MATHEMATICS

Yu. I. GILDERMAN

THE FOURIER TRANSFORM FOR ABSTRACT SET FUNCTIONS

(Presented by Academician S. L. Sobolev, 23 I 1962)

In the paper \((^1)\), S. L. Sobolev, in connection with embedding theorems for abstract functions, considered the \(B\)-spaces \(\Phi_p(\Omega)\), \(p \geqslant 1\), of abstract additive set functions \(\varphi(E)\), defined for all \(L\)-measurable sets \(E\) from some bounded domain \(\Omega \in R_n\). Here, for any \(L\)-measurable set \(E \in R_n\), the function \(\varphi(E)\) is defined by the equality \(\varphi(E)=\varphi(E \cap \Omega)\). In the present note we construct the \(B\)-space \(\Phi_p(R_n)\) of abstract additive set functions \(\varphi(E)\), defined for all \(L\)-measurable sets \(E \in R_n\) of finite measure (in the case \(p=1\) the measure of \(E\) may also be infinite), and then, for these spaces, as well as for certain subspaces \(\Psi_1^*(R_n) \subset \Phi_1(R_n)\) and \(\Psi_2^*(R_n) \subset \Phi_2(R_n)\), we consider the Fourier transform. The theorems proved are analogous to the Riemann—Lebesgue theorem on the Fourier transform for the space \(L_1(R_n)\) and to Plancherel’s theorem on the Fourier transform for the space \(L_2(R_n)\).

Let \(X\) be a \(B\)-space in which multiplication by complex numbers is defined. Let \(\varphi(E)\) be an abstract additive function of \(L\)-measurable sets \(E \in R_n\), taking, for each \(E\), values in \(X\), such that \(\|\varphi(E)\|_X \leqslant A_E < \infty\). In particular, \(\|\varphi(R_n)\|_X \leqslant A_{R_n} < \infty\).

By introducing the norm \(\|\varphi(E)\|_M = \sup\limits_{E \in R_n} \|\varphi(E)\|_X\), from the vector manifold of such functions one can single out the \(B\)-space \(M(R_n)\). The norm of \(M(R_n)\) is equivalent to the norm
\(\|\varphi\|_{\Phi_1(R_n)} = \sup\limits_{E_1 \cap E_2=0} \|\varphi(E_1)-\varphi(E_2)\|_X\).

The totality \(\overline{\Phi}_1(R_n)\) of normal additive functions \(\varphi(E)\) constitutes a subspace of the space \(\Phi_1(R_n)\). Normality means the convergence to zero of the norm \(\|\varphi(E_k)\|_X\) as \(k \to \infty\), where \(\{E_k\}\) is a vanishing sequence of sets.

In contrast to the space \(\Phi_1(\Omega)\) of set functions from a bounded domain \(\Omega \in R_n\), absolute continuity and even continuity with respect to translation of a function \(\varphi(E) \in \Phi_1(R_n)\) do not imply the normality of \(\varphi(E)\). Moreover, in contrast to the space of numerical functions \(L_1(R_n)\), the totality of finite functions is dense not in all of \(\Phi_1(R_n)\), but in the space \(\Psi_1(R_n)\) of functions \(\varphi(E) \in \Phi_1(R_n)\) continuous with respect to translation.

Theorem 1. The totality of finite functions \(\varphi(E) \in \Phi_1(R_n)\) is dense in \(\overline{\Phi}_1(R_n)\).

Considering integrals of the form
\[ \int_{R_n} \omega(x)\,d\varphi(E), \]
where \(\varphi(E) \in \Phi_1(R_n)\), and \(\omega(x)\) is a bounded step function taking only a finite number of values, one can obtain the estimate
\[ \left\|\int_{R_n} \omega(x)\,d\varphi(E)\right\|_X \leqslant 2 \max_x |\omega(x)|\,\|\varphi(E)\|_{\Phi_1(R_n)}. \]

Hence, by passage to the limit, the integral \(\int_{R_n}\omega(x)\,d\varphi(E)\) is defined,

where \(\omega(x)\) is an arbitrary bounded measurable complex-valued function of the real variable \(x\).

Consider the space \(\Psi_1^*(R_n)=\Phi_1^*(R_n)\cap \Psi_1(R_n)\), which is the closure, in the metric \(\Phi_1(R_n)\), of the collection of abstract finite continuous point functions.

Theorem 2. To every function \(\varphi(E)\in \Psi_1^*(R_n)\) one can assign its Fourier transform

\[ F(u)=\frac{1}{(2\pi)^{n/2}}\int_{R_n} e^{i(u,x)}\,d_x\varphi(E). \]

The function \(F(u)\) is uniformly continuous, \(\|F(u)\|_X\to 0\) as \(|u|\to\infty\).

Remark. The Fourier transform \(F(u)\) can also be constructed for any function \(\varphi(E)\in \Phi_1(R_n)\). However, it is easy to indicate an example of a function \(\varphi(E)\in \Phi_1(R_n)\) whose Fourier transform is not a continuous function and does not tend to zero in norm as \(|u|\to\infty\).

Alongside the space \(\Phi_1(R_n)\), one may consider the spaces \(\Phi_p(R_n)\), \(p>1\), of abstract additive functions \(\varphi(E)\), defined on the collection of bounded \(L\)-measurable sets \(E\in R_n\) with values in a \(B\)-space \(X\) with bounded norm

\[ \|\varphi\|_{\Phi_p(R_n)} = \sup_{\omega\in L_{p'}(R_n)} \frac{ \left\|\int_{R_n}\omega(x)\,d\varphi(E)\right\|_X }{ \|\omega\|_{L_{p'}(R_n)} }, \qquad p>1,\quad \frac{1}{p}+\frac{1}{p'}=1. \]

Let us dwell in more detail on the case \(p=2\). The space \(\Phi_2(R_n)\) is a \(B\)-space. Every function \(\varphi(E)\in \Phi_2(R_n)\) is absolutely continuous in the metric \(\Phi_1(R_n)\), i.e. continuous in the norm \(X\), and is defined not only for bounded, but also for unbounded \(L\)-measurable sets \(E\in R_n\) of finite measure. The collection \(\Phi_2^*(R_n)\) of finite functions from \(\Phi_2(R_n)\) is dense neither in \(\Phi_2(R_n)\) nor in \(\Psi_2(R_n)\), where \(\Psi_2(R_n)\) is the space of functions from \(\Phi_2(R_n)\) that are continuous with respect to shifts.

Denote by \(\Psi_2^*(R_n)\) the intersection \(\Phi_2^*(R_n)\cap \Psi_2(R_n)\). The space \(\Psi_2^*(R_n)\) is the closure of the set of finite continuous abstract point functions in the metric \(\Phi_2(R_n)\). There exist functions \(\varphi(E)\in \Psi_2^*(R_n)\) that do not belong to \(\Phi_1(R_n)\). On the other hand, there exist functions \(\varphi(E)\in \Psi_1^*(R_n)\cap \Phi_2(R_n)\) that do not belong to \(\Phi_2^*(R_n)\).

Finally, let us note one more feature of the space \(\Phi_p(R_n)\), \(p>1\), distinguishing this space from the space \(\Phi_p(\Omega)\). There exist abstract point functions \(\varphi(x)\in \Phi_p(R_n)\), continuous in the norm \(X\), which at the same time are not continuous with respect to shifts in the norm \(\Phi_p(R_n)\).

Simple considerations show that the Fourier transform for a function \(\varphi(E)\in \Phi_2(R_n)\) in the form of the integral

\[ \int_{R_n} e^{-i(u,x)}\,d_x\varphi(E) \]

may have no meaning. At the same time, the following is valid:

Theorem 3. To every function \(\varphi(E)\in \Phi_2(R_n)\) one can assign its Fourier transform

\[ \Psi(E')=\frac{1}{(2\pi)^{n/2}}\int_{R_n}\left[\int_{E'} e^{i(u,x)}\,du\right]d_x\varphi(E) \]

so that \(\Psi(E')\in \Phi_2(R_n)\), \(\|\Psi\|_{\Phi_2(R_n)}=\|\varphi\|_{\Phi_2(R_n)}\), and

\[ \varphi(E)=\frac{1}{(2\pi)^{n/2}}\int_{R_n}\left[\int_E e^{i(u,x)}\,dx\right]d_u\Psi(E'). \]

If \(\varphi(E)\in \Psi_2^*(R_n)\), then also \(\Psi(E')\in \Psi_2^*(R_n)\).

As an example, let us construct the Fourier transform for the function

\[ \frac{1}{\sqrt{2\pi}} e^{iux}. \]

1. We form the set function \(\varphi(E)=\int_E \frac{1}{\sqrt{2\pi}} e^{iux}\,dx\). For any bounded \(E \in R_1\), the function \(\varphi(E)\) takes its value in \(X=L_2(R_1)\). It is easy to see that \(\varphi(E)\in \Phi_2(R_1)\). The Fourier transform for it is the abstract function \(\Psi(E')=\chi_{E'}(x)\in \Phi_2(R_1)\); \(\Psi(E')\in L_2(R_1)\) for each fixed \(E'\) of finite Lebesgue measure.

If the range of variation of \(u\) is a bounded domain \(\Omega\in R_1\), then instead of a set function one may consider the point function

\[ \frac{1}{\sqrt{2\pi}} e^{iux}. \]

1. Consider the space \(\mathcal L=L_1(R_1)\cap L_2(R_1)\) with norm \(\|f\|_{\mathcal L}=\|f\|_{L_1}+\|f\|_{L_2}\), and let \(X=\mathcal L^*\), where \(\mathcal L^*\) is the space of linear functionals on \(\mathcal L\). Then the function \(\frac{1}{\sqrt{2\pi}}e^{iux}\) may be regarded as an abstract function of the point \(u\) with values in \(X=\mathcal L^*\). In this case the functional is defined by the formula

\[ l(u\mid f)=\frac{1}{\sqrt{2\pi}}\int_{R_n} e^{iux} f(x)\,dx. \]

The function \(l(u)\) belongs to \(\Psi_2^*(R_1)\). The Fourier transform for it is the abstract function \(\Psi(E')\) with values in \(X=\mathcal L^*\),

\[ \Psi(E'\mid f)=\int_{E'} f(x)\,dx. \]

In conclusion, I express my gratitude to Academician S. L. Sobolev for his attention to this work.

Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
17 I 1962

REFERENCES

1. S. L. Sobolev, Fund. Math., 47, 3, 278 (1959).
2. V. I. Smirnov, A Course of Higher Mathematics, 5, Moscow, 1959.
3. N. I. Akhiezer, Lectures on Approximation Theory, Moscow–Leningrad, 1947.