Levinson’s condition in the theory of entire functions: Equivalent statements

In terms of distribution functions of zeros of an entire function of exponential type, we prove assertions equivalent to the bilogarithmic Levinson condition for the Borel transform of the function. As an application, we present solutions of two problems related to Pavlov-Korevaar-Dixon interpolation.     

Classification theorems for operators preserving zeros in a strip

We characterize all linear operators which preserve certain spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip to real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a “strong universal factor” as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.     

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

We establish new differential inequalities for the entire functions of finite degree with a majorant an entire function without zeros in the lower half-plane, for the entire functions with constraints on zeros and, as a consequence, for the rational functions with prescribed poles. All cases of equality in the main results are found. The estimates obtained generalize and strengthen some inequalities by Bernstein, Gardner, and Govil for entire functions of finite degree; by Smirnov, Aziz, and Shah for algebraic polynomials; and by Borwein and Erdelyi, Aziz and Shah, and the others for rational functions.     

Zeros of entire functions and a problem of Ramanujan

We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation.     

Entire Approximations to the Truncated Powers

For a variation diminishing function $g$ which is analytic on a set containing the real line and any real polynomial $P$, we prove that $g+P$ has at most $\text{deg}(P)+2$ real zeros. Based on this estimate, we present a way to construct entire approximations $G_n$ to the truncated powers $x_+^n$ for $n\in{\bf N}_0$. Here $x_+^n=x^n$ for $x>0$ and $x_+^n=0$ for $x<0$. The function $G_n$ is constructed in such a way that \[ G_n(x)-x_+^n=F(x)H_n(x)\] holds, where $F$ is entire and $H_n$ has no zeros on the real line. The function $G_n$ can be viewed as an interpolant of $x_+^n$ with a nodal set that is given by the (real) zeros of $F$. As an application of this method, we give explicit formulas for best $L^1({\bf R})$-approximation and best one-sided $L^1({\bf R})$-approximation from the class of entire functions with given exponential type $\eta$ to $x_+^n$. These approximations are given in terms of the logarithmic derivative of the Euler Gamma function.     

一类整函数零点的渐近公式

借助Rouché定理、留数定理及渐近分析的方法,给出了整函数f(z)=zmsinz-α(0≠α∈R,m∈Z+)零点的渐近公式及渐近迹.这种方法也适用于其它整函数的零点估计.     

Sums of entire functions having only real zeros

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.

     

Polynomial Asymptotics of Entire Functions of Finite Order

We obtain new asymptotic relations for entire functions of finite order with zeros on a ray under the condition of regular growth for the counting function of their zeros. These relations improve the well-known results of Valiron.     

高阶复微分方程解的超级的角域分布

设f1,f2,…,fn是复方程f(n)+An-1f(n-1)+…+A0f=0的n个线性无关解,其中A0,A1,…,An-1是不全为多项式,且至少有一个为无限级整函数,σ2(Aj)=0(j=1,2,…,n-1).假设E=f1,f2,…,fn.研究了微分方f(n)+An-1f(n-1)+…+A0f=0的解在角域中的零点分布,获得E的超级为+∞的Borel方向与零点聚值线的关系.     

On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.     

Limit Curves for Zeros of Sections of Exponential Integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\) , and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.     

Snapback repellers and semiconjugacy for iterated entire mappings

A theorem of Poincaré guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we related the multiplicity of the zeros of the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two.     

Two problems in the value distribution theory

This paper includes two parts. In the first part, we deal with the relations between the orders, the lower orders and the number of Julia directions of entire functions. In the second part, we deal with the existence of the zeros of a class of meromorphic functions. This class of meromorphic functions has an interesting physical background.     

A note on exponent of convergence of zeros of entire functions

If f(z) is an entire function with ρ ₁ > 0 as its exponent of convergence of zeros and if 0 ≤ α < ρ ₁, then we prove the existence of entire functions each having α as its exponent of convergence of zeros.      

The Delsarte extremal problem for the Jacobi transform

We give the solution of the Delsarte extremal problem for even entire functions of exponential type that are Jacobi transforms and prove the uniqueness of the extremal function. The quadrature Markov formula on the half-line with zeros of the modified Jacobi function are used.     

Krein Orthogonal Entire Matrix Functions and Related Lyapunov Equations: A State Space Approach

For a class of entire matrix valued functions of exponential type new necessary and sufficient conditions are derived in order that these functions are Krein orthogonal functions. The conditions are stated in terms of certain operator Lyapunov equations. These equations arise by using infinite dimensional state space representations of the entire matrix functions involved. As a corollary, using a recent operator inertia theorem, we give a new proof of the Ellis-Gohberg-Lay theorem which relates the number of zeros of a Krein orthogonal function in the open upper half plane to the number of negative eigenvalues of the corresponding selfadjoint convolution operator.     

Concertina-like movements of the error curve in the alternation theorem

If a continuous function f is approximated by elements of a Haar space in the maximum norm on an interval, the error curve of the best approximation has well known alternation properties. It is shown that if f is adjoined to the Haar space all zeros of the error function are monotonously increasing functions of the endpoints, and that under an additional hypothesis, the entire graph of the error curve is shifted to the left or right when the endpoints are moved accordingly.     

On the Uniform Stability of Recovering Sine-Type Functions with Asymptotically Separated Zeros

Mathematical Notes - We obtain the uniform stability of recovering entire functions of special form from their zeros. To such a form, we can reduce the characteristic determinants of strongly...     

Microformal Geometry and Homotopy Algebras

We give a solution to the Bohman extremal problem for nonnegative even entire functions of exponential type that are Jacobi transforms of compactly supported functions. We prove that the extremal function is unique. The Gauss quadrature formula on the half-line over zeros of the Jacobi function is used.