Reports of the Academy of Sciences of the USSR
1967. Volume 173, No. 4

UDC 517.522 + 519.48

MATHEMATICS

S. P. GEISBERG

COMPLETENESS OF TRANSLATES OF A FUNCTION IN CERTAIN SPACES

(Presented by Academician V. I. Smirnov on 28 V 1966)

1°. The completeness* of the system of functions \(\{f_x(\tau)\}\), \(f_x(\tau)=f(x+\tau)\), \(-\infty<x<\infty\), in the spaces \(L_\alpha\),

\[ \|f\|=\int_{-\infty}^{\infty}|f(\tau)|e^{\alpha(\tau)}\,d\tau, \]

where \(\alpha(\tau)=\alpha(-\tau)>0\), \(\alpha(\tau)\) is nondecreasing, \(\alpha(\tau)/\tau\) is nonincreasing, \(\alpha(\tau_1+\tau_2)\leq \alpha(\tau_1)+\alpha(\tau_2)\), was studied in \((^{1-3})\). It turned out that the conditions for completeness depend on the rate of growth of \(\alpha(\tau)\). Shilov \((^2)\) showed that, for \(\alpha(\tau)=o(\ln \tau)\), the system \(\{f_x(\tau)\}\) is complete in \(L_\alpha\) if and only if

\[ \tilde f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(\tau)e^{ix\tau}\,d\tau \ne 0, \qquad -\infty<x<\infty . \tag{1} \]

Korenblum \((^3)\) studied the case \(\alpha(\tau)=c\tau\). He found that, instead of (1), the necessary and sufficient conditions** are

\[ \tilde f(z)\ne 0,\qquad |\operatorname{Im} z|\leq c,\qquad -\infty<\operatorname{Re} z<\infty, \]

\[ \varlimsup_{x\to\infty}\ln|\tilde f(x)|/e^{\pi x/2c}=0,\qquad \varlimsup_{x\to-\infty}\ln|\tilde f(x)|/e^{-\pi x/2c}=0. \tag{2} \]

The question naturally arises: for which \(\alpha(\tau)\) is condition (1) sufficient for the completeness of the system \(\{f_x(\tau)\}\) in \(L_\alpha\). Let \(N\) be the class of such functions \(\alpha(\tau)\). In the present paper a sufficient condition is obtained for \(\alpha(\tau)\) to belong to the class \(N\), which is necessary if \(\alpha(\tau)=o(\tau)\) and \(\tau\alpha'(\tau)/\alpha(\tau)\) is nonincreasing. It is shown that this condition is at the same time, in an analogous way, necessary and sufficient for the regularity of \(L_\alpha\), regarded as a normed ring with multiplication given by convolution. For nonregular rings \(L\), necessary completeness conditions are obtained which are analogous to conditions (2).

2°. Theorem 1. If

\[ \int_0^\infty \frac{\alpha(\tau)\,d\tau}{1+\tau^2}<\infty, \tag{3} \]

then the ring \(L_\alpha\) is regular and the system \(\{f_x(\tau)\}\) is complete in \(L_\alpha\) if and only if (1) holds.

Proof. We first show the regularity of \(L_\alpha\). From the description of the maximal ideals of \(L_\alpha\) obtained in \((^4)\) it follows that it suffices, for arbitrary \(x_0\) and \(\varepsilon>0\), to construct a function \(\varphi(x)\in L_\alpha\) such that \(\tilde\varphi(x_0)\ne 0\) and \(\tilde\varphi(x)=0\) for \(|x-x_0|\geq \varepsilon\). We may assume that \(x_0=0\) and

\[ \alpha(x)\leq \int_1^x \frac{\alpha(\tau)\,d\tau}{\tau} \]

* A system \(\{x_\nu\}\) of elements of a Banach space \(X\) is called complete in \(X\) if the closure of the linear span of the system coincides with \(X\).

** In what follows, \(\tilde f(x)\) denotes the Fourier transform of the function \(f(x)\), defined by the equality in formula (1).

for \(x \geqslant 2\). Put

\[ \beta(\tau)=\alpha(\tau)\left(\int_{\tau}^{\infty}\frac{\alpha(\theta)\,d\theta}{1+\theta^2}\right)^{-1/2},\qquad \tau \geqslant 0,\qquad \beta(-\tau)=\beta(\tau). \]

Using the condition imposed on \(\alpha(\tau)\), we obtain, for \(A>1\),

\[ \int_{0}^{\infty}\frac{\beta(\tau)\,d\tau}{1+\tau^2}<\infty;\qquad \beta(A\tau)\leqslant A^2\beta(\tau);\qquad \lim_{x\to\infty}\left(\int_{1}^{x}\frac{\alpha(\tau)\,d\tau}{\tau} -\int_{1}^{x/A}\frac{\beta(\tau)\,d\tau}{\tau}\right)=-\infty . \tag{4} \]

Next, let \(\omega(\tau)\) be the function inverse to \(\beta(\tau)\), and for \(x \geqslant 1\)

\[ K(x)=\exp\left(\int_{1}^{x}\frac{\alpha(\tau)\,d\tau}{\tau}\right),\qquad N(x)=\exp\left(\int_{0}^{\omega(x)}\beta'(\tau)\ln \tau\,d\tau\right),\qquad F(x)=\min_{\tau\geqslant 1} N(\tau)x^{-\tau}. \]

Then \(N'(\beta(x))/N(\beta(x))=\ln x\), and therefore \(F(x)=N(\beta(x))x^{-\beta(x)}\), i.e.

\[ F(x)=\exp\left(-\int_{1}^{x}\frac{\beta(\tau)\,d\tau}{\tau}\right). \tag{5} \]

Moreover, for \(x \geqslant 1\) we have

\[ \frac{N'(x)}{N(x)}=\ln \omega(x),\qquad \ln\frac{N(x+1)}{N(x)}\geqslant \ln\omega(x),\qquad \frac{N(x)}{N(x+1)}\leqslant \int_{x-1}^{x}\frac{d\theta}{\omega(\theta)}, \]

hence, by virtue of (4),

\[ \sum_{n=2}^{\infty}\frac{N(n)}{N(n+1)} \leqslant \int_{0}^{\infty}\frac{d\theta}{\omega(\theta)} =O(1)+\int_{0}^{\infty}\frac{\beta(\tau)\,d\tau}{1+\tau^2}<\infty . \tag{6} \]

Now put

\[ \mu_2=\mu_3=1/N(2),\qquad \mu_j=N(j-3)/N(j-2), \]

\[ 4\leqslant j<\infty \quad\text{and}\quad \psi(x)=\prod_{n=2}^{\infty}\frac{\sin \mu_n x}{\mu_n x},\qquad \psi_\delta(x)=\psi(\delta x). \]

By (6), \(\psi(x)\) is an entire function of some finite degree. Since
\[ |\psi(x)|=O(1)\min_{j\geqslant 2} N(j-1)|x|^{-j}, \]
we have \(|\psi(x)|=O(F(|x|))\). Hence, from (4), (5) it follows that

\[ \int_{-\infty}^{\infty}|\psi_\delta(x)|e^{\alpha(x)}\,dx =O(1)\int_{1}^{\infty}F(\delta x)\exp\left(\int_{1}^{x}\frac{\alpha(\tau)\,d\tau}{\tau}\right)\,dx<\infty . \]

Thus \(\psi_\delta(x)\in L_\alpha\) for \(\delta>0\), and from the Wiener–Paley theorem we obtain that \(\widetilde{\psi}_\delta(x)=0\) for \(|x|\geqslant \delta\sigma\), while \(\widetilde{\psi}_\delta(0)=0\). Therefore the function \(\varphi(x)=\psi_{\varepsilon/\sigma}(x)\) is the desired one, and the regularity of \(L_\alpha\) is proved.

From the regularity of \(L_\alpha\) and the results of G. E. Shilov \({}^{(2)}\) it follows that, in order to prove the second assertion, it is enough to show the density in \(L_\alpha\) of functions with finite Fourier transforms. Let \(p(x)\in L_\alpha\). Put

\[ p_A(x)=Aq\int_{-\infty}^{\infty}p(x-t)\psi_A(t)\,dt,\qquad q=\left(\int_{-\infty}^{\infty}\psi(t)\,dt\right)^{-1}. \]

Obviously, \(p_A(x)\in L_\alpha\) and \(\widetilde{p}_A(x)\) is finite. We shall show that

\[ \lim_{A\to\infty}\|p-p_A\|=0. \]

For \(\mu>0\) and \(A\to\infty\) we have

\[ \int_{-\infty}^{\infty}\int_{|t|>\mu}|p(x)-p(x-t)|\,|\psi_A(t)|\,Aq\,dt\,e^{\alpha(x)}\,dx = O(1)\|p\|\int_{|t|>A\mu}|\psi(t)|e^{\alpha(t)}\,dt, \]

\[ \int_{-\infty}^{\infty}\int_{|t|<\mu}|p(x)-p(x-t)|\,|\psi_A(t)|\,Aq\,dt\,e^{\alpha(x)}\,dx\leqslant \]

\[ \leqslant \int_{|t|<\mu} |\psi_A(t)|\, A q \int_{-\infty}^{\infty} \bigl|p(x)e^{\alpha(x)} - p(x-t)e^{\alpha(x-t)}\bigr|\, dx\, dt + \]

\[ + \max_{|x|<\infty,\ |t|<\mu} \bigl|e^{\alpha(x)-\alpha(x-t)} - 1\bigr| \int_{|t|<\mu} |\psi_A(t)|\, A q\, dt = o(1) + O(\mu). \]

Consequently,

\[ \overline{\lim}_{A\to\infty} \|p-p_A\| = O(\mu). \]

Letting \(\mu \to 0\), we obtain

\[ \lim_{A\to\infty} \|p-p_A\| = 0, \]

as was required.

\(3^\circ\). Theorem 2. Let \(\alpha(\tau)=o(\tau)\), let \(\tau \alpha'(\tau)/\alpha(\tau)\) be nondecreasing,

\[ \int_0^\infty \frac{\alpha(\tau)\, d\tau}{1+\tau^2}=\infty. \]

Then the Fourier transforms of functions from \(L_\alpha\) form a quasi-analytic class of functions, and the ring \(L_\alpha\) is nonregular.

Theorem 3. If \(\alpha(\tau)\) satisfies the conditions of Theorem 2 and the system \(\{f_x(\tau)\}\) is complete in \(L_\alpha\), then (1) holds and, for real \(A\),

\[ \overline{\lim}_{x\to\infty} \frac{\ln |\widetilde f(x)|}{\xi\bigl(\tfrac12\pi(x-A)\bigr)} = 0, \qquad \overline{\lim}_{x\to-\infty} \frac{\ln |\widetilde f(x)|}{\xi\bigl(\tfrac12\pi(x+A)\bigr)} = 0, \tag{7} \]

where \(\xi(x)\) is the inverse function of the function

\[ \eta(x)=\int_0^x \frac{\alpha(\tau)\, d\tau}{1+\tau^2}. \]

Let us note that (7) coincides with (2) when \(\alpha(x)=cx\). Denote by \(I_{\gamma,A}\) the set of functions \(f\in L_\alpha\) for which

\[ \overline{\lim}_{x\to\infty} \frac{\ln |\widehat f(x)|}{\xi\bigl(\tfrac12\pi(x-A)\bigr)} \leqslant \gamma < 0. \]

Theorem 3 follows from the following assertion, which is of independent interest.

Theorem 4. Let \(\alpha(\tau)\) satisfy the conditions of Theorem 2. Then for arbitrary \(\gamma\) and \(A\) the set \(I_{\gamma,A}\) contains a function whose Fourier transform does not vanish, and there exists a function \(g(t)\), for which

\[ 0< \operatorname{vrai\,sup}_{-\infty<\tau<\infty} \bigl|g(\tau)e^{-\alpha(\tau)}\bigr| < \infty, \qquad \int_{-\infty}^{\infty} g(x-t) f(t)\, dt \equiv 0 \tag{8} \]

for all \(f\in I_{\gamma,A}\).

We shall need the following lemmas.

Lemma 1. Let \(\omega(t)\) be the inverse function for the function \(2\alpha(t)/\pi\),

\[ z=x+iy,\qquad \Phi(z)=\prod_{n=1}^{\infty}\left(1+\frac{z}{\omega(n)}\right)e^{-z/\omega(n)}. \]

Then we have

\[ \frac{c_1}{1+|y|^{c_2}} \leqslant \Phi(iy)e^{-\alpha(y)} \leqslant \Phi_1(y), \qquad \Phi_1(y)=\Phi_1(-y); \]

\[ \Phi_1(y)\uparrow,\qquad \int_0^\infty \frac{\ln \Phi_1(y)}{1+y^2}\, dy < \infty, \tag{9} \]

and, for \(x\geqslant 0\),

\[ |\Phi(x)| \leqslant c_4 e^{c_3 x} \exp\left( -\frac{2x}{\pi}\int_0^x \frac{\alpha(\tau)\, d\tau}{1+\tau^2} \right), \tag{10} \]

\[ |\Phi(z)| \geqslant \frac{c_5}{1+|y|^{c_6}} e^{-c_7 x} \exp\left( -\frac{2x}{\pi}\int_0^x \frac{\alpha(\tau)\, d\tau}{1+\tau^2} \right), \tag{11} \]

where \(c_i\) are positive constants.

\[ \text{* } 1/g(t)\ \text{depends on } \gamma \text{ and } A. \]

Lemma 2. If \(f\in I_{\gamma,A}\), then for \(x\geqslant 0\)

\[ \int_0^\infty |\widetilde f(p)|e^{xp}\,dp \leqslant c_9\exp\left(\frac{2x}{\pi}\int_0^x\frac{\alpha(\tau)\,d\tau}{1+\tau^2}+c_8x\right), \]

where \(c_8\) depends only on \(\gamma,A\).

Proof of Theorem 3. Put

\[ R(y)=\left(\frac{\sin y}{y}\right)^B\frac{1}{\Phi(iy)}. \]

By virtue of 9, for sufficiently large \(B\), \(R(y)\in L_\alpha\), and from (11) it follows that, for \(x>0,\ \sigma>0\),

\[ |\widetilde R(x)|\leqslant \exp\left(-\sigma x+c_{10}\sigma+\frac{2\sigma}{\pi}\int_0^\sigma \frac{\alpha(\tau)\,d\tau}{1+\tau^2}\right). \]

Taking \(\sigma=\xi(1/2\pi(x-H))\), for large \(H\) we obtain

\[ |\overline R(x)|\leqslant \exp\left(-\frac12 H\,\xi(1/2\pi(x-H))\right). \]

It is not difficult to show that, for any \(\delta>0\),

\[ \lim_{x=\infty}\frac{\xi(x-\delta)}{\xi(x)}=0. \]

Therefore one can choose \(\eta\) so that

\[ \widetilde H(y)=e^{i\eta y}R(y)\in I_{\gamma,A}, \]

and \(\widetilde H(y)\) does not vanish. The first assertion of the theorem is proved.

Next, put

\[ \mu(z)=\frac{2}{\pi}\int_{-\infty}^{\infty} \frac{\ln\Phi_1(\rho)\,d\rho}{i\rho-(z+1)}. \]

Then \(\mu(z)\) is holomorphic for \(\operatorname{Re}z>-1\), and by virtue of (9)

\[ \operatorname{Re}\mu(z)\leqslant -\frac{z\ln\Phi_1(z)}{\pi} \int_{\rho>|z|}\frac{(x+1)\,d\rho}{(\rho-y)^2+(x+1)^2}. \]

The function \(s(z)=e^{\mu(z)}\) is holomorphic and bounded for \(\operatorname{Re}z\geqslant0\), and

\[ |s(iy)|\leqslant \frac{1}{\Phi_1(|y|)}. \]

Put

\[ f\in I_{\gamma,A} \quad\text{and}\quad f_1(x)=\int_{-\infty}^{\infty}e^{-(x-\tau)^2}f(\tau)\,d\tau. \]

In this case

\[ \max_{-\infty<x<\infty}|f_1(x)e^{\alpha(x)}|<\infty,\qquad f_1(x)\in I_{\gamma,A}, \]

and, by Lemma 2, for \(x\geqslant0\),

\[ |f_1(iz)|=O(1)\left|\int_{-\infty}^{\infty}\widetilde f_1(p)e^{zp}\,dp\right| =O(1)\exp\left(\frac{2x}{\pi}\int_0^x \frac{\alpha(\tau)\,d\tau}{1+\tau^2}+c_8x\right). \]

Hence, and from (9), it follows that for any real \(\beta\) the function

\[ Q_\beta(z)=\Phi(z)s(z)e^{-2c_8z}f_1(iz+\beta) \]

is holomorphic in the right half-plane, bounded on the axis \(y\) and on the positive semiaxis \(x\), continuous for \(x\geqslant0\), and of order \(\rho<2\). Applying the Phragmén–Lindelöf principle, we find that \(Q_\beta(z)\) is bounded in the right half-plane. Then, by Cauchy’s theorem,

\[ \int_{-\infty}^{\infty}\frac{Q_\beta(iy)\,dy}{(1+iy)^2}=0. \]

Consequently, for \(-\infty<\beta<\infty\),

\[ \int_{-\infty}^{\infty}\Phi(iy)s(iy)e^{-2c_8iy}f_1(\beta-y)\frac{dy}{(1-iy)^2} = \int_{-\infty}^{\infty}e^{-(\beta-\tau)^2} \int_{-\infty}^{\infty} f(\tau-v)g(v)\,dv\,d\tau=0, \tag{12} \]

where

\[ g(v)=\Phi(iv)s(iv)e^{-2ic_8v}(1+iv)^{-2}. \]

From (12), by the usual device, we derive

\[ \int_{-\infty}^{\infty} f(\tau-v)g(v)\,dv=0. \]

Since \(g\) does not depend on \(f\) and, by virtue of (10),

\[ 0< \max_{-\infty<v<\infty}|g(v)e^{-\alpha(v)}|<\infty, \]

the last equality is equivalent to the last assertion of Theorem 3.

Leningrad Civil Engineering Institute

Received
18 V 1966

References

1. N. Wiener, Ann. Math., 33, No. 1 (1932).
2. G. E. Shilov, On regular normed rings, Moscow–Leningrad, 1947.
3. B. I. Korenblum, Tr. Moscow Math. Soc., 7, 121 (1958).
4. I. M. Gelfand, D. A. Raikov, G. E. Shilov, Uspekhi Mat. Nauk, 2 (12) (1946).