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E. B. VUL
ON THE UNIQUENESS OF REPRESENTATION OF CERTAIN CLASSES OF POSITIVE-DEFINITE GENERALIZED FUNCTIONS
(Presented by Academician M. V. Keldysh, 8 VIII 1960)
In the work of I. M. Gel'fand and S. Do-shin 1, questions connected with representations of positive-definite even linear functionals defined on the spaces \(Z\) and \(Z_2^2\) of basic functions were considered. Below, some of these questions are considered for other spaces of basic functions.
Following the book of I. M. Gel'fand and G. E. Shilov 2, we introduce the space \(W_M^\Omega\) of entire functions \(\varphi(z)=\varphi(x+iy)\) satisfying the inequalities
\[ |\varphi(x+iy)|\leq ce^{-M(a|x|)+\Omega(b|y|)}, \]
where the positive numbers \(a\), \(b\), and \(c\) depend on the function \(\varphi(z)\), and \(M\) and \(\Omega\) are increasing functions having continuous increasing derivatives. Denote by \(\overset{0}{W}{}_M^\Omega\) the subspace of the space \(W_M^\Omega\) consisting of even functions. Conditions for the nontriviality of such spaces were investigated in the works of S. Mandelbrojt 3 and K. I. Babenko 4.
Consider on the space \(\overset{0}{W}{}_M^\Omega\) a linear functional \(T\), given in the form
\[ (T,\varphi)= \int_0^\infty \int_{-\infty}^{\infty} \cos x\lambda\, \varphi(x)\, dx\, d\mu(\lambda) + \int_0^\infty \int_{-\infty}^{\infty} \operatorname{ch} x\lambda\, \varphi(x)\, dx\, d\sigma(\lambda), \tag{1} \]
where the measures \(\sigma(\lambda)\) and \(\mu(\lambda)\) are such that
\[ \int_0^\infty \left|\int_{-\infty}^{\infty} \cos x\lambda\, \varphi(x)\, dx\right| d\mu(\lambda)<\infty, \]
\[ \int_0^\infty \left|\int_{-\infty}^{\infty} \operatorname{ch} x\lambda\, \varphi(x)\, dx\right| d\sigma(\lambda)<\infty \]
for all functions \(\varphi\in \overset{0}{W}{}_M^\Omega\). It is easy to see that the functional \(T\) satisfies the relation \(T(\varphi * \varphi^*)\geq 0\), where \(\varphi\in \overset{0}{W}{}_M^\Omega\), \(\varphi^*(z)=\varphi(-\bar z)\), and is therefore positive-definite. Functionals of this type were studied in the work of A. G. Kostyuchenko and B. S. Mityagin 5. We shall be interested in the question of when, in the expansion (1), the measures \(\mu(\lambda)\) and \(\sigma(\lambda)\) are determined uniquely.
Let us pass to the Fourier transform of the space \(\overset{0}{W}{}_M^\Omega\). It is shown in 2 that the Fourier transform of the space \(W_M^\Omega\) is the space
\(W_{\Omega_1}^{M_1}\), where \(\Omega_1\) is the function conjugate in the sense of Young* to \(M\), and \(M_1\) is the function conjugate in the sense of Young to \(\Omega\). It is obvious that the Fourier transform of the space \(\overset{0}{W}{}_{M}^{\Omega}\) will be the space \(\overset{0}{W}{}_{\Omega_1}^{M_1}\). The linear functional \(T\), defined on the space \(\overset{0}{W}{}_{M}^{\Omega}\) by formula (1), is transformed into the linear functional \(T^*\) on the space \(\overset{0}{W}{}_{\Omega_1}^{M_1}\), written in the form
\[ (T^*,\psi)=\int_0^\infty \psi(x)\,d\mu(x)+\int_0^\infty \psi(iy)\,d\sigma(y), \tag{2} \]
where
\[ \int_0^\infty |\psi(x)|\,d\mu(x)<\infty,\qquad \int_0^\infty |\psi(iy)|\,d\sigma(y)<\infty . \]
The question of the uniqueness of the representation of the functional \(T\) in the form (1) has thereby been reduced to the question of the uniqueness of the representation of the functional \(T^*\) in the form (2). Under the assumption that the finite limit \(\lim_{x\to\infty}\dfrac{xM_1'}{M_1}\) exists, Theorems 1 and 2 hold.
Theorem 1. If
\[ \int_1^\infty \frac{M_1(x)}{x^3}\,dx=\infty, \]
then to every positive-definite functional \(T\) on the space \(\overset{0}{W}{}_{M}^{\Omega}\), given in the form (1), there correspond unique measures \(\mu(\lambda)\) and \(\sigma(\lambda)\).
Theorem 2. If
\[ \int_1^\infty \frac{M_1(x)}{x^3}<\infty, \]
then there exist measures \(\mu(\lambda)\ne\mu_1(\lambda)\) and \(\sigma(\lambda)\ne\sigma_1(\lambda)\) such that, for any function \(\psi(z)\in \overset{0}{W}{}_{\Omega_1}^{M_1}\),
\[ \int_0^\infty \psi(x)\,d\mu(x)+\int_0^\infty \psi(iy)\,d\sigma(y) = \int_0^\infty \psi(x)\,d\mu_1(x)+\int_0^\infty \psi(iy)\,d\sigma_1(y) \tag{3} \]
and the integrals in (3) converge absolutely.
Proof of Theorem 1. It suffices to consider only the spaces \(W_{M_1}^{M_1}\), since from the condition that the space is nonempty it follows that \(W_{M_1}^{M_1}\in W_{\Omega_1}^{M_1}\) (4). Suppose that the representation of the functional \(T^*\) in the form (2) is not unique. Then there exist such absolutely additive set functions \(\tilde{\mu}\) and \(\tilde{\sigma}\) that
\[ \int_0^\infty \psi(x)\,d\tilde{\mu}(x)+\int_0^\infty \psi(iy)\,d\tilde{\sigma}(y)=0, \tag{4} \]
where \(\psi(z)\) is an arbitrary function from \(\overset{0}{W}{}_{M_1}^{M_1}\), and \(\tilde{\mu}\) and \(\tilde{\sigma}\) have bounded variation on every finite interval and the integral in (4) converges absolutely. To each function \(\psi(z)\in \overset{0}{W}{}_{M_1}^{M_1}\) assign the function
* If \(g(x)\) is the inverse function to \(\Omega'(x)\), then the function \(\displaystyle \int_0^x g(t)\,dt\) is called the function conjugate in the sense of Young to the function \(\Omega(x)\).
\(\psi_1(z)=\psi(\sqrt{\overline z})\). Then for the function \(\psi_1(z)\) the inequalities
\[ \begin{aligned} |\psi_1(x)| &\leq c e^{-M_1(a\sqrt{x})}, && x>0;\\ |\psi_1(x)| &\leq c' e^{M_1(b\sqrt{|x|})}, && x<0. \end{aligned} \tag{5} \]
hold.
It follows from (4) that, for any entire function \(\psi_1(z)\) satisfying the inequalities (5),
\[ \int_{-\infty}^{\infty}\psi_1(x)\,d\nu(x)=0, \tag{6} \]
where
\[ d\nu(x)= \begin{cases} d\tilde{\mu}(x^2), & x\geq 0;\\ d\tilde{\sigma}(x^2), & x<0. \end{cases} \]
Moreover, the integral in (6) converges absolutely for any function \(\psi_1(x)\), if \(\psi_1(z^2)\in \overset{0}{W}{}_{M_1}^{M_1}\).
It can be shown that for any \(c>0\) there exists a function \(\varphi(z)\) from \(\overset{0}{W}{}_{M_1}^{M_1}\) satisfying the inequality
\[ |\varphi(iy)|>e^{cM_1(|y|)}. \tag{7} \]
It then follows from (6) that
\[ \int_{-\infty}^{0} e^{cM_1(\sqrt{|y|})}\,d\nu(y)<\infty \]
for any \(c>0\). Put \(d\nu_1(x)=\psi_1^2(x)d\nu(x)\), where \(\psi_1(z^2)\in \overset{0}{W}{}_{M_1}^{M_1}\). If \(P(x)\) is a polynomial, then \(P(x)\psi_1^2(x)\) again satisfies the inequalities (5), and
\[ \int_{-\infty}^{\infty}P(x)\,d\nu_1(x)=0. \tag{8} \]
We shall now show that it follows from the hypothesis of the theorem that
\[ \int_{-\infty}^{\infty}f(x)\,d\nu_1(x)=0, \tag{9} \]
where \(f(x)\) is an arbitrary continuous finite function. Indeed, using S. N. Bernstein’s theorem (see the survey article \({}^{6}\)), for any \(\varepsilon>0\) we find a polynomial \(P(x)\) such that for all \(x\)
\[ |P(x)-f(x)|e^{-M_1(\sqrt{|x|})}<\varepsilon. \]
Then
\[ \left|\int_{-\infty}^{\infty} f(x)\,d\nu_1(x)\right| = \left|\int_{-\infty}^{\infty}(f(x)-P(x))\,d\nu_1(x)\right| \leq \varepsilon\int_{-\infty}^{\infty} e^{M_1(\sqrt{|x|})}\,d\nu_1(x) = \varepsilon\cdot \mathrm{const}. \]
In view of the arbitrariness of \(\varepsilon\), we have \(\int_{-\infty}^{\infty} f(x)\,d\nu_1(x)=0\). Consequently, \(d\nu_1=d\nu=0\). The theorem is proved.
Proof of Theorem 2. Let the function \(\widetilde M(x)\) be such that
\[ \lim_{x\to\infty}\frac{\widetilde M(x)}{M(x)}=\infty,\qquad \int_1^\infty \frac{\widetilde M(x)}{x^3}\,dx = 2\int_1^\infty \frac{\widetilde M(\sqrt{x})}{x^2}\,dx<\infty. \]
In (7), for a nondecreasing function \(\widetilde M\) for which
\[ \int_1^\infty \frac{\widetilde M(\sqrt{x})}{x^2}\,dx<\infty, \]
an entire function \(F(z)\) is constructed which takes real values on the real axis and satisfies, for all \(z\), the inequality
\[ |F(z)|\le ce^{\beta|y|-\widetilde M(\sqrt{|z|})},\qquad \beta<0. \]
Consider the function \(F_1(z)=F(z)e^{i\beta z}\). In the upper half-plane it satisfies the inequality
\[ |F_1(z)|<ce^{-\widetilde M(\sqrt{|z|})}, \]
and therefore, for any function \(\psi_1(z)\), \(\psi_1(z^2)\in W_{\Omega}^{M_1}\),
\[ \lim_{R\to\infty}\int_{C_R'} \psi(z)F_1(z)\,dz=0 \]
(where \(C_R\) is the semicircle \(\operatorname{Im} z\ge 0,\ |z|=R\)), whence it follows that
\[ \int_{-\infty}^{\infty}\psi_1(x)F_1(x)\,dx=0. \tag{10} \]
Carrying out analogous arguments for the function \(F_2(z)=F(z)e^{-i\beta z}\) in the lower half-plane, we obtain that
\[ \int_{-\infty}^{\infty}\psi_1(x)F_2(x)\,dx=0. \]
Adding (10) and (11), we have
\[ \int_{-\infty}^{\infty}\psi_1(\lambda)F(x)\cos\beta x\,dx=0. \]
Let
\[ dv_1=\max(0,F(x)\cos\beta x)\,dx, \]
\[ dv_2=\max(0,-F(x)\cos\beta x)\,dx. \]
Then
\[ \int_{-\infty}^{\infty}\psi_1(x)\,dv_1(x) = \int_{-\infty}^{\infty}\psi_1(x)\,dv_2(x) \]
and the measures \(\sigma(x^2)\), \(\mu(x^2)\) and \(\sigma_1(x^2)\), \(\mu_1(x^2)\), constructed from the measures \(\nu_1\) and \(\nu_2\), define one and the same functional \(T\).
I express my deep gratitude to I. M. Gelfand and M. A. Evgrafov for their constant attention to this work.
Received
2 VIII 1960
References
- I. M. Gelfand, S. Ya. Doshin, UMN, 15, issue 1 (1960).
- I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 2, Moscow, 1958.
- S. Mandelbrojt, C. R., 249, 2465 (1959).
- K. I. Babenko, DAN, 132, No. 6 (1960).
- A. G. Kostyuchenko, B. S. Mityagin, DAN, 131, No. 1 (1960).
- S. N. Mergelyan, UMN, 11, issue 5, 107 (1956).
- S. Mandelbrojt, Mathematics, Collection of Translations, 2, 3 (1958).