S. Favorov
We give a complete description of zero sets for some well-known subclasses of entire functions of exponential growth (bounded on real axis, Cartwright's class)
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We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type.
We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.
In the early 1960s, P. Malliavin and L. A. Rubel gave a complete description of pairs of distributions of positive points $Z$ and $W$ such that for each entire function of exponential type $g\neq 0$ that vanishes on $W$, there is an entire function of exponential type $f\neq 0$ such that $f$ vanishes on $Z$ and satisfies the inequality $|f|\leq |g|$ everywhere on the imaginary axis. We extend this result to much more general distributions of complex points $Z$ and $W$ lying outside of some pair of vertical angles containing the imaginary axis as the points approach $\infty$.
On common zeros of entire functions of exponential growth
本文研究 $\mathbb C^n$ 中指数增长整函数方程组的无限公共零点，利用凸集几何刻画其渐近分布，将 Kushnirenko–Bernstein–Khovanskii 型零点计数结果推广到渐近密度情形。
In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($δ$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at origin on the complex plane $\mathbb C$. In this third part of our work, we use only the case $q=1$ when $S$ is a pair of oppositely directed rays, i.e., $S$ is a straight line as the point set, and balayage is made from both sides of this line. We consider measures and subharmonic functions of finite type of order $1$. This bilateral balayage of genus $1$ will be ap…
Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type that vanishes on $ Z$ and satisfies the constraint $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$, $y\to \pm\infty$. Our results generalize and develop a joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions on the complex plane, as well as to the completeness…
Let $Z$ and $W$ are distributions of points on the complex plane $\mathbb C$. The following problem goes back to the studies of F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane and others. For which $Z$ and $W$ for an entire function $g\neq 0$ of exponential type vanishing on $W$, there is an entire function $f\neq 0$ of exponential type vanishing on $Z$ such that $|f|\leq |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $Z$ and $W$ lying only on the positive semiaxis. A number of …
Suppose that an almost periodic in Besicovitch's sense function $f(x)$ of several variables is the restriction to the real hyperplane of an entire function of exponential type $b$. Then its spectrum is contained in the ball of radius $b$ with the center in the origin.
We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the real axis is deduced. A connection is established to the positive definite functions.
Abstract. In this work we use an elementary method to derive an upper bound on the right half-plane for genus 0 entire functions if it has only negative zeros. The bound only uses information of the function on the positive real axis. Applications to the modified Bessel functions and the Airy function are also provided.
Taylor coefficients and zeroes of entire functions of exponential type
研究全纯指数型函数的泰勒系数与零点分布，证明若零点计数函数增长线性则函数为指数函数，或系数部分为单位模时同样成立，扩展了卡尔松经典结果。
An extremal problem and inequalities for entire functions of exponential type
本文研究指数型整函数 Carathéodory–Fejér–Turán 问题的两种变体，分别涉及径向递减条件和导数推广，并建立了受 Duffin–Schaeffer、Landau 及 Hardy–Littlewood 启发的若干不等式。
A weighted space of entire functions rapidly decreasing on the real line is considered in the paper. A growth of these functions along the imaginary axis is controlled by some system of weight functions. The Fourier transform of functions of this space is studied. Equivalent description of the considered space in terms of estimates on derivatives of functions on real line is obtained.
Let $Z$ and $W$ be a pair of point distributions of finite upper density on the complex plane $\mathbb C$ with the real axis $\mathbb R$. We give several variants of necessary and at the same time sufficient conditions for their arrangement, under which for every entire function of exponential type $g\neq 0$ vanishing on $W$, there is, respectively, either an entire function of exponential type $f\neq 0$ vanishing on $Z$ and satisfies one of the two variants of the constraints: 1) $|f(iy)|\leq |g(iy)|$ at each $y\in \mathbb R$, i.e. everywhere on the imaginary axis $i\mathbb R$, 2) $\ln |f(iy)…
In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary $b\mathcal U$ of the Siegel upper half-space $\mathcal U$ and it is fundamental that $b\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. We consider entire functions in $\mathbb C^{n+1}$ of exponential type with respect to the hypersurface $b\mathca…
We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studded.
Previously, we developed the technique of balayage of measures or charges and ($δ$-)subharmonic functions of finite order onto an closed system of rays $S$ with a vertex at zero on the complex plane $\mathbb C$. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order $1$ and their differences. First, it is a classical balayage of the genus $q=0$ on a system of four closed rays: positive and negative, and real and imaginary semi-axis $\mathbb R^+$, $-\mathbb R^+$, $i\mathbb R^+$, $-i\mathbb R$. Second, it is tw…
Let $σ>0$. For $1\le p\le \infty$, the Bernstein space $B^p_σ$ is a Banach space of all $f\in L^p(R)$ such that $f$ is bandlimited to $σ$; that is, the distributional Fourier transform of $f$ is supported in $[-σ, σ]$. We study the approximation of\ $f\in B^p_σ by finite trigonometric sums \[ P_τ(x)=χ_τ(x) \sum_{|k|\le στ/π}c_{k,τ} e^{i\fracπτk x } \] in $L^p$ norm on $R$ as\ $τ\to\infty$,\ where\ $χ_τ$ denotes the indicator function of $[-τ, τ]$.
We investigate the conjugate indicator diagram or, equivalently, the indicator function of (frequently) hypercyclic functions of exponential type for differential operators. This gives insights into growth conditions of these functions on particular rays or sectors. Our research extends known results in several respects.
Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions
本研究针对整指数型函数的Hankel变换，通过复平面围道平移方法，在特定宇称和参数条件下得到无次主导项的精确终止Poincaré展开，证明其渐近行为等价于1/λ的多项式。
For $σ\in (0,+\infty)$, denote by $B_σ^{\infty}$ the Bernstein space (of type $σ$) of all entire functions of exponential type $\leq σ$ bounded on real axis $\R$. Let $I_d\subset \R$ be a segment of length $d>0$. We announce complete description of non-uniqueness sequences of points for $B_σ^\infty$ and criteria of completeness of exponential system in $C(I_d)$ or $L^p(I_d)$ to within one or two exponential functions.
In 1979 I. Ciorănescu and L. Zsidó have proved a minimum modulus theorem for entire functions dominated by the restriction to the positive half axis of a canonical product of genus zero, having all roots on the positive imaginary axis and satisfying a certain condition. Here we prove that the above result is optimal: if a canonical product ω of genus zero, having all roots on the positive imaginary axis, does not satisfy the condition in the 1979 paper, then always there exists an entire function dominated by the restriction to the positive half axis of ω, which does not satisfy the desired mi…
In the space of holomorphic functions in a convex domain it is studied the interpolation problem by means of sums of the series of exponentials converging uniformly on all compact sets of the domain. The discrete set of the interpolation nodes with multiplicities is located on the real axis in the domain and it has the only finite limit point. It is obtained a criterion for solvability of the problem in the terms of distribution of limit directions of exponents of exponentials at infinity.
On the Zeros of Certain Entire Functions
本研究证明了一类整q-函数存在无穷多非零根且根的模至少指数增长，并将结果应用于含平面波函数q-类似物在内的级数展开定义的整q-函数，丰富了q-特殊函数的零点理论。
Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ are two subharmonic functions in the complex plane $\mathbb C$ with the Riesz measures $ν_u$ and $μ_M$ such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$. If the growth of a function $M$ in some sense exceeds the growth of a function $u$ on some straight line, then we can expect measure $μ_M$ to dominate measure $ν_u$ in some sense. We give quantitative forms of such dominance. The main results are illustrated by a new uniqueness theorem for entire functions of exponential type.
We study the influence of the multipliers $ξ(n)$ on the angular distribution of zeroes of the Taylor series \[ F_ξ(z) = \sum_{n\ge 0} ξ(n) \frac{z^n}{n!}\,. \] We show that the distribution of zeroes of $ F_ξ$ is governed by certain autocorrelations of the sequence $ ξ$. Using this guiding principle, we consider several examples of random and pseudo-random sequences $ξ$ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product we show that if $ξ$ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps…
The "pits effect" for entire functions of exponential type and the Wiener spectrum
本文研究了整个函数的零点等分布问题，提出一个简单的谱条件，适用于多种随机和伪随机序列，并揭示了梅比乌斯函数在二元 Chowla 猜想下的性质。
Uniqueness sets with angular density for spaces of entire functions, II: $ρ=1$
本文研究指数型整函数空间中具有角密度的唯一性集合，在放弃Lindelöf条件限制的情况下展开分析，并保持内容自足可独立阅读。
We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang circle theorem from statistical mechanics. Also, we state several open problems.
It is known that the famous Heins Theorem (also known as the de Branges Lemma) about the minimum of two entire functions of minimal type does not extend to functions of finite exponential type. We study in detail pairs of entire functions $f, g$ of finite exponential type satisfying $\sup_{z\in\mathbb{C}}\min\{|f(z)|,|g(z)|\}<\infty.$ It turns out that $f$ and $g$ have to be bounded on some rotating half-planes. We also obtain very close upper and lower bounds for possible rotation functions of these half-planes.
Meromorphic Solutions of Certain Kind of Non-Linear Differential equation with finite sums of Exponential Functions
本研究利用Nevanlinna理论，探讨含有限和指数函数的非线性微分方程亚纯解的存在性，证明可容许解必为指数型，并强化了现有结果。
Asymptotic evaluation of the Sinc transform of entire exponential type function resulting to exact polynomial asymptotic behavior
本文研究指数型偶函数的 Sinc 变换在大参数 λ 下的渐近行为，在 λ 超过与函数型 τ 相关的阈值时，推导出表现为 λ 正幂次多项式的终止渐近展开式。
In the space of all entire functions it is solved the problem of interpolation taking into account multiplicities by sums of the series of exponentials with the exponents from a given set. It is found a criterion of solubility of the problem in the case when all infinite portions of interpolation nodes are situated on a finite system of rays. It is also disclosed that the problem is equivalent to particular problems of simple interpolation, as well as of point-wise crude approximation, by sums of the series of exponentials.
In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $η=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion theorem, where we expand the function in $q$-analogs of Lidstone polynomials, i.e., q-Bernoulli and $q$-Euler polynomials as in the classical case. We also raise and solve a $q$-extension of the problem of representing an entire function of the form $f(z)=g(z+1)-g(z)$, where $g(z)$ is also an entire function of the same order as $f(z)$.
The celebrated Paley-Wiener theorem naturally identifies the spaces of bandlimited functions with subspaces of entire functions of exponential type. Recently, it has been shown that these spaces remain invariant only under composition with affine maps. After some motivation demonstrating the importance of characterization of range spaces of bandlimited functions, in this paper we identify the subspaces of $ L^2 (\mathbb{R}) $ generated by these action. Extension of these theorems where Paley-Wiener spaces are replaced by the deBranges-Rovnyak spaces are given.
As an analogue of the Hadamard quotient problem in number theory, the quotient problem (in the sense of complex entire functions) for two sequences $F(n)=a_0+a_1f_1^n+\cdots+a_lf_l^n$ and $ G(n)=b_0+b_1g_1^n+\cdots+b_mg_m^n$, has been solved, where the $f_i$ and $g_j$ are nonconstant entire functions and $a_i$ and $b_j$ are non-zero constants except that $a_0$ can be zero. In this paper, we consider the generalization of this problem in which we allow $a_i$ and $b_j$ to be small growth entire functions with respect to $(g_1, \cdots, g_m)$ by modifying the second main theorem with moving target…
Supposing that $A(z)$ is an exponential polynomial of the form $$ A(z)=H_0(z)+H_1(z)e^{ζ_1z^n}+\cdots +H_m(z)e^{ζ_mz^n}, $$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric location of the leading coefficients $ζ_1,\ldots,ζ_m$ play a key role in the oscillation of solutions of the differential equation $f''+A(z)f=0$. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragmén-Lindelöf indicator function. In addition to results in the whole complex plane, results on sector…
We construct and apply the classic balayage (sweeping out) of measures and subharmonic functions on closed system of rays in the complex plane with vertex at the origin, including measures and subharmonic functions and infinite order. The need for such a procedure occurs in the study of the behavior of entire and subharmonic functions on systems of rays. The results apply to the complete regularity of the growth of subharmonic and entire functions on a system of rays, the study of the distribution of zeros of entire functions of exponential type of class A, touches upon the incompleteness of e…
An entire function that violates quasiminimality
针对 Koiran 关于将指数函数替换为任意一元整函数后拟极小性是否仍成立的问题，本文利用全纯逼近理论构造反例，给出否定回答。
We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane. We obtain integral representations of some particular cases of these operators and apply these results to obtain explicit solutions to some Cauchy problems for diffusion equations with nonconstant drift term.
Let $θ$ be a Young function and consider the space $\mathcal{F}_θ(\C)$ of all entire functions with $θ$-exponential growth. In this paper, we are interested in the solutions $f\in \mathcal{F}_θ(\C)$ of the convolution equation $T\star f=0$, called mean-periodic functions, where $T$ is in the topological dual of $\mathcal{F}_θ(\C)$. We show that each mean-periodic function can be represented in an explicit way as a convergent series of exponential polynomials.
A scale of the Frechet spaces of exponential type entire functions of one complex variable is considered. Certain special properties of subsets of these spaces consisting of Laguerre entire functions, which are obtained as uniform limits on compact subsets of the complex plane of polynomials with real nonpositive zeros only, are described. A class of infinite order differential operators which act between these Frechet spaces is inrtoduced and studied. In particular, it is shown that this class preserves the set of Laguerre entire functions. The results obtained are then used to obtain and to …
We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions with only finitely many singularities of the inverse, counting multiplicity, all of which either escape exponentially fast or are pre-periodic. For these functions we are able to decide, whether the function is recurrent of not. In the case that the Julia set is not the entire plane we also obtain estimates for the measure of the Julia set.
Hilbert's tenth problem for lacunary entire functions of finite order
本文针对单变量复整函数环上希尔伯特第十问题的未解情形，证明了在原点具有缺项幂级数展开的有限阶复整函数环的类比问题有否定解，填补了多项式与有限阶指数多项式之外的新情形。
Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights
研究高斯解析函数在幂指数权重下的零点分布，揭示在特定条件下零点趋于避开特定区域的孔现象，扩展了Ghosh和Nishry关于高斯整个函数的研究。
We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity `slowly', and which have Hausdorff dimension equal to 1. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.
Sum of the exponential and a polynomial: Singular values and Baker wandering domains
本文研究形如指数函数复合与多项式之和的整函数，证明其无穷远点为Baker遗漏值，具有无限临界值且无有限渐近值，并证明此类函数无Baker游荡域，为相关猜想提供候选研究对象。
The Julia set of the exponential family $E_κ:z\mapstoκe^z$, $κ>0$ was shown to be the entire complex plane when $κ>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<κ\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points…
We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a polynomial. The proof in the one dimensional case relies on Lidstone expansion of the function. For $n$ variables, we need $n+1$ points, having the property that the differences of $n$ of them with the remaining one give a basis of ${\mathbb C}^n$. The proof is by reduction to the one variable situation.
On a dense set of functions determined by sampled Gabor magnitude
研究通过采样离散集上的Gabor变换幅度恢复函数的唯一性，证明特定类信号可通过足够密集的采样集恢复，主要贡献是证明指数型整个函数在第二类整个函数中由模值唯一确定。
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