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MATHEMATICS
N. MEIMAN
ON THE COMPLETENESS OF THE ZEROS OF ENTIRE FUNCTIONS
(Presented by Academician L. S. Pontryagin on 3 VII 1961)
1°. In this paper it is shown that for the set of zeros of one important class of entire functions of exponential type one can introduce a characteristic playing the same role as the degree of a polynomial does for a polynomial. The terminology and notation are the same as in the note (¹).
2°. Definition. A monotone sequence of real points \(\{\xi_\nu\}\), \(\nu=\pm1,\pm2,\ldots\), will be called regular with characteristic \((\sigma;p)\) if the points \(\xi_\nu\) admit the representation
\[ \xi_\nu=(\nu+\delta(\nu))\frac{\pi}{\sigma}\quad \text{for } \nu<0;\qquad \xi_\nu=(\nu-\delta(\nu))\frac{\pi}{\sigma}\quad \text{for } \nu>0, \tag{1} \]
where the shifts \(\delta(\nu)\) satisfy the conditions
\[ -\infty<-a\leq \delta(\nu)\leq b<+\infty,\qquad |\nu(\delta(\nu+1)-\delta(\nu))|<c<+\infty, \tag{2} \]
and \(p\) is the smallest integer for which the quantity
\[ \chi(N)=\sum_{\nu=1}^{N}\frac{\delta(-\nu)+\delta(\nu)-1}{\nu}+2p\ln N \tag{3} \]
is bounded below. If \(\chi(N)\) is also bounded above, then the sequence \(\{\xi_\nu\}\) will be called completely regular.
Lemma (see (¹)). Let \(\{\xi_\nu\}\) be a regular sequence with characteristic \((\sigma;p)\), and let \(v(z;\{\xi_\nu\})\) be the function with zeros at the points \(\xi\)
\[ v(z;\{\xi_\nu\})= \prod_{\nu=1}^{\infty} \left(1+\frac{\sigma z-\delta(-\nu)\pi}{\nu\pi}\right) e^{-\frac{\sigma z-\delta(-\nu)\pi}{\nu\pi}} \times \]
\[ {}\times \prod_{\nu=1}^{\infty} \left(1-\frac{\sigma z+\delta(\nu)\pi}{\nu\pi}\right) e^{\frac{\sigma z+\delta(\nu)\pi}{\nu\pi}}. \tag{4} \]
The function \(v(z;\{\xi\})\) has the following properties:
1) \(v(z)\) is a function of type \(\sigma\) with indicator \(h(\varphi)=\sigma|\sin\varphi|\).
2) On the real axis
\[ v(x;\{\xi\})=(\xi_j^- -x)\cdots(\xi_1^- -x)(\xi_1^+ -x)\cdots(\xi_j^+ -x)\,B(x;\{\xi\})\times \]
\[ {}\times \exp\left( \sum_{\nu=1}^{[\sigma|x|/\pi]} \frac{\delta(-\nu)+\delta(\nu)-1}{\nu} \right), \tag{5} \]
where \(j=\max([a]+1,[b]+1)\); \(\xi^-\), \(\xi^+\) are the points \(\xi_\nu\) nearest to \(x\) on the left and on the right; \(B(x)\) is positive and bounded on both sides
\[ 0<\alpha(a,b,c)\leq B(x;\{\xi\})\leq \beta(a,b,c)<+\infty. \tag{6} \]
3) The ratio \(v'(x') : v(x'')\), where the points \(x'\), \(x''\) lie outside the \(\varepsilon\)-neighborhoods of the points \(\xi_\nu\), is bounded in modulus from above by a constant \(\gamma=\gamma(\varepsilon,a,b,c)\);
\[ |v(x'')|>d\,(|x''|+1)^{-2p};\qquad d=d(\varepsilon,a,d,c)>0. \]
The sequence \(\{\eta_\nu\}\),
\[ \eta_\nu=\frac{\xi_{\nu+1}+\xi_\nu}{2} \]
is also regular, with characteristic \((\sigma,p)\).* Introduce the functions
\[ u(z;\{\xi\})=v(z;\{\eta\}),\qquad v_1(z)=R_{2p}(z)v(z),\qquad u_1(z)=Q_{2p}(z)u(z), \]
where \(R_{2p}(z)\) and \(Q_{2p}(z)\) are arbitrary real polynomials of degree \(2p\) with complex zeros, assuming values of different sign on the real axis. According to Theorem VI.18 from (2), the function \(\omega(z)=u_1(z)+iv_1(z)\) belongs to the class \(B\) (see, for example, \((2,3)\)) with \(p\) zeros in the upper half-plane.
Theorem 1. Let \(f(z)\) be a real entire function of exponential type not exceeding \(\sigma\), \(L_f<\infty\), and let all points of a \((\sigma;p)\)-regular sequence \(\{\xi\}\) be zeros of the function \(f(z)\). If the function \(f(z)\) does not vanish identically, then \(f(z)\) has no more than \(2p\) zeros distinct from the points \(\{\xi\}\).
Proof. Let the number of zeros of the function \(f(z)\) distinct from \(\{\xi\}\) be greater than \(2p\). Choose \(2p\) such zeros and construct a function \(\hat v_1(z;\{\xi\})\) in which \(2p\) arbitrary complex zeros are replaced by these zeros. Obviously,
\[ \varphi(z)=f(z):\hat v_1(z) \]
is an entire function of no more than minimal type. If \(\varphi(z)\) does not reduce to a constant, then \(\lim |\varphi(x)|=\infty\) as \(x\to\infty\) on a set of positive relative density and, by virtue of (4) and (5), \(L_f\) cannot be bounded.
This theorem is an analogue of the property that the number of roots of a polynomial coincides with its degree. Especially important is the special case of the theorem for \(p=0\).
3°. Let \(\{a\}\) be the sequence of zeros of some function \(f(z)\). If from this sequence one can remove \(q\) points so that the remaining zeros will alternate with the sequence \(\{\xi\}\), then we shall say that the number of nontrivial zeros of the function \(f(z)\) relative to the sequence \(\{\xi\}\) is equal to \(d\).
Theorem 2. Let \(f(z)\) be a real entire function of exponential type \(\leqslant\sigma\) and \(L_f<+\infty\). If at all points of some \((\sigma;p)\)-regular sequence \(\{\xi\}\), for which the function
\[ u(z)=v_1\left(z;\left\{\frac{\xi_\nu+\xi_{\nu+1}}{2}\right\}\right) \]
the inequality
\[ f(\xi):u_1(\xi)\leqslant 1, \tag{7} \]
is satisfied, then all zeros of the difference \(u(z)-f(z)\), with the possible exception of \(2p\) zeros, are real and alternate with the points \(\xi\).
Proof. Let \(T\) be the closure of the image of the real axis under the mapping \(t=\omega(z):f(z)\) \((\omega(z)=u_1(z)+iv_1(z))\). We shall prove that the half-closed interval \(0\leqslant t<1\) lies inside one of the domains \(D\) complementary to \(T\). Suppose the contrary: let \(t_0\in T\), \(0\leqslant t_0<1\). Then there exists a sequence of real points \(\{x_n\}\) for which
\[ \lim(\omega(x_n):f(x_n))=t_0 \]
or
\[ \frac{f(x_n)}{u(x_n)}\to t_0^{-1}>1,\qquad \frac{v(x_n)}{f(x_n)}\to 0. \]
Consequently, \(x_n\to\infty\), since otherwise \(x_n\to\xi\), which would contradict inequality (7). From \(v(x_n)\to0\) and relation (4) it follows that, for sufficiently large \(n\), \(x_n\) is arbitrarily close to \(\xi_-(x)\) or \(\xi_+(x)\)—denote this point by \(\xi(x)\).
* For \(\nu=1\), \(\eta_1=\dfrac{\xi_{-1}+\xi_1}{2}\).
** It is sufficient to require boundedness in modulus of the ratio \(f'(x):u(\xi)\), when \(x\) belongs to some system of \(\varepsilon'\)-neighborhoods of \(\xi_\nu\), and boundedness of
\[ f(x)\exp\left(-\sum_{\nu=1}^{[|x|\sigma/\pi]}\frac{\delta(-\nu)+\delta(\nu)-1}{\nu}\right). \]
For \(L_f<+\infty\) this condition is certainly satisfied.
From the lemma it follows that \(\dfrac{f(x_n)}{u(x_n)}-\dfrac{f(\xi)}{u(\xi)}\to 0\), which is impossible. From the fact that the interval \(0 \leqslant t < 1\) belongs to \(D\), by Theorem 2 of \((^1)\) it follows that the function \(\omega_t(z)=[u_1(z)-t f(z)]+i v_1(z)\) belongs to the class \(B\) and has \(p\) zeros in the upper half-plane, while the component \(u_1(z)-t f(z)\) has no more than \(2p\) nontrivial zeros. This property, by continuity, is preserved as \(t\to 1\).
Remark. If zero is not an asymptotic value of the function \([u_1(z)-f(z)]:v_1(z)\), then the number of nontrivial zeros is equal to \(2p\).
Theorem 3. If at two neighboring points \(\xi\) the function \(f(z)\) assumes values of different signs or, more generally, if at all points \(\xi\) at which \(f(\xi)\ne 0\), the ratio \(f(\xi):u_1(\xi)\) has one and the same sign, then the function \(f(z)\) has no more than \(2p\) nontrivial zeros.
Proof. Choose \(\lambda\) so that \(0 \leqslant \lambda f(\xi):u_1(\xi)<1\); then the function \(f_1(z)=u_1(z)-\lambda f(z)\) satisfies the conditions of Theorem 1 and \(f(z)=\lambda^{-1}[u_1(z)-f_1(z)]\) has no more than \(2p\) nontrivial zeros.
Remark. If zero is not an asymptotic value of the function \(f(z):v_1(z)\), then the function \(f(z)\) has \(2p\) nontrivial zeros.
Theorem 4. Let \(f(z)\) be a real entire function of exponential type not exceeding \(\sigma\), \(L_f<+\infty\) (see the footnote to Theorem 2), and let \(\{\xi\}\) be a \((\sigma;0)\)-regular sequence. If the function \(f(z)\) assumes values of alternating signs on a set consisting of the points \(x_1,x_2\) and the sequence \(\{\xi\}\), then \(f(z)\) is identically equal to zero.
Proof. Remove two neighboring points from the set \(x_1,x_2,\{\xi\}\). The remaining sequence will be \((\sigma;0)\)-regular, and the number of nontrivial zeros of \(f(z)\) with respect to this sequence is not less than 2, which is impossible.
This theorem is analogous to the fact that a polynomial of degree \(n\) cannot assume values of alternating signs at \(n+2\) points.
\(4^\circ\). In the theory of approximation of continuous functions by polynomials, a well-known theorem of Vallée-Poussin plays an important role. The preceding theorems make it possible to obtain analogues of this theorem for entire functions. From Theorem 4 it follows:
Theorem 5. Let \(f(z)\) and \(\varphi(z)\) be two real entire functions of exponential type not exceeding \(\sigma\); \(L_f,L_\varphi<\infty\) (see the footnote to Theorem 2); let \(\{\xi\}\) be a \((\sigma;0)\)-regular sequence, and let \(F(x)\) be an arbitrary real function. If for some \(x_1\) and \(x_2\) at the points of the sequence \(\{x_1,x_2,\{\xi\}\}\) the difference \(F(x)-\varphi(x)\) assumes values of alternating signs, then at least at one point of this sequence
\[ [F(x)-f(x)]:[F(x)-\varphi(x)]>1. \tag{9} \]
From Theorem 3 it follows
Theorem 6. Let \(\{\xi\}\) be a \((\sigma;p)\)-regular sequence and let \(A\) be such a family of real entire functions of exponential type not exceeding \(\sigma\), bounded on the real axis (see the footnote to Theorem 2), that the difference of any two functions from \(A\) has at least \(2p+1\) nontrivial zeros with respect to \(\{\xi\}\). Let \(F(x)\) be an arbitrary real function and \(\varphi(z)\in A\). If the difference \(F(x)-\varphi(x)\) has alternating signs at the points \(\{\xi\}\), then for any function \(f(z)\) from \(A\) at least at one point \(\{\xi\}\)
\[ [F(\xi)-f(\xi)]:[F(\xi)-\varphi(\xi)]>1. \tag{9} \]
Example of a family \(A\). The family \(A\) consists of all real entire functions of exponential type not exceeding \(\sigma\) satisfying the following conditions: 1) the values of the successive derivatives of orders \(k_1,k_1+1,\ldots,k_1+2l_1-1,\ldots,k_\mu,k_\mu+1,\ldots,k_\mu+2l_\mu-1\) at the real points \(x_1,\ldots,x_\mu\) have prescribed values; 2) the functions
or their derivatives of certain orders take prescribed values at the \(2j\) complex points \(\xi_1,\bar{\xi}_1,\ldots,\xi_j,\bar{\xi}_j\).
According to Theorem 6 of [1], the difference of two functions of this family has at least \(2(j+l_1+\cdots+l_\mu)\) nontrivial zeros with respect to any sequence not containing the points \(x_1,x_2,\ldots,x_\mu\). As the sequence \(\{\xi\}\) one may take any such \((\sigma,\rho)\)-regular sequence for which \(p<j+l_1+\cdots+l_\mu\).
Institute of Theoretical and Experimental Physics
Academy of Sciences of the USSR
Received
28 VI 1961
References
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- N. G. Chebotarev, N. N. Meiman, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 19, 241 (1949).
- N. N. Meiman, DAN, 124, No. 6 (1959).