Showing papers on "Entire function published in 1989"

Unbiased estimation of a nonlinear function a normal mean with application to measurement err oorf models

Abstract: Let W be a normal random variable with mean μand known variance σ2. Conditions on the function f(·) are given under which there exists an unbiased estimator, f(W), of f(μ) for all real μ. In particular it is shown that f(·) must be an entire function over the complex plane. Infinite series solutions for F(·) are obtained which are shown to be valid under growth conditions of the derivatives, fk( ·), of f(·). Approximate solutions are given for the cases in which no exact solution exists. The theory is applied to nonlinear measurement-error models as a means of finding unbiased score functions when measurement error is normally distributed. Relative efficiencies comparing the proposed method to the use of conditional scores (Stefanski and Carroll, 1987) are given for the Poisson regression model with canonical link.

An indirect method for numerical optimization using the Kreisselmeir-Steinhauser function

Abstract: A technique is described for converting a constrained optimization problem into an unconstrained problem. The technique transforms one of more objective functions into reduced objective functions, which are analogous to goal constraints used in the goal programming method. These reduced objective functions are appended to the set of constraints and an envelope of the entire function set is computed using the Kreisselmeir-Steinhauser function. This envelope function is then searched for an unconstrained minimum. The technique may be categorized as a SUMT algorithm. Advantages of this approach are the use of unconstrained optimization methods to find a constrained minimum without the draw down factor typical of penalty function methods, and that the technique may be started from the feasible or infeasible design space. In multiobjective applications, the approach has the advantage of locating a compromise minimum design without the need to optimize for each individual objective function separately.

On the zeros of the derivatives of real entire functions and Wiman’s conjecture

Abstract: Let f be an entire function of finite order, real on the real axis, and possessing only real zeros. A classical problem, proposed by G. P6lya [9], [10] and A. Wiman [1], is to determine, from the Hadamard canonical representation of the function, the number of nonreal zeros of the higher derivatives of f. In this paper we shall prove the following conjecture of Wiman, which has been open since 1915:

Geometric function theory

Abstract: We consider the basic problems, notions and facts in the theory of entire functions of several variables, i. e. functions J(z) holomorphic in the entire n space 1 the zero set of an entire function is not discrete and therefore one has no analogue of a tool such as the canonical Weierstrass product, which is fundamental in the case n = 1. Second, for n> 1 there exist several different natural ways of exhausting the space

Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros

Abstract: If two meromorphic functions f and g share 0 and ∞ CM. then g=f exp ∞ for a certain entire function ∞ In this paper we show. If for n=1…6. the derivatives f (n) and g (n) share 0 CM. then ∞ is cons...

Exponential series in smirnov spaces and interpolation by entire functions of special classes

Abstract: This paper studies problems of the Dirichlet series expansions of elements of Smirnov spaces in convex domains, and related theorems on interpolation by entire functions of exponential type. Bibliography: 22 titles.

A probabilistic approach to the wiman—valiron theory for functions of several complex entire

Abstract: The classical theorems of Wiman Valiron and Clunie on the relationship between the maximum modulus, maximum term and central index of an entire function of one complex variable are generalized to the case of several complex variables. The method of proof is based on the probabilistic approach of Rosenbloom along with some new results on exponential families. The ideas presented should also allow extensions of the theory to analytic functions defined on arbitrary domains and represented by power series, Dirichiet series or general Laplace transforms.

Fourier transforms and the Hermite-Biehler theorem

Abstract: A new necessary and sufficient condition for real entire functions, represented by Fourier transforms, to have only real zeros is proved. An appli- cation of this result to the Riemann {-function is also given.

Rational Variance Functions

Abstract: Exponential dispersion models play an important role in the context of generalized linear models, where error distributions, other than the normal, are considered. Any statistical model expressible in terms of a variance-mean relation $(V, \Omega)$ leads to an exponential dispersion model provided that $(V, \Omega)$ is a variance function of a natural exponential family: Here $\Omega$ is the domain of means and $V$ is the variance function of the natural exponential family. Therefore, it is of a particular interest to examine whether a pair $(V, \Omega)$ can serve as the variance function of a natural exponential family. In this study we consider the case where $\Omega$ is bounded and examine whether $V$ can be the restriction to $\Omega$ of a rational function vanishing at the boundary points of $\Omega$. The class of such functions is large and contains the important subclass of polynomials. It is shown that, apart from the binomial family (possessing a quadratic variance function) and affine transformations thereof, there exists no natural exponential family with variance function belonging to this class. Such a result implies, in particular, that the only variance functions of natural exponential families among polynomials of at least third degree are those restricted to unbounded domains $\Omega$.

Radial entire solutions of the linear equation $\Delta u+łambda p(\vert x\vert )u=0$

Abstract: We are concerned with radial entire solutions of the linear elliptic differential equation Δu+λp(|x|)u=0, x∈R N , where Δ is the N-dimensional Laplacian, |x| denotes the Euclidean length of x∈R N , and λ is a positive parameter. We always assume that N≥3 and p satisfies p∈C[0, ∞), p(t)≥0 on [0, ∞), and p(t)÷ on [T, ∞) for every T≥0

On Picard’s theorem for entire quasiregular mappings

Abstract: Several refinements of Picard's theorem for entire functions in the complex plane have been proved by many authors in connection with the theory of Picard sets We prove a result of this type for entire quasiregular mappings in euclidean n-space in the case when the "Picard set" consists of a sequence ak on a ray emanating from 0 with lak = 2k

Approximation of homogeneous subharmonic functions

Abstract: Let be a positive homogeneous subharmonic function, i.e., , , , and let be its associated measure. Let the function be such that Then there exists an entire function for which Bibliography: 6 titles.

A weighted version of the Paley–Wiener theorem

Abstract: A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

Generalized Padé Approximants of Kakehashi’s Type and Meromorphic Continuation of Functions

Abstract: Converse theorems related to interpolating sequences of rational functions with a fixed number of free poles are proved. These theorems give sufficient conditions for the meromorphic continuability of functions.

Maximum modulus convexity and the location of zeros of an entire function

Abstract: Let f be an entire function with non-negative Maclaurin coefficients and let b(r) = r(rf'(r)/f(r))' . It is shown that if all the zeros of f lie in the angle I arg zI 1 cosec2 2a. In particular, we always have lim SUpr-,o b(r) > 1 for such functions.

Domains of exponentiated fractional Jacobi operators : characterizations, classifications, expansion results

Abstract: The main topic of this paper is the study of bases of Jacobi polynomials in topological vector spaces of entire functions of slow growth. These topological vector spaces are weightedL2-spaces and inductive-projective limits of these. One of the side results is a characterization of the analyticity and entireness domains of general fractional Jacobi operators.