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UDC 513.881:519.46
MATHEMATICAL PHYSICS
S. S. SANNIKOV
SPACES OF TEST AND GENERALIZED FUNCTIONS OF EXPONENTIAL TYPE
(Presented by Academician N. N. Bogolyubov, April 7, 1967)
1. Since the introduction by Dirac of the concept of the $\delta$-function, test and generalized functions have played a fundamental role in theoretical physics. For example, in constructing the mathematical foundations of quantum mechanics ($^1$) (the spectral theory of linear operators) it was found that the framework of Hilbert space is too narrow for the needs of this theory; extensions of Hilbert spaces to spaces of generalized functions ($^2$) or rigged Hilbert spaces ($^3$) are necessary. As is well known, the spectral theory of operators of the form $p^k$ ($p$ is a differential operator, $k$ is a number) is well served by a certain type of spaces of test and generalized functions introduced by L. Schwartz ($^2$). We shall call the original Schwartz spaces spaces of test and generalized functions of power type.
In studying infinite-dimensional representations of compact Lie groups, in particular the rotation group $R_3$ ($^4$), we shall need a new type of spaces of test and generalized functions, which we shall call spaces of exponential type. This type of space is well adapted to the study of operators of the form $K^p$ ($K$ is any complex number). The present note is devoted to a description of this type of space.
2. Let $\Phi$ be a linear space. In the abstract model the elements $\varphi$ of the space $\Phi$ are infinite numerical sequences
$\varphi = (\varphi_0, \varphi_1, \varphi_2, \ldots)$. Consider the scalar product
\[ (\varphi,\psi)_K=\sum_{n=0}^{\infty} |K|^{2n}\varphi_n\overline{\psi}_n, \tag{1} \]
where $K$ is any complex number. We shall consider in $\Phi$ only a countable system of scalar products of the form (1): $|K| = 1, 2, \ldots$ (so that the first axiom of countability is satisfied). Obviously, for any $\varphi \in \Phi$ the inequalities
\[ (\varphi,\varphi)_1 \leq (\varphi,\varphi)_2 \leq \cdots \leq (\varphi,\varphi)_K \leq \cdots \tag{2} \]
hold.
It can be shown that the norms generated by the scalar products (2) are pairwise compatible (see the examples). Then the completions $H_K$ of the space $\Phi$ with respect to the scalar products $(\varphi,\psi)_K$ are connected by the inclusion
($H_K$, $|K| = 1,2,\ldots$ are complete Hilbert spaces)
\[ H_1 \supset H_2 \supset \cdots \supset H_K \supset \cdots \supset \Phi. \]
The complete countably Hilbert space $\Phi$ is the intersection of all $H_K$:
\[ \Phi=\bigcap_{|K|=1}^{\infty} H_K. \]
The topology in $\Phi$ is introduced as follows. A complete countable system of neighborhoods of zero of the space $\Phi$ is given by the sets $U_{K,1/m}(0)$,
\(|K| = 1, 2, \ldots;\ m = 1, 2, \ldots,\) consisting of all \(\varphi \in \Phi\) for which \((\varphi,\varphi)_p < 1/m,\ |p| = 1, 2, \ldots, |K|\).
A sequence \(\varphi_N \in \Phi\) converges to \(\varphi \in \Phi\) in the (original) topology in \(\Phi\) if \((\varphi_N - \varphi, \varphi_N - \varphi)_K \to 0\) for every \(K\) as \(N \to \infty\).
Definition 1. A countably Hilbert space in which the topology is given by a countable set of (compatible) scalar products of the form (1) will be called a space of test functions of exponential type.*
It is known that every complete countably normed space can be made into a complete Fréchet metric space \((^5)\) if one defines in it the distance
\[ \rho(\varphi,\psi)=\sum_{k=1}^{\infty}\frac{1}{2^k}\, \frac{\|\varphi-\psi\|_k}{1+\|\varphi-\psi\|_k}. \]
In our case \(\|\varphi\|_K=\sqrt{(\varphi,\varphi)_K}\).
We now consider the space \(\Phi'\), conjugate to \(\Phi\) with respect to the scalar product \((f,\varphi)_1,\ f \in \Phi',\ \varphi \in \Phi\). It is the union of an increasing sequence of complete Hilbert spaces \(H'_K\), conjugate to \(H_K\):
\[ H'_1 \subset H'_2 \subset \ldots \subset H'_K \subset \ldots \subset \Phi', \quad \text{i.e.} \quad \Phi'=\bigcup_{|K|=1}^{\infty} H'_K; \]
\(\Phi'\) is the space of linear functionals on \(\Phi\).
Definition 2. The space \(\Phi'\), conjugate to the space of test functions of exponential type \(\Phi\), will be called the space of generalized functions of exponential type.
In \(\Phi'\) we shall consider only the weak topology. A weak neighborhood of zero in \(\Phi'\) is specified by a number \(\varepsilon>0\) and by an arbitrary finite set \(\varphi_1,\varphi_2,\ldots,\varphi_n \in \Phi\) as the totality of all \(f \in \Phi'\) for which \(|(f,\varphi_p)_1| < \varepsilon,\ p=1,2,\ldots,n\). A sequence \(f_N \in \Phi'\) converges weakly to \(f\) if \((f_N,\varphi)_1 \to (f,\varphi)_1\) for all \(\varphi \in \Phi\).
The space \(\Phi'\) can be made into a metric space \((^6)\) if one defines in it the distance
\[ \rho(f,g)=\sum_{k=1}^{\infty}\frac{1}{2^k}\, \frac{p_k(f-g)}{1+p_k(f-g)}, \qquad f,g \in \Phi', \]
where \(p_k(f)=|(f,\varphi_k)_1|,\ k=1,2,\ldots\), is a countable number of seminorms defining the weak topology.
Next, in each of the \(H_K\) consider the orthonormal systems
\[ {}^{(K)}e_n=\frac{1}{|K|^n}\,{}^{(1)}e_n, \qquad ({}^{(K)}e_n,{}^{(K)}e_{n'})_K=\delta_{nn'}. \]
\(\{{}^{(K)}e_n\}_{n=0}^{\infty}\) is a complete system in \(H_K\), so that every element \(\varphi \in H_K\) can be represented in the form
\[ \varphi=\sum_{n=0}^{\infty}(\varphi,{}^{(K)}e_n)_K\,{}^{(K)}e_n. \]
For \(|K|<|K'|\) the mapping \(T^{K}_{K'}\) of the space \(\tilde H_K\) into \(\tilde H_{K'}\) has the form
\[ T^{K}_{K'}\,{}^{(K)}\varphi = \sum_{n=0}^{\infty}\lambda_n (\varphi,{}^{(K)}e_n)_K\,{}^{(K')}e_n \in H_{K'}, \qquad {}^{(K)}\varphi \in H_K, \]
* In the abstract model, Schwartz’ original spaces are specified by a countable system of scalar products of the form
\[ (\varphi,\varphi)_k=\sum_{n=0}^{\infty} n^{2k}|\varphi_n|^2, \qquad k=0,1,2,\ldots . \]
The synthesis of these scalar products and the scalar products of the form (1) is
\[ (\varphi,\varphi)_{K,p} = \sum_{n=0}^{\infty} |K|^{2n} n^{2p} |\varphi_n|^2, \qquad |K|=1,2,\ldots;\ p=0,1,2,\ldots . \]
where \(\lambda_n=\left|\dfrac{K}{K'}\right|^n\). Since the series \(\sum_{n=0}^{\infty}\lambda_n=\dfrac{|K'|}{|K'|-|K|}\) converges, the mapping \(T_{K'}^{K}\) is nuclear \((^3)\). Thus, the following holds:
Theorem. Countably Hilbert spaces of exponential type are nuclear.
This fact means that all bounded sets in \(\Phi\) are compact (such spaces are called perfect); weak convergence in \(\Phi\) coincides with strong convergence (the latter, as is known \((^5)\), in the case of countably normed spaces coincides with the original one); a countable sum of bounded sets in \(\Phi\) does not give the whole of \(\Phi\) (bounded sets in \(\Phi\) are nowhere dense).
3. Examples.
A. Consider the Hilbert space \(H^+\) of square-integrable functions \(f(x)\) on the interval \([0,2\pi]\):
\((f,f)=\int_{0}^{2\pi}|f(x)|^2dx\). The system of functions \(\{e^{inx}\}_{n=0}^{\infty}\) forms a complete system (an orthonormal basis) in \(H^+\), so that any function \(f(x)\in H^+\) can be represented in the form
\[ f(x)=\sum_{n=0}^{\infty} f_n e^{inx}, \qquad (f,f)=\sum_{n=0}^{\infty}|f_n|^2 . \]
In this realization, the space of test functions of exponential type \(\Phi\) is formed by functions
\[ \varphi(x)=\sum_{n=0}^{\infty}\varphi_n e^{inx}, \]
square-integrable on the interval \([0,2\pi]\) under any of its parallel translations in the complex plane \((x)^*\), i.e., if \(\varphi\in\Phi\), then also \(L_K\varphi(x)=\varphi(x-i\ln K)\in\Phi\), where \(0\le |K|<\infty\), and
\[ (\varphi,\varphi)=\sum_{n=0}^{\infty}|\varphi_n|^2, \qquad (L_K\varphi,L_K\varphi)=\sum_{n=0}^{\infty}|K|^{2n}|\varphi_n|^2 =(\varphi,\varphi)_K . \]
(An example of such a function is \(\varphi(x)=\exp(e^{ix})\).)
If we denote \(e^{ix}=z\), then \(\Phi\) is formed by all entire analytic functions \(\varphi(z)\) of the complex variable \(z\) (these functions are square-integrable on the circle of any radius \(|z|=|K|\)). It follows from this that the scalar products (1), (2) are pairwise consistent. Indeed, if a sequence of entire analytic functions \(\varphi_N(z)\) converges to \(\varphi(z)\), this means that for any \(K\), \(\|\varphi_N-\varphi\|_K\to0\) as \(N\to0\). Thus, if \(|K|>|K'|\) and \(\|\varphi_N\|_{K'}\to0\), then also \(\|\varphi_N\|_K\to0\), i.e., the scalar products (2) are consistent.
In the realization under consideration, the space of generalized functions \(\Phi'\) (linear functionals on \(\Phi\)), in addition to entire functions, also includes analytic functions \(f(z)\) with singularities concentrated in any part of the complex plane \(z=e^{ix}\). Here the elements \(f\in\Phi'\) are germs of analytic functions \(f(z)\), considered only in their Mittag-Leffler star \((^7)\), where they are represented by series
\[ f(z)=\sum_{n=0}^{\infty} f_n z^n \]
with coefficients \(|f_n|\to |K|^n\) as \(n\to\infty\), \(0\le |K|<\infty\). Series of this type we shall call summable in the sense of convergen-
* In the realization under consideration, the original Schwartz space of test functions \((^2)\) is formed by functions \(\varphi(x)\) square-integrable together with all their derivatives on \([0,2\pi]\). An example of such a function may be
\[ \varphi(x)=(1-ae^{ix})^{-1},\qquad |a|<1. \]
in \(\Phi'\).* An example may be the function \(f(z)=(1-Az)^{-1}\), \(0\le |A|<\infty\).
B. Another example of a space \(\Phi'\supset H\supset \Phi\) is the class of entire functions of the complex variable \(z=x+iy\), considered in \((^7,^8)\). In this realization the Hilbert space \(H\) is formed by entire analytic functions of order \(\rho\le 2\) and type \(\tau\) \((0\le \tau<\infty\), if \(\rho<2\), and \(0\le \tau<1/2\), if \(\rho=2)\).
The scalar product in \(H\) is the form
\[
(f,g)=\int f(z)\overline{g}(z)\,d\mu(z),
\quad \text{where } d\mu(z)=\frac{1}{\pi}e^{-|z|^2}\,dz,\quad dz=dx\,dy.
\]
The space of basic functions of exponential type \(\Phi\) is formed by entire analytic functions of order \(\rho<2\) and type \(0\le \tau<\infty\), while the conjugate space \(\Phi'\) is formed by entire analytic functions of order \(\rho\le 2\) and type \(0\le \tau<\infty\).
- In conclusion let us make several remarks concerning infinite-dimensional irreducible representations of compact Lie groups. The basis for introducing the topology considered above in the representation space \(\Phi_\lambda\) (\(\lambda\) is a set of complex numbers specifying an irreducible representation \(D(\lambda)\) of the Lie group \(G\)) is the existence of a bilinear functional (4) (cf. \((^{10})\))
\[ [f^{(\lambda)},\varphi^{(\bar\lambda)}]_\lambda = (f^{(\lambda)}, I_{\bar\lambda}\varphi^{(\bar\lambda)})_\lambda \tag{3} \]
with the following properties: a) \(f^{(\lambda)}\in\Phi'_\lambda\), \(\varphi^{(\bar\lambda)}\in\Phi_{\bar\lambda}\), where \(\Phi_{\bar\lambda}\) and \(\Phi'_\lambda\) are spaces of basic and generalized functions of the type considered (\(\Phi_{\bar\lambda}\) is a countably normed space; \(\Phi'_\lambda\) is the space of linear continuous functionals on \(\Phi_\lambda\); with respect to (3) the representations \(D(\lambda)\) and \(D(\bar\lambda)\) are conjugate to one another); b) (3) is a nondegenerate functional, and
\[ [f^{(\lambda)},\varphi^{(\bar\lambda)}]_\lambda = [f^{(\bar\lambda)},\varphi^{(\lambda)}]_{\bar\lambda}; \]
c)
\[ [f^{(\lambda)},T_{\bar\lambda}^{-1}(g)\varphi^{(\bar\lambda)}]_\lambda = [T_\lambda(g)f^{(\lambda)},\varphi^{(\bar\lambda)}]_\lambda, \]
although \(T_{\bar\lambda}^{-1}(g)\varphi^{(\bar\lambda)}\notin\Phi_{\bar\lambda}\), where \(g\to T_\lambda(g)\), \(g\in G\), is the representation \(D(\lambda)\) of the group \(G\) in \(\Phi'_\lambda\).
For real \(\lambda\), (3) turns into a strongly indefinite scalar product.**
Physical-Technical Institute
Academy of Sciences of the Ukrainian SSR
Received
4 IV 1967
CITED LITERATURE
- I. Neumann, Mathematical Foundations of Quantum Mechanics, Moscow, 1964.
- L. Schwartz, Application of Generalized Functions to the Study of Elementary Particles in Relativistic Quantum Mechanics, Moscow, 1964.
- I. M. Gel'fand, N. Ya. Vilenkin, Generalized Functions, vol. 4, Moscow, 1961.
- S. S. Sannikov, Nuclear Physics, 2, 570 (1965).
- I. M. Gel'fand, G. E. Shilov, Generalized Functions, vol. 2, Moscow, 1958.
- Yu. P. Ginzburg, I. S. Iokhvidov, Uspekhi Mat. Nauk, 17, 3 (1962).
- G. Hardy, Divergent Series, Moscow, 1951.
- S. S. Sannikov, Doklady AN, 176, No. 4 (1967).
- V. Bargmann, Comm. Pure and Appl. Math., 14, 187 (1961).
- I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions, vol. 5, Moscow, 1962.
* The method of summation presented is a functional analogue of the classical summation methods of Le Roy and Mittag-Leffler \((^7)\).
** The remarks given concern only the so-called discrete series of representations. We do not touch upon the continuous series of representations here.