Showing papers on "Entire function published in 2010"

Fast escaping points of entire functions

Abstract: Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

Dynamics of Entire Functions

Abstract: Complex dynamics of iterated entire holomorphic functions is an active and exciting area of research. This manuscript collects known background in this field and describes several of the most active research areas within the dynamics of entire functions.

Some extremal functions in Fourier analysis. II

Abstract: We obtain the best approximation in L1(ℝ), by entire functions of exponential type, for a class of even functions that includes e−λ|x|, where λ>0, log |x| and |x|α, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

Uniqueness and periodicity of meromorphic functions concerning the difference operator

Abstract: In this paper, we investigate the uniqueness problems of difference polynomials of meromorphic functions that share a value or a fixed point. We also obtain several results concerning the shifts of meromorphic functions and the sufficient conditions for periodicity which improve some recent results in Heittokangas et al. (2009) [10] and Liu (2009) [11].

Noncolliding Squared Bessel Processes

Abstract: We consider a particle system of the squared Bessel processes with index $ u > -1$ conditioned never to collide with each other, in which if $-1 < u < 0$ the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function $J_{ u}$ is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.

Entire extensions and exponential decay for semilinear elliptic equations

Abstract: We consider semilinear partial differential equations in ℝ n of the form $$ \sum\limits_{\frac{{|\alpha |}} {m} + \frac{{|\beta |}} {k} \leqslant 1} {c_{\alpha \beta } x^\beta D_x^\alpha u = F(u)} , $$ where k and m are given positive integers. Relevant examples are semilinear Schrodinger equations $$ - \Delta u + V(x)u = F(u), $$ , where the potential V(x) is given by an elliptic polynomial. We propose techniques, based on anisotropic generalizations of the global ellipticity condition of M. Shubin and multiparameter Picard type schemes in spaces of entire functions, which lead to new results for entire extensions and asymptotic behaviour of the solutions. Namely, we study solutions (eigenfunctions and homoclinics) in the framework of the Gel’fand-Shilov spaces S µ (ℝ n ). Critical thresholds are identified for the indices µ and ν, corresponding to analytic regularity and asymptotic decay, respectively. In the one-dimensional case −u″ + V(x)u = F(u), our results for linear equations link up with those given by the classical asymptotic theory and by the theory of ODE in the complex domain, whereas for homoclinics, new phenomena concerning analytic extensions are described.

Random Complex Zeroes and Random Nodal Lines

Abstract: In these notes, we describe the recent progress in understanding the zero sets of two remarkable Gaussian random functions: the Gaussian entire function with invariant distribution of zeroes with respect to isometries of the complex plane, and Gaussian spherical harmonics on the two-dimensional sphere.

Rate of growth of frequently hypercyclic functions

Abstract: We study the rate of growth of entire functions that are frequently hypercyclic for the differentiation operator or the translation operator. Moreover, we prove the existence of frequently hypercyclic harmonic functions for the translation operator and we study the rate of growth of harmonic functions that are frequently hypercyclic for partial differentiation operators.

On the set of hypercyclic vectors for the differentiation operator

Abstract: Let D be the differentiation operator Df = f′ acting on the Frechet space H of all entire functions in one variable with the standard (compact-open) topology. It is known since the 1950’s that the set H(D) of hypercyclic vectors for the operator D is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepulveda whether the set H(D) contains (up to the zero function) a non-trivial subalgebra of H or an infinite-dimensional closed linear subspace of H. In the present article both questions are answered affirmatively.

Are Devaney hairs fast escaping

Abstract: Beginning with Devaney, several authors have studied transcendental entire functions for which every point in the escaping set can be connected to infinity by a curve in the escaping set. Such curves are often called Devaney hairs. We show that, in many cases, every point in such a curve, apart from possibly a finite endpoint of the curve, belongs to the fast escaping set. We also give an example of a Devaney hair which lies in a logarithmic tract of a transcendental entire function and contains no fast escaping points.

Fixed-points and uniqueness of meromorphic functions

Abstract: In the paper, we study the uniqueness and the shared fixed-points of meromorphic functions and prove two main theorems which improve the results of Fang and Fang and Qiu.

Correlation functions for random complex zeroes: strong clustering and local universality

Abstract: We prove strong clustering of k-point correlation functions of zeroes of Gaussian Entire Functions. In the course of the proof, we also obtain universal local bounds for k-point functions of zeroes of arbitrary nondegenerate Gaussian analytic functions. In the second part of the paper, we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of Gaussian Entire Functions, in particular, of the number of zeroes in measurable domains of large area. This complements our recent results from the paper "Fluctuations in random complex zeroes" (arXiv:1003.4251v1).

Some Extremal Functions in Fourier Analysis, III

Abstract: We obtain the best approximation in L 1(ℝ), by entire functions of exponential type, for a class of even functions that includes e −λ|x|, where λ>0, log |x| and |x| α , where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

Asymptotics of the Hole Probability for Zeros of Random Entire Functions

Abstract: Consider the random entire function where the ϕ n are independent and identically distributed (i.i.d.) standard complex Gaussian variables. The zero set of this function is distinguished by invariance of its distribution with respect to the isometries of the plane. We study the probability P H (r) that f has no zeros in the disk {|z| < r} (hole probability). Improving a result of Sodin and Tsirelson, we show that as r → ∞. The proof does not use distribution invariance of the zeros, and can be extended to other Gaussian Taylor series. If ϕ n are compactly supported random variables instead of Gaussians, we get a very different result: there exists r 0 so that every random function of the form (★) must vanish in the disk {|z| < r 0 }.

Complex Brownian Motion Representation of the Dyson Model

Abstract: Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of $h$-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.

Density of hyperbolicity for classes of real transcendental entire functions and circle maps

Abstract: We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salomao and Vargas. We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded.

Entire functions for which the escaping set is a spider's web

Abstract: We construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.

On the Hausdorff dimension of the Julia set of a regularly growing entire function

Abstract: We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

On the equation P(f) = Q(g), where P, Q are polynomials and f, g are entire functions

Abstract: In 1922 Ritt described polynomial solutions of the functional equation $P(f) = Q(g).$ In this paper we describe solutions of the equation above in the case when $P, Q$ are polynomials while $f, g$ are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation $s = P(f) = Q(g),$ where $s, f, g$ are entire functions and $P, Q$ are arbitrary rational functions. As an application we solve the problem of description of "strong uniqueness polynomials" for entire functions.

The structure of spider's web fast escaping sets

Abstract: Building on recent work by Rippon and Stallard, we explore the intricate structure of the spider's web fast escaping sets associated with certain transcendental entire functions. Our results are expressed in terms of the components of the complement of the set (the 'holes' in the web). We describe the topology of such components and give a characterisation of their possible orbits under iteration. We show that there are uncountably many components having each of a number of orbit types, and we prove that components with bounded orbits are quasiconformally homeomorphic to components of the filled Julia set of a polynomial. We also show that there are singleton periodic components and that these are dense in the Julia set.

Dual of the function algebra $A^{-\infty }(D)$ and representation of functions in Dirichlet series

Abstract: In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-space ( being either a bounded -smooth convex domain in , with , or a bounded convex domain in ) as an (FS)-space of entire functions satisfying a certain growth condition; an explicit construction of a countable sufficient set for ; and a possibility of representating functions from in the form of Dirichlet series.

Analyticity of extremisers to the Airy Strichartz inequality

Abstract: We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by using the linear profile decomposition. Furthermore we show that, if $f$ is an extremiser, then $f$ is extremely fast decaying in Fourier space and so $f$ can be extended to be an entire function on the whole complex domain. The rapid decay of the Fourier transform of extremisers is established with a bootstrap argument which relies on a refined bilinear Airy Strichartz estimate and a weighted Strichartz inequality.

A landing theorem for dynamic rays of geometrically finite entire functions

Abstract: A transcendental entire function f is called geometrically finite if the intersection of the set S(f) of singular values with the Fatou set F(f) is compact and the intersection of the postsingular set P(f) with the Julia set J (f) is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then F(f) is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z0 to ��. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions.

Localization and perturbation of zeros of entire functions

Abstract: Finite Matrices Inequalities for eigenvalues and singular numbers Inequalities for convex functions Traces of powers of matrices A relation between determinants and resolvents Estimates for norms of resolvents in terms of the distance to spectrum Bounds for roots of some scalar equations Perturbations of matrices Preservation of multiplicities of eigenvalues An identity for imaginary parts of eigenvalues Additional estimates for resolvents Gerschgorin's circle theorem Cassini ovals and related results The Brauer and Perron theorems Eigenvalues of Compact Operators Banach and Hilbert spaces Linear operators Classification of spectra Compact operators in a Hilbert space Compact matrices Resolvents of Hilbert-Schmidt operators Operators with Hilbert-Schmidt powers Resolvents of Schatten-von Neumann operators Auxiliary results Equalities for eigenvalues Proofs of Theorems 261 and 281 Spectral variations Preservation of multiplicities of eigenvalues Entire Banach-valued functions and regularized determinants Some Basic Results of the Theory of Analytic Functions The Rouche and Hurwitz theorems The Caratheodory inequalities Jensen's theorem Lower bounds for moduli of holomorphic functions Order and type of an entire function Taylor coefficients of an entire function The theorem of Weierstrass Density of zeros An estimate for canonical products in terms of counting functions The convergence exponent of zeros Hadamard's theorem The Borel transform Polynomials Some classical theorems Equalities for real and imaginary parts of zeros Partial sums of zeros and the counting function Sums of powers of zeros The Ostrowski-type inequalities Proof of Theorem 451 Higher powers of real parts of zeros The Gerschgorin type sets for polynomials Perturbations of polynomials Proof of Theorem 491 Preservation of multiplicities Distances between zeros and critical points Partial sums of imaginary parts of zeros Functions holomorphic on a circle Bounds for Zeros of Entire Functions Partial sums of zeros Proof of Theorem 511 Functions represented in the root-factorial form Functions represented in the Mittag-Leffler form An additional bound for the series of absolute values of zeros Proofs of Theorems 551 and 553 Partial sums of imaginary parts of zeros Representation of ezr in the root-factorial form The generalized Cauchy theorem for entire functions The Gerschgorin-type domains for entire functions The series of powers of zeros and traces of matrices Zero-free sets Taylor coefficients of some infinite-order entire functions Perturbations of Finite-Order Entire Functions Variations of zeros Proof of Theorem 612 Approximations by partial sums Preservation of multiplicities Distances between roots and critical points Tails of Taylor series Functions of Order Less than Two Relations between real and imaginary parts of zeros Proof of Theorem 711 Perturbations of functions of order less than two Proof of Theorem 731 Approximations by polynomials Preservation of multiplicities of in the case p(f) < 2 Exponential-Type Functions Application of the Borel transform The counting function The case a(f) < Variations of roots Functions close to cos z and ez Estimates for functions on the positive half-line Difference equations Quasipolynomials Sums of absolute values of zeros Variations of roots Trigonometric polynomials Estimates for quasipolynomials on the positive half-line Differential equations Positive Green functions of functional differential equations Stability conditions and lower bounds for quasipolynomials Transforms of Finite-Order Entire Functions and Canonical Products Comparison functions Transforms of entire functions Relations between canonical products and Sp Lower bounds for canonical products in terms of Sp Proof of Theorem 1041 Canonical products and determinants Perturbations of canonical products Polynomials with Matrix Coefficients Partial sums of moduli of characteristic values An identity for sums of characteristic values Imaginary parts of characteristic values of polynomial pencils Perturbations of polynomial pencils Multiplicative representations of rational pencils The Cauchy type theorem for polynomial pencils The Gerschgorin type sets for polynomial pencils Estimates for rational matrix functions Coupled systems of polynomial equations Vector difference equations Entire Matrix-Valued Functions Preliminaries Partial sums of moduli of characteristic values Proof of Theorem 1221 Imaginary parts of characteristic values of entire pencils Variations of characteristic values of entire pencils Proof of Theorem 1251 An identity for powers of characteristic values Multiplicative representations of meromorphic matrix functions Estimates for meromorphic matrix functions Zero free domains Matrix-valued functions of a matrix argument Green's functions of differential equations Bibliography Index

A power of an entire function sharing one value with its derivative

Abstract: In this paper, we investigate uniqueness problems of entire functions that share one value with one of their derivatives. Let f be a non-constant entire function, n and k be positive integers. If f^n and (f^n)^(^k^) share 1 CM and n>=k+1, then f^n=(f^n)^(^k^), and f assumes the form f(z)=ce^@l^n^z, where c is a non-zero constant and @l^k=1. This result shows that a conjecture given by Bruck is true when F=f^n, where n>=2 is an integer.

Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities

Abstract: This is a continuation of our papers [3] and [4]. In those papers we obtained estimates for finite differences (Δ K f)(A) = f(A+K)-f(A) of the order 1 and of the order m for certain classes of functions f, where A and K are bounded self-adjoint operators. In this paper we extend results of [3] and [4] to the case of unbounded self-adjoint operators A. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs of the main results of [3]. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers [3] and [4] were announced in [2].

Poincar\'{e} functions with spiders' webs

Abstract: For a polynomial p with a repelling fixed point w, we consider Poincare functions of p at w, i.e. entire functions L which satisfy L(0)=w and p(L(z))=L(p'(w)*z) for all z in the complex plane. We show that if the component of the Julia set of p that contains w equals {w}, then the (fast) escaping set of L is a spider's web; in particular it is connected. More precisely, we classify all linearizers of polynomials with regards to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point.

Algebraic values of transcendental functions at algebraic points

Abstract: In this paper, the authors will prove that any subset of $\overline{\QQ}$ can be the exceptional set of some transcendental entire function. Furthermore, we could generalize this theorem to a much more general version and present a unified proof.