7
$\begingroup$

On $\mathbb{R}^n$, let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{D}(\mathbb{R}^n)$ be the space of smooth, compactly supported functions.

According to p.145 of the book by Reed & Simon, tempered distributions have "polynomially bounded growth" while elements of $\mathcal{D}'(\mathbb{R}^n)$ have no restriction on growth at infinity.

Now, I wonder if we can think of any "intermediate" spaces of test functions between these two examples.

For example, is it possible to consider some space of test functions growing faster than any "exponential rate" at infinity and a continuous dual of this space? If so, can this space of test functions be given some nuclear Frechet topology?

I guess somebody must have thought of these notions already, but I cannot find relevant information in standard textbooks.

Could anyone please help me?

Improve this question
$\endgroup$
5
  • 2
    $\begingroup$ The first two volumes of Gel'fand-Shilov's Generalized Functions consider some of these intermediate test function spaces and their corresponding topological duals = "spaces of generalized functions", as far as I remember. It's the first place I'd look for that, at least. $\endgroup$ Commented May 25, 2024 at 0:04
  • 1
    $\begingroup$ I seem to recall that the book of J. Ian Richards and Heekyung Youn titled (IIRC) Theory of Distributions has something about this. They addressed the question: When can two generalized functions be multiplied by each other? If one of them is a smooth compactly supported function and they other is any generalized function, then it can be done. Their idea was that the less well behaved a function gets (with smooth compactly supported functions as the most well behaved of all), the fewer generalized functions it can be multiplied by. $\endgroup$ Commented May 25, 2024 at 17:43
  • 1
    $\begingroup$ ${} \qquad\uparrow\qquad$I think some of the ones considered by Richards and Youn were less well behaved than those in $\mathcal S. \qquad$ $\endgroup$ Commented May 25, 2024 at 17:47
  • 1
    $\begingroup$ The claim that tempered distributions have polynomially bounded growth is misleading: Since every bounded measurable function is a tempered distribution also its (distributional) derivative is tempered. In particular, $e^x\cos(e^x)= \sin(e^x)'$ is a tempered distribution. $\endgroup$ Commented May 26, 2024 at 12:02
  • $\begingroup$ @JochenWengenroth I see. Reed & Simon must have meant what you describe. $\endgroup$ Commented May 26, 2024 at 12:30

1 Answer 1

Reset to default
7
$\begingroup$

If $T$ is a positive (say $T\geq Id$) unbounded self-adjoint operator on Hilbert space, then $\bigcap D(T^n)$ ( intersection of the domains of the powers) has a natural Fréchet space structure. The Schwartz space you refer to is the special case of the standard one-dimensional Schrödinger operator. You can easily create spaces of the type you require by replacing this $T$ by $f(T)$ for suitable functions $f$, using the functional calculus. It is not clear precisely what kind of condition you want but you have complete freedom in the choice of growth conditions on $f$.

The resulting space will be Fréchet and nuclear under rather non-restrictive growth conditions on $f$. This follows from a theorem of Pietsch which states that the space generated by $T$ is nuclear if and only if the latter has discrete spectrum, with eigenvalues $(\lambda_n)$ which are such that $\Sigma \dfrac 1{\lambda_n^\alpha }<\infty$ for some positive $\alpha$.

The details are in „Über die Erzeugung von (F)-Räumen durch selbst-adjungierte Operatoren“ (in Math, Ann. vol. 164 but easily available online), MR196495, Zbl 0139.31003. The explicit form of the Schrödinger operator used is the final example there

Improve this answer
$\endgroup$
5
  • $\begingroup$ 1. Could you clarify what you mean by Schrodinger operator? What exactly is the potential term here? $\endgroup$ Commented May 24, 2024 at 22:04
  • $\begingroup$ 2. My original intention was to find a test function space larger than $\mathcal{D}$ on which the function $e^{x^2}$ is well-defined as a distribution. $\endgroup$ Commented May 24, 2024 at 22:07
  • $\begingroup$ 3. Could you provide me a reference to the Pietsch Theorem you mentioned? $\endgroup$ Commented May 24, 2024 at 22:08
  • 1
    $\begingroup$ Answer to 3. - have a look at A. Pietsch, Nuclear Locally Convex Spaces (Springer-Verlag, 1972). H. Jarchow's Locally Convex Vector Spaces (B.G. Teubner, 1981), the second volume of G. Köthe's book on topological vector spaces and the fourth volume of Gel'fand's "Generalized Functions" (with N. Ya. Vilenkin as the second author) should also have information on that. $\endgroup$ Commented May 25, 2024 at 0:08
  • 1
    $\begingroup$ @Isaac. You can take $f$ to be $\exp(x^2)$ or even $\exp\circ\exp \dots \exp$ to be on the safe side. $\endgroup$ Commented May 25, 2024 at 8:36

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.