:49 -0700
In article <6mtsue$fs8@charity.ucr.edu>, john baez wrote:
>1) The space of distributions on S^1. This is the topological dual
>of the space of infinitely differentiable functions on S^1. It's
>a Frechet space.
>2) The space of hyperfunctions on S^1. This is the topological dual of
>the space of real-analytic functions on S^1. Again it's a Frechet space.
Thanks go to Allan Adler for helping me realize that I'd screwed up here.
With their usual weak topologies these spaces are not Frechet spaces; they
are something a bit more general: locally convex topological vector spaces.
I'm away from my books so some of the following definitions may deviate from
the standard ones, but apart from that I think everything will be okay.
Probably if I screw up someone will catch me.
A "topological vector space" is a vector space with a topology
such that the vector space operations (addition, scalar multiplication)
are continuous. A topological vector space is "locally convex" if
the origin has a basis of neighborhoods that are convex.
A "seminorm" |.| on a vector space is something with all the properties
of a "norm" except that |v| = 0 needn't imply v = 0. In particular,
note that for any seminorm, the set {v: |v - w| < c} is convex, so if we
use these as a basis of neighborhoods for the point w, we get a topology
on our vector space making it a locally convex topological vector space.
More generally, if we have any collection of seminorms |.|_i, we can use
them to make our vector space into a locally convex topological vector space
having the sets {v: |v - w|_i < c} as a subbasis of neighborhoods of w.
(I.e., finite intersections of such sets form a basis.) To avoid a
non-Hausdorff topology, let us demand that for any nonzero v, at least
one of our seminorms has |v|_i > 0. If in addition our collection of
seminorms is countable we call our locally convex topological vector
space a "Frechet space".
Let D = -i d/d theta as a self-adjoint operator on L^2(S^1). The space
of infinitely differentiable functions on the circle becomes a Frechet
space with the seminorms
|f|_n = ||D^n f||
where ||.|| is the usual L^2 norm. (The sup norm, or any other L^p norm
where p >= 1, will do just as well.) I don't know a similar trick to
make the space of analytic functions on the circle into a Frechet space.
However one can use the following trick to make them into a locally
convex topological vector space. For any n > 0, there is a Banach
space V_n consisting of real-analytic functions on the unit circle
that analytically continue to L^2 functions on an annulus of width 1/n
centered about the unit circle; the norm of this Banach space is
just the L^2 norm of this analytic continuation. We have V_n
contatined in V_{n+1} for all n, and the space of all analytic
functions on the circle is just the union of these nested spaces
V_n. Any increasing union of Banach spaces becomes a locally
convex topological vector space with something called the "inductive
limit topology" - so use that. For details on this how this topology
works, see the references given in my previous post. It's easy
to describe convergence of sequences in this topology: a sequence
of functions converges if it eventually stays in some V_n and converges
as a sequence in the Banach space V_n.
The dual V* of a locally convex topological vector space V naturally
becomes a locally convex topological vector space with the so-called
"weak topology". Here there is one seminorm |.|_v on V* for each element
v of V; the seminorm |.|_v is defined by
|f|_v = |f(v)|.
By this means the space of distributions and the space of hyperfunctions
on the circle become locally convex topological vector spaces. There
are also other popular topologies, but the weak topology is probably
the most important.
I can easily imagine hyperfunctions being useful in digital signal
processing, because the idea of analytically continuing functions
from the unit circle to a neighborhood thereof is basic to this subject.
However, I don't know if anyone in digital signal processing has ever
applied hyperfunctions. I'm pretty sure they've thought to apply
distributions, since distributions are widely used in engineering.
==============================================================================
From: [email removed upon request] (N Christopher Phillips)
Newsgroups: sci.math.research
Subject: Definition of Frechet space; was Re: Rigorous theory of digital signal processing?
Date: 1 Jul 1998 14:16:48 -0700
In article <2g90mfiouh.fsf@hera.wku.edu>,
Allen Adler wrote:
>
>There is one detail in John Baez' informative posting that I'm
>confused about. My vague recollection is that a Frechet space is
>defined to be a topological vector space whose topology is defined
>by a translation invariant metric. Is that the correct definition?
This is a response to Allen Adler's article referenced above,
and also to one by Stefan Larsson about the definition of a Frechet
space.
According to Rudin's Functional Analysis (1973 edition),
Definition 1.8, an F-space is a topological vector space whose topology
is given by a _complete_ invariant metric. A Frechet space is a locally
convex F-space. Both are required to be complete.
I don't think this terminology is universal--some people do not
require their Frechet spaces to be locally convex. But they are always
(as far as I know) required to be complete.
A topological vector space is a Frechet space in Rudin's sense
if and only if it is complete and its topology is determined by a
countable family of (continuous) seminorms. (Completeness makes sense
for arbitrary topological vector spaces.)
---Chris Phillips
==============================================================================
From: baez@math.ucr.edu (john baez)
Newsgroups: sci.math.research
Subject: Re: Rigorous theory of digital signal processing?
Date: 2 Jul 1998 16:12:11 -0700
In article <6najl4$cqp$1@zingo.tninet.se>,
Stefan Larsson wrote:
>Is this the standard definition?
>In fact, the requirement of countability here means that you can prove that
>such a topology can always be defined by a translation-invariant metric.
>My understanding was that a Frechet space is a locally convex Hausdorff
>space, the topology of which can be defined by a metric, _such that the
>space is complete as a metric space_?
Ugh, you're right, I left out completeness, which is very important
if you want to actually do any analysis with one of these guys!