Abstract
Full Text
MATHEMATICS
A. D. MYSHKIS and A. Ya. LEPIN
ON THE DEFINITION OF GENERALIZED FUNCTIONS
(Presented by Academician S. L. Sobolev on 31 V 1957)
The concept of a generalized function was first introduced by S. L. Sobolev \((^{1,2})\) as a linear functional in a space of “fundamental functions.” At present there are a number of approaches to the definition of a generalized function (see, for example, \((^{3-9})\)). The aim of this note is, starting from the definition of Mikusiński—Korevaar \((^5)\), to give a general definition of various classes of generalized functions.
By a class of estimates \(\mathfrak M\) we shall mean any collection of nonnegative functions (all functions under consideration are assumed to be real-valued and continuous on the axis \(-\infty < x < \infty\)) having the following properties:
1) if \(0 \le m(x) \le m_1(x) \in \mathfrak M\), then \(m(x) \in \mathfrak M\);
2) if \(m_1(x) \in \mathfrak M\) and \(m_2(x) \in \mathfrak M\), then \(m_1(x) + m_2(x) \in \mathfrak M\);
3) \(1 \in \mathfrak M\);
4) if \(m(x) \in \mathfrak M\), then
\[
\max_{0 \le t \le 1} m(tx) \in \mathfrak M;
\]
5) if \(m(x) \in \mathfrak M\), then \(|x|\,m(x) \in \mathfrak M\).
Let a class of estimates \(\mathfrak M\) be given. We shall say that a sequence of functions \(F_n(x)\) \((n=1,2,\ldots)\) \(\mathfrak M\)-almost uniformly converges to a function \(F(x)\), and write \(F_n(x) \to F(x)\), if this sequence converges to \(F(x)\) uniformly on every finite interval and, moreover,
\[
\sup_n |F_n(x)| \in \mathfrak M .
\]
A sequence \(f\) of functions \(f=\{f_n(x)\}\) is called \(\mathfrak M\)-fundamental if there exists an integer \(k \ge 0\) and an \(\mathfrak M\)-almost uniformly convergent sequence of \(k\)-times continuously differentiable functions \(F=\{F_n(x)\}\), for which
\[
F_n^{(k)}(x)=f_n(x)\quad (n=1,2,\ldots)
\]
(or, in shorter notation, \(F^{(k)}=f\)).
We shall say that two \(\mathfrak M\)-fundamental sequences
\[
f=\{f_n(x)\}\quad \text{and}\quad \varphi=\{\varphi_n(x)\}
\]
are \(\mathfrak M\)-equivalent if there exists an integer \(k \ge 0\) and sequences \(F=\{F_n(x)\}\) and \(\Phi=\{\Phi_n(x)\}\), for which
\[
F^{(k)}=f,\qquad \Phi^{(k)}=\varphi,\qquad F_n(x)-\Phi_n(x)\to 0.
\]
The introduced notion of \(\mathfrak M\)-equivalence is symmetric, reflexive, and transitive; hence the totality of all \(\mathfrak M\)-fundamental sequences decomposes into classes of \(\mathfrak M\)-equivalence, each of which, by definition, is an \(\mathfrak M\)-generalized function in the sense of Mikusiński—Korevaar. We denote the set of these generalized functions (the symbol \(\mathfrak M\) will be omitted) by \(M_{\mathfrak M}\). This definition reduces to the definition of Mikusiński—Korevaar if \(\mathfrak M\) is the collection of all nonnegative functions.
If \(\bar f \in M_{\mathfrak M}\), \(\bar\varphi \in M_{\mathfrak M}\), then by \(\alpha \bar f+\beta \bar\varphi\), by definition, is meant the generalized function containing \(\alpha f+\beta \varphi\) \((f\in \bar f,\ \varphi\in \bar\varphi)\). Thus the set \(M_{\mathfrak M}\) becomes a linear space.
Every generalized function \(\bar f\in M_{\mathfrak M}\) contains a sequence \(\omega\) consisting of infinitely differentiable functions; by definition, the derivative \(\bar f^{(l)}\) \((l=1,2,\ldots)\) is understood to be the generalized function containing \(\omega^{(l)}\).
If \(f(x)\) is a function for which the sequence \(\{f(x), f(x),\ldots\}\) is fundamental (this will be so if and only if there exists an integer \(k\geq 0\) and a \(k\)-times continuously differentiable function \(F(x)\) for which \(|F(x)|\in\mathfrak M\), \(F^{(k)}(x)=f(x)\)), then by \(\hat f\) is meant the generalized function containing this sequence. Thus a one-to-one mapping is defined of the linear space \(C_{\mathfrak M}\) of such functions \(f(x)\) onto the subspace \(\hat C_{\mathfrak M}\) of the space \(M_{\mathfrak M}\).
Let us note some properties of generalized functions. \(\hat f^{(l)}=\hat\varphi\) if and only if the function \(f(x)\) is \(l\) times continuously differentiable and \(f^{(l)}(x)=\varphi(x)\). If \(\bar f'=\hat F\), then
\[ \bar f=\widehat{\int_0^x F(t)\,dt}+\hat C, \]
where \(C\) is some constant.
Every generalized function can be represented as a derivative of some order of a generalized function from \(\hat C_{\mathfrak M}\) (“Schwartz’s theorem”).
We introduce the definition of (strong) convergence in the space \(M_{\mathfrak M}\) in the same way, following Mikusiński. If \(\bar f_n\in M_{\mathfrak M}\) \((n=1,2,\ldots)\) and \(\bar f\in M_{\mathfrak M}\), then we shall write that \(\bar f_n\to\bar f\) if, for some integer \(k\geq 0\), there exist functions \(F_n(x)\) \((n=1,2,\ldots)\) and \(F(x)\) for which \(F_n(x)\to F(x)\) in \(\mathfrak M\), \(\hat F_n^{(k)}=\bar f_n\) \((n=1,2,\ldots)\), \(\hat F^{(k)}=\bar f\). If \(\bar f_n\to\bar f\), then \(\bar f'_n\to\bar f'\). The subspace \(\hat C_{\mathfrak M}\) is everywhere dense in \(M_{\mathfrak M}\).
We now introduce the collection \(O_{\mathfrak M}\) of “\(\mathfrak M\)-basic functions” \(s(x)\), infinitely differentiable and such that
\[ s^{(k)}(x)m(x)\to 0\quad (k=0,1,\ldots) \]
as \(|x|\to\infty\) for every \(m(x)\in\mathfrak M\). It is obvious that \(O_{\mathfrak M}\) is a linear space (with respect to the usual operations).
If \(\bar f\in M_{\mathfrak M}\), \(\bar f=\hat F^{(k)}\), \(|F(x)|\in\mathfrak M\), and \(s(x)\in O_{\mathfrak M}\), then the integral
\[ (s,\bar f)=(-1)^k\int_{-\infty}^{\infty}s^{(k)}(x)F(x)\,dx \]
converges and depends only on \(\bar f\) and \(s(x)\), and if \(\bar f_n\to\bar f\), then \((s,\bar f_n)\to(s,\bar f)\).
Define convergence in the space \(O_{\mathfrak M}\) by saying that \(s_n(x)\to s(x)\) \((s_n(x)\in O_{\mathfrak M};\ n=1,2,\ldots;\ s(x)\in O_{\mathfrak M})\) if, for every \(k=0,1,\ldots\), the sequence \(s_n^{(k)}(x)\) converges to \(s^{(k)}(x)\) uniformly and if
\[ m(x)\sup_n |s_n^{(k)}(x)|\to 0 \]
as \(|x|\to\infty\) for every estimate \(m(x)\in\mathfrak M\). (In the case where \(\mathfrak M\) consists of all nonnegative functions, the space \(O_{\mathfrak M}\) coincides with Schwartz’s space of finite functions; Schwartz \((^3)\) and I. M. Gel'fand and G. E. Shilov \((^4)\) also considered some other concrete spaces of basic functions.)
The notion of convergence introduced is weak convergence with respect to the scalar product \((s,\bar f)\), as is stated by the following theorem.
Theorem 1. Let \(s_n(x)\in O_{\mathfrak M}\) \((n=1,2,\ldots)\), \(s(x)\in O_{\mathfrak M}\). In order that, for every generalized function \(\bar f\in M_{\mathfrak M}\),
\[ (s_n,\bar f)\to (s,\bar f), \]
it is necessary and sufficient that
\[ s_n(x)\xrightarrow{\mathfrak M} s(x). \]
The scalar product \((s,\bar f)\) makes it possible to introduce in the space \(M_{\mathfrak M}\), besides strong convergence, also weak convergence: namely, for \(\bar f_n\in M_{\mathfrak M}\) \((n=1,2,\ldots)\), \(\bar f\in M_{\mathfrak M}\), we shall say that \(\bar f_n \xrightarrow{\mathrm{sl}} \bar f\) if \((s,\bar f_n)\to (s,\bar f)\) for every function \(s(x)\in O_{\mathfrak M}\). If \(\bar f_n \xrightarrow{\mathrm{sl}} \bar f\), then \(\bar f'_n \xrightarrow{\mathrm{sl}} \bar f'\). We note that the weak limit is unique.
The scalar product \((s,\bar f)\) is a linear continuous functional. It turns out that there are no other linear continuous functionals in the space \(M_{\mathfrak M}\).
Theorem 2. Every linear functional \(k(\bar f)\) \((\bar f\in M_{\mathfrak M})\), continuous with respect to strong convergence, can be represented in the form \((s,\bar f)\) for some function \(s(x)\in O_{\mathfrak M}\), which is determined uniquely by the formula
\[ s(a)=\frac{1}{2}k\!\left(\widehat{(x-a)+|x-a|''}\right)\quad (-\infty<a<\infty). \]
This theorem establishes a natural one-to-one correspondence between the space of linear continuous functionals on the space \(M_{\mathfrak M}\) and the space of basic functions \(O_{\mathfrak M}\). It is an isomorphism with respect to linear operations and, if the derivative of a functional \(k(\bar f)\) is defined by the formula \(k'(\bar f)=-k(\bar f')\), also with respect to differentiation. If weak convergence is introduced in the space of linear continuous functionals on \(M_{\mathfrak M}\), then the indicated correspondence will preserve convergence.
By the Schwartz space \(S_{\mathfrak M}\) we shall mean the collection of linear continuous functionals on the space \(O_{\mathfrak M}\), this collection being endowed with the usual linear operations, differentiation according to the formula \(T'(s)=-T(s')\) \((s(x)\in O_{\mathfrak M})\), and weak convergence.
Theorem 1 shows that, for fixed \(\bar f\in M_{\mathfrak M}\), the scalar product \((s,\bar f)\) is a linear continuous functional on the space \(O_{\mathfrak M}\), i.e., an element of the space \(S_{\mathfrak M}\), which we shall denote by \(\bar f^{\,*}\), so that \(\bar f^{\,*}(s)=(s,\bar f)\). This relation defines a natural mapping of the space \(M_{\mathfrak M}\) onto a subspace \(M_{\mathfrak M}^{*}\) of the space \(S_{\mathfrak M}\). This mapping is one-to-one and is an isomorphism with respect to linear operations and the operation of differentiation. Moreover, this mapping is continuous, since the convergence \(\bar f_n^{\,*}\to \bar f^{\,*}\) is equivalent to the convergence \(\bar f_n \xrightarrow{\mathrm{sl}} \bar f\). The space \(M_{\mathfrak M}^{*}\) consists of all functionals of finite order, i.e., functionals of the form \(T^{(k)}(s)\) \((k=0,1,\ldots)\), where
\[ T(s)=\int_{-\infty}^{\infty} F(x)s(x)\,dx \quad (|F(x)|\in \mathfrak M). \]
We shall say that a class of estimates \(\mathfrak M\) has a countable basis if there exists a sequence \(m_i(x)\in\mathfrak M\) \((i=1,2,\ldots)\) such that any estimate \(m(x)\in\mathfrak M\) does not exceed \(m_i(x)\) for all \(x\in(-\infty,\infty)\) for some \(i=1,2,\ldots\).
Theorem 3. If \(\mathfrak M\) has a countable basis, then \(M_{\mathfrak M}^{*}=S_{\mathfrak M}\).
Thus, when the class of estimates has a countable basis, each of the spaces \(M_{\mathfrak M}\) and \(O_{\mathfrak M}\) is isomorphic to the space of linear discontinuous functionals on the other.
In the case where there is a countable basis, by introducing a norm one can turn the space \(O_{\mathfrak M}\) into a countably normed linear space. (In this case the spaces \(O_{\mathfrak M}\) and \(S_{\mathfrak M}\) become a special case of a broader class of spaces studied by I. M. Gelfand and G. E. Shilov (unpublished).) Using Theorem 3, one could obtain Theorems 1 and 2 by means of the general theory of countably normed linear spaces.
Kharkov Aviation Institute
Received
27 X 1956
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