Showing papers on "Entire function published in 1993"

Iteration of meromorphic functions

Abstract: This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

Harmonic analysis and boundary value problems in the complex domain

Abstract: 1 Preliminary results. Integral transforms in the complex domain -- 1.1 Introduction -- 1.2 Some identities -- 1.3 Integral representations and asymptotic formulas -- 1.4 Distribution of zeros -- 1.5 Identities between some Mellin transforms -- 1.6 Fourier type transforms with Mittag-Leffler kernels -- 1.7 Some consequences -- 1.8 Notes -- 2 Further results. Wiener-Paley type theorems -- 2.1 Introduction -- 2.2 Some simple generalizations of the first fundamental Wiener-Paley theorem -- 2.3 A general Wiener-Paley type theorem and some particular results -- 2.4 Two important cases of the general Wiener-Paley type theorem -- 2.5 Generalizations of the second fundamental Wiener-Paley theorem -- 2.6 Notes -- 3 Some estimates in Banach spaces of analytic functions -- 3.1 Introduction -- 3.2 Some estimates in Hardy classes over a half-plane -- 3.3 Some estimates in weighted Hardy classes over a half-plane -- 3.4 Some estimates in Banach spaces of entire functions of exponential type -- 3.5 Notes -- 4 Interpolation series expansions in spacesW1/2,?p,?of entire functions -- 4.1 Introduction -- 4.2 Lemmas on special Mittag-Leffler type functions -- 4.3 Two special interpolation series -- 4.4 Interpolation series expansions -- 4.5 Notes -- 5 Fourier type basic systems inL2(0, ?) -- 5.1 Introduction -- 5.2 Biorthogonal systems of Mittag-Leffler type functions and their completeness inL2(0, ?) -- 5.3 Fourier series type biorthogonal expansions inL2(0, ?) -- 5.4 Notes -- 6 Interpolation series expansions in spacesWs+1/2,?p,?of entire functions -- 6.1 Introduction -- 6.2 The formulation of the main theorems -- 6.3 Auxiliary relations and lemmas -- 6.4 Further auxiliary results -- 6.5 Proofs of the main theorems -- 6.6 Notes -- 7 Basic Fourier type systems inL2spaces of odd-dimensional vector functions -- 7.1 Introduction -- 7.2 Some identities -- 7.3 Biorthogonal systems of odd-dimensional vector functions -- 7.4 Theorems on completeness and basis property -- 7.5 Notes -- 8 Interpolation series expansions in spacesWs,?p,?of entire functions -- 8.1 Introduction -- 8.2 The formulation of the main interpolation theorem -- 8.3 Auxiliary relations and lemmas -- 8.4 Further auxiliary results -- 8.5 The proof of the main interpolation theorem -- 8.6 Notes -- 9 Basic Fourier type systems inL2spaces of even-dimensional vector functions -- 9.1 Introduction -- 9.2 Some identities -- 9.3 The construction of biorthogonal systems of even-dimensional vector functions -- 9.4 Theorems on completeness and basis property -- 9.5 Notes -- 10 The simplest Cauchy type problems and the boundary value problems connected with them -- 10.1 Introduction -- 10.2 Riemann-Liouville fractional integrals and derivatives -- 10.3 A Cauchy type problem -- 10.4 The associated Cauchy type problem and the analog of Lagrange formula -- 10.5 Boundary value problems and eigenfunction expansions -- 10.6 Notes -- 11 Cauchy type problems and boundary value problems in the complex domain (the case of odd segments) -- 11.1 Introduction -- 11.2 Preliminaries -- 11.3 Cauchy type problems and boundary value problems containing the operators $$ {\mathbb{L}_{s + 1/2}}$$ and $$ \mathbb{L}_{s + 1/2}^*$$ -- 11.4 Expansions inL2{?2s+1(?)} in terms of Riesz bases -- 11.5 Notes -- 12 Cauchy type problems and boundary value problems in the complex domain (the case of even segments) -- 12.1 Introduction -- 12.2 Preliminaries -- 12.3 Cauchy type problems and boundary value problems containing the operators $${{\mathbb{L}}_{s}} $$ and $$ \mathbb{L}_{s}^*$$ -- 12.4 Expansions inL2{?2s(?)} in terms of Riesz bases -- 12.5 Notes.

On the limit functions of iterates in wandering domains

Abstract: In this paper, it is shown that finite limit functions of iterates of entire functions in wandering domains are limit points of the forward orbits of the finite singularities of the inverse function. From this the absence of wandering domains for some classes of entire functions is deduced.

Newton's method and a class of meromorphic functions without wandering domains

Abstract: Let N be the class of meromorphic functions f with the following properties: f has finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points of f are zeros of f′ and vice versa, with finitely many exceptions; f has finite order. It is proved that if f ∈ N, then f does not have wandering domains. Moreover, if f ∈ N and if ∞ is among the limit functions of fn in a cycle of periodic domains, then this cycle contains a singularity of f−1. (Here fn denotes the nth iterate of f) These results are applied to study Newton's method for entire functions g of the form where p and q are polynomials and where c is a constant. In this case, the Newton iteration function f(z) = z − g(z)/g′(z) is in N. It follows that fn(z) converges to zeros of g for all z in the Fatou set of f, if this is the case for all zeros z of g″. Some of the results can be extended to the relaxed Newton method.

Iteration of meromorphic functions

Abstract: This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

The iteration of entire functions of small growth

Abstract: We show that if a transcendental entire function has order zero and sufficiently small growth or has order ρ < ½ and regular growth then its set of normality has no unbounded components.

Theory of entire and meromorphic functions : deficient and asymptotic values and singular directions

Abstract: Preface The Nevanlinna theory The singular directions The deficient value theory The asymptotic value theory The relationship between deficient values and asymptotic values of an entire function The relationship between deficient values of a meromorphic function and direct transcendental singularities of its inverse functions Some supplementary results References.

On the explicit determination of certain solutions of periodic differential equations

Abstract: We consider the class of equations of the form (∗)w′′ + A(z)w = 0, where A(z) is a nonconstant entire periodic function (where, for convenience, we take the period to be 27φi), and where we assume that A(z) is a rational function of ez . The solutions of such equations are entire functions of infinite order of growth, but such equations can possess one or two linearly independent solutions each of whose zero-sequences has a finite exponent of convergence. In this paper, we develop a method which involves using only the simple technique of undetermined coefficients, which can test any equation (∗) for the existence of a solution whose zero-sequence has a finite exponent of convergence. The method will also produce explicitly any such solution.

Completeness of systems of functions, quasianalyticity and subharmonic majorants

Abstract: In the paper one investigates completeness questions for certain frequently encountered systems of entire functions (for example, the completeness of the system of all entire functions of order not exceeding a given number σ) in Φ-spaces (for example, in C(E), E being a closed subset of R). Dual problems are the problems regarding the quasianatyticity of classes of functions, having Fourier transforms supported on a thin set. These latter problems are solved with the use of potential theory technique and with the aid of the investigation of conformal mappings of the upper half-plane onto special comb-like domains.

Spectral synthesis in a complex domain for a differential operator with constant coefficients. i: a duality theorem

Abstract: The problem of spectral synthesis in a complex domain for a differential operator with symbol , is reduced to the problem of a local description of the closed submodules of a module (of entire functions of exponential type) over the ring of polynomials of the form , .

Using sums of squares to prove that certain entire functions have only real zeros

Abstract: It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions J�(z) when � ≥ −1, confluent hypergeometric functions 0F1(c;z) when c > 0 or 0 > c > −1, Laguerre polynomials L � (z) when � ≥ −2, and Jacobi polynomials P (�,�) n (z) when � ≥ −1 and � ≥ −1. Besides yielding new inequalities for |F(z)| 2 , where F(z) is one of these functions, the derived identities lead to inequalities for @|F(z)| 2 /@y and @ 2 |F(z)| 2 /@y 2 , which also give new proofs of the reality of the zeros.

On the Value Distribution of Composite Meromorphic Functions

Abstract: Let f and g be transcendental entire functions and let p be a nonconstant polynomial. A recent result of Bergweiler [2] says that the function f(g(z))− p(z) has infinitely many zeros, confirming a conjecture of Gross [11] dealing with the special case p(z) = z. The case that f(g) is of finite order follows from either of the earlier results of Gol’dberg and Prokopovich [8], Goldstein [9], Gross and Yang [15], and Mues [19]. In fact, various generalizations are obtained in these papers. In particular, it follows from each of these papers that if f and g are entire, p is a nonconstant polynomial, and f(g(z)) − p(z) has only finitely many zeros, then either f is linear or there exists a polynomial q such that p = q(g), provided f(g) is transcendental and of finite order (the restriction on the order not being essential, as shown in [4]). This latter result does not hold for meromorphic f , even if f(g) has finite order, as shown by the example f(z) = i √ z tan √ z, g(z) = z, and p(z) = z. It is natural to conjecture, however, that the function f(g(z)) − R(z) has infinitely many zeros, if f is meromorphic and transcendental, g is entire and transcendental, and R is rational and nonconstant. As proved in [3], this is in fact the case if R(z) = z and hence for any Mobius transformation R. The method used in [3], however, does not seem to be suitable to handle the case that the degree of R is greater than one. In this paper, we give an affirmative answer to the above question in the case that f(g) has finite order.

Nuclear Fréchet Spaces of Entire Functions with Transitive Differentiation

Abstract: A continuous . linear operator on an infinite dimensional complex topological vector space is called transitive, if it has no closed invariant subspaces which are different from the zero space and the whole space. The existence of transitive operators on certain nonreflexive Banach spaces was established by Enflo [6] and Read [14J. On the other hand, the existence of such operators on a reflexive Banach space, in particular on a Hilbert space, is still an open problem. As shown in [2], the problem for Hilbert spaces is equivalent to the problem of whether there exists a Hilbert space of entire functions with reproducing kernel, on which the differentiation operator is continuous and transitive. In this paper we consider the transitivity problem of the differentiation operator on certain nuclear Frechet spaces of entire functions. Our main result is that, for 0 < a < 1, the vector space Fa of all entire functions f for which the norms

Analysis in ℂ p

Abstract: This chapter tries to give the reader a taste of what analysis in ℂ p is like. Rather than attempt to be exhaustive, which would violate the goals of this book, we try to touch on a few remarkable points: the theory of Newton polygons, the p-adic Weierstrass Preparation Theorem, the description of entire functions. As usual, the first step is to re-appropriate all the results we obtained earlier. We then go on to consider how to extend the p-adic valuation to polynomials and power series. This will yield a norm on the spaces of polynomials and of power series, which will prove to be an important tool. We then go on to proving the main theorems themselves.

Area of Julia sets of holomorphic self-maps onC *

Abstract: It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC*. We'll prove thatJ(f) has positive area, wheref:C*→C*,f(z)=zmcP(z)+Q(1/z),P(z) andQ(z) are monic polynomials of degreed, andm is an integer.

On the deficiencies of composite entire functions

Abstract: For any sequence (a j ) of complex numbers and for any ρ > ½, we construct an entire function F with the following properties. F has order ρ, mean type, each a j is a deficient value of F , and F is given by F (z)= f (g(z)), where f and g are transcendental entire functions. This complements a result of Goldstein. We also construct, for any ρ>½, an entire function G of order p , mean type, such that liminf,→ ∞ T ( r, G )/T( r, G ′)>1.

On the complex oscillation theory of f(k) + Af = F

Abstract: In this paper, we investigate the complex oscillation theory ofwhere A, F≢0 are entire functions, and obtain general estimates of the exponent of convergence of the zero-sequence and of the order of growth of solutions for the above equation.

A generalized potential in the theory of the Rabi and E(X) epsilon Jahn-Teller system

Abstract: The eigenvalue problem for the Rabi and the E(X) epsilon Jahn-Teller Hamiltonian in Bargmanns Hilbert space of analytical functions is a system of two first-order differential equations for the two-component wavefunctions, whose entire solutions (the eigenstates) are sought. We show that each eigenstate is a terminating series in the derivatives of a scalar entire function D(z), called the generalized potential, which satisfies a higher-order differential equation. The coefficients of the terminating series depend on the physical parameters and are polynomials in the independent variable z. The coefficients are identical in all eigenstates.

On a theorem of Craven, Csordas and Smith

Abstract: We get a simple and direct proof of the following known result. Let f(z)-eαz2 g(z) where α← 0 and g is a real entire function of genus at most one. If f has only a finite number of nonreal zeros, then its derivatives, from a certain one onward, have only real zeros.

Weighted polynomial approximation of entire functions on unbounded subsets of the complex plane

Abstract: Abstract We study the asymptotic behavior of the n-widths of a class of entire functions in weighted approximation on subsets of the complex plane.