Abstract
Full Text
UDC 517.535
MATHEMATICS
V. E. KATSNELSON
ON THE LOCATION IN THE COMPLEX PLANE OF THE ZEROS OF THE FOURIER TRANSFORM
(Presented by Academician S. N. Bernstein on 29 I 1966)
In this note the distribution of the zeros of functions
\[ f(z)=\int_{\sigma}^{\sigma+T}\varphi(t)e^{itz}\,dt, \tag{1} \]
where \(\varphi(t)\) is continuous and \(\varphi(\sigma)\ne 0\), is studied.
Theorem 1. If \(\varphi(t)\) is continuous at \(t=\sigma\) and of bounded variation on the segment \([\sigma,\sigma+T]\), then some half-plane \(y\ge h\) is free of zeros of the function \(f(z)\).
Proof. Integrating by parts, we obtain, as \(y\to\infty\), uniformly in \(x\),
\[ f(z)=-iz^{-1}e^{i\sigma z}[1+o(1)]. \]
This asymptotic equality implies Theorem 1.
Yu. I. Lyubich posed the following question: is it possible that, for a continuous function \(\varphi(t)\), \(\varphi(\sigma)\ne 0\), the function \(f(z)\) defined in (1) should have zeros with arbitrarily large imaginary parts? The following theorem answers this question.
Theorem 2. Let \(\psi(\delta)\), \(0<\delta<\infty\), be a function satisfying the conditions: a) \(\psi(\delta)>0\) if \(\delta>0\); b) \(\lim_{\delta\to0}\psi(\delta)=0\), c) \(\lim_{\delta\to0}\delta^{-1}\psi(\delta)=\infty\); d) \(\psi(\delta)\) increases, and \(\delta^{-1}\psi(\delta)\) decreases, as \(\delta\) increases.
Then there exists a function \(\varphi(t)\), satisfying the condition \(\varphi(-\pi)\ne 0\), with modulus of continuity
\[ \omega_\varphi(\delta)\le \psi(\delta) \]
and such that the entire function
\[ f(z)=\int_{-\pi}^{\pi}\varphi(t)e^{itz}\,dt \tag{2} \]
has an infinite set of zeros \(z_k=x_k+iy_k\), for which the real and imaginary parts are related by
\[ y_k=|x_k|\psi(|x_k^{-1}|)+o(1). \]
Proof. We shall construct the function \(\varphi(t)\) in the form of an absolutely convergent lacunary Fourier series
\[ \varphi(t)=\frac{1}{20}\sum_{k=0}^{\infty}(-1)^{n_k}c_k e^{-in_k t}, \tag{3} \]
* An analogous fact was noted earlier by T. M. Karaseva in the paper (5), § 3. In that paper the restrictions on \(\varphi(t)\) are stronger. An analogous theorem from the theory of almost-periodic functions is also well known.
where \(n_k\) are integers. Substituting (3) into (2) and integrating term by term, we obtain
\[ f(z)=\frac{1}{10}\sum_{k=0}^{\infty}\frac{\sin z}{z-n_k}. \tag{4} \]
The series in (4) converges absolutely and uniformly on every compact set in the complex plane.
We shall construct the numbers \(c_k\) and \(n_k\) inductively. Put \(c_0=1,\ n_0=1\). Suppose that the numbers \(c_0,c_1,\ldots,c_k\) and \(n_0<n_1<\cdots<n_k\) have already been constructed, and that the following conditions are satisfied:
-
\(\ |c_l|<2\psi(n_l^{-1})<2^{-l+1}(l+1)^{-1}\quad (l=1,2,\ldots,k).\)
-
\[ \sum_{s=l+1}^{k}|c_s|<\bigl(2^{-1}-2^{-(k-l+1)}\bigr)|c_l| \quad (l=0,1,\ldots,k-1). \]
-
\[ \sum_{s=0}^{l-1} n_s |c_s|<n_l|c_l| \quad (l=1,2,\ldots,k). \]
-
In each disk
\[ K_s=\{z:\ |z-n_s(1+i\psi(n_s^{-1}))|<2^{-s-1}\} \]
there is a zero of the function
\[ f_l(z)=\frac{1}{10}\sin z\sum_{j=0}^{l}c_j(z-n_j)^{-1} \quad (s=1,2,\ldots,l;\ l=1,2,\ldots,k). \]
Put
\[ c_{k+1}=-i\psi(n_{k+1}^{-1})[1+i\psi(n_{k+1}^{-1})]\sum_{s=0}^{k}c_s \tag{5} \]
and choose \(n_{k+1}\). By virtue of properties b) and c) of the function \(\psi(\delta)\) and our choice of \(c_{k+1}\), there exists \(N_1\) such that, for \(n_{k+1}>N_1\), conditions 1, 2, 3 of the inductive construction will be satisfied with \(k\) replaced by \(k+1\). Using condition 1 with \(l=k+1\), property b) of the function \(\psi(\delta)\), and Rouché’s theorem, we obtain that there exists \(N>N_1\) such that, for \(n_{k+1}>N\), the function \(f_{k+1}(z)\) will have a zero \(z_{k+1,s}\) in the disk \(K_s\) \((s=1,2,\ldots,k)\). Choose \(n_{k+1}>N\) so that the inequalities
\[ \psi(n_{k+1}^{-1})<2^{-k-6}\left(\sum_{s=0}^{k} n_s\right)^{-1}, \tag{6} \]
\[ \left|f_k(z)-\frac{1}{10}\frac{\sin z}{z}\sum_{s=0}^{k}c_s\right| \leq \frac{1}{5}\frac{|\sin z|}{|z|^2}\sum_{s=0}^{k}n_s \quad (2|z|>n_{k+1}). \tag{7} \]
Then the function \(f_{k+1}(z)\) will have a zero \(z_{k+1,k+1}\) in the disk \(K_{k+1}\). Indeed, on the circle
\[
|z-n_{k+1}(1+i\psi(n_{k+1}^{-1}))|=2^{-k-2}
\]
the inequality
\[ \left|\sum_{s=0}^{k}c_s z^{-1}+c_{k+1}(z-n_{k+1})^{-1}\right| >2^{-k-5}n_{k+1}^{-2}\psi^{-1}(n_{k+1}^{-1}) \tag{8} \]
holds, and the function
\[
z^{-1}\sum_{s=0}^{k}c_s+(z-n_{k+1})^{-1}c_{k+1}
\]
vanishes at the point
\[
z=n_{k+1}(1+i\psi(n_{k+1}^{-1})).
\]
From (9), (10), and (11), by Rouché’s theorem it follows that the function \(f_{k+1}(z)\) has a zero \(z_{k+1,k+1}\) in the disk \(K_{k+1}\), and thus we have satisfied condition 4 of the inductive construction with \(k\) replaced by \(k+1\). This completes the inductive construction.
By virtue of this construction the function (4) is defined, and there exists a sequence \(\{z_l\}_{l=1}^{\infty}\) of its zeros satisfying the condition
\[ |z_l-n_l(1+i\psi(n_l^{-1}))|\leq 2^{-l-1}. \]
The coefficients \(c_l\) satisfy conditions 1, 2, and 3 with \(k=\infty\). The condition \(\varphi(-\pi)\ne 0\) is fulfilled, since, by condition 1,
\[ 20|\varphi(-\pi)|>|c_0|-\sum_{k=1}^{\infty}|c_k|>1-\sum_{k=1}^{\infty}(k+1)^{-1}\cdot 2^{-k}>2^{-1}. \]
Let us estimate the modulus of continuity \(\omega_\varphi(\delta)\) of the function \(\varphi(t)\). Let \(m\) be a natural number such that \(n_k\delta<1\) for \(k\leq m\) and \(n_k\delta>1\) for \(k>m\). From conditions 1, 2, and 3, used with \(k=\infty\), we obtain
\[ \omega_\varphi(\delta)\ll \frac1{10}\sum_{k=1}^{m}|c_k|\cdot |e^{-in_k\delta}-1| +\frac2{10}\sum_{k=m+1}^{\infty}|c_k| \leq \frac{\delta}{10}\sum_{k=1}^{m}|c_k|n_k+ \]
\[ +\frac2{10}\sum_{k=m+1}^{\infty}|c_k| \leq \frac2{10}\delta |c_m|n_m+\frac3{10}|c_{m+1}| \ll \frac25 n_m\psi(n_m^{-1})+\frac3{10}\psi(n_{m+1}^{-1}). \]
Since \(n_{m+1}^{-1}<\delta<n_m^{-1}\), it follows from condition d) of Theorem 2 that
\[ \omega_\varphi(\delta)\ll \psi(\delta). \]
Theorem 2 is proved.
This theorem is in a certain sense sharp, as is shown by
Theorem 3. The function \(f(z)\), defined in (1), has no zeros in the region
\[ y>C|\varphi(\sigma)|^{-1}|x|\omega_\varphi(|x|^{-1})+C_1, \]
where \(C\) is an absolute constant; \(C_1\) depends on \(\varphi(t)\), and \(\omega_\varphi(\delta)\) is the modulus of continuity of the function \(\varphi(t)\).
The following theorem shows that, for any smoothness of the function \(\varphi(t)\) near the left endpoint of the segment \([\sigma,\sigma+T]\), the function \(f(z)\), defined in (1), may have zeros with arbitrarily large imaginary part.
Theorem 4. Let the function \(u(x)\), \(0<x<\infty\), satisfy the condition \(\lim_{x\to\infty}u(x)=0\), \(0<T_0<T<\infty\), and let \(h(x)\) be a solution of the equation
\[ x^{-1}h(x)\exp[T_0h(x)]=u(x). \]
Then there exists a function \(\varphi(t)\), continuous on \([\sigma,\sigma+T]\), satisfying the condition \(\varphi(t)=\varphi(\sigma)\ne0\), \(\sigma\leq t\leq T_0+\delta\), and such that the function \(f(z)\) (1) has a sequence of zeros approaching without bound the curve \(y=h(x)\).
The sharpness of Theorem 4 is shown by
Theorem 5. If \(\varphi(t)\) is continuous on the segment \([\sigma,\sigma+T]\) and of bounded variation on the segment \([\sigma,\sigma+T_0]\), \(0<T_0<T<\infty\), \(\varphi(\sigma)\ne0\), then there exists a function \(h(t)\) satisfying the condition
\[ \lim_{x\to\infty}x^{-1}h(x)\cdot \exp[T_0h(x)]=0 \]
such that the region \(y>h(x)\) is free of zeros of the function \(f(z)\), defined by equality (1).
The author expresses sincere gratitude to Yu. I. Lyubich for formulating the problem and to B. Ya. Levin for help in preparing the present article.
Kharkov State University
named after A. M. Gorky
Received
8 I 1966
CITED LITERATURE
- N. I. Akhiezer, Lectures on the Theory of Approximation, “Nauka,” 1965.
- A. Zygmund, Trigonometric Series, Mir, 1965.
- B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956.
- B. M. Levitan, Almost-Periodic Functions, Moscow, 1953.
- T. M. Karaseva, Zap. Kharkovsk. matem. obshch., 25, 1957.