Showing papers on "Entire function published in 1981"

The iteration of polynomials and transcendental entire functions

Abstract: The iterative behaviour of polynomials is contrasted with that of small transcendental functions as regards the existence of unbounded domains of normality for the sequence of iterates.

The singularity expansion method: background and developments

Abstract: The singularity expansion method (SEM) arose from the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids. In the complex frequency plane, these damped sinusoids are poles of the Laplace-transformed response. The question is then one of characterizing the object response (time and frequency domains) in terms of all the singularities (poles, branch cuts, entire functions) in the complex frequency plane (hence singularity expansion method). Building on the older concept of natural frequencies, formulae were developed for the pole terms from an integral-equation formulation of the scattering process. The resulting factoring of the pole terms has important application consequences. Later developments include the eigenmode expansion method (EEM) which diagonalizes the integral-equation kernels and which can be used as an intermediate step in ordering the SEM terms. Additional concepts which have appeared include e...

Evidence that bears on the left half plane asymptotic behavior of the sem expansion of surface currents

Abstract: The issues which have persisted in connection with the so-called “entire function contribution” and in connection with alternative coupling coefficient form interrelate closely with the larges asymptotic behavior in the left half plane in SEM representations. To date, no generally applicable rigorous information has been gleaned about this asymptotic behavior. On the other hand, the specific scattering geometries of the sphere and the wire loop yield analytic solutions which can be analyzed asymptotically. Further information can be discerned on a numerical basis or through a procedure based on the discretization of an integral equation. All of this evidence form a mutually-consistent picture of the asymptotic behavior in question. The principal conclusion which results is that the observed behavior taken with the Mittag-Leffler-type expansion theory for complex functions leads to SEM representations which are free from entire function constituents.

An extension of Liouville's theorem on integration in finite terms

Abstract: In this paper we give an extension of the Liouville theorem [RISC69, p. 169] and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone.Our main result generalizes Liouville's theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral to appear in the integral of an elementary function. The basic conclusion is that these functions, if they appear, appear linearly.

The zeros of the second derivative of the reciprocal of an entire function

Abstract: Let f be a real entire function of finite order with only real zeros. Assuming that f' has only real zeros, we show that the number of nonreal zeros of f" equals the number of real zeros of F", where F = I/f. From this, we show that F" has only real zeros if and only if f(z) = exp(az2 + bz + c), a > 0, or f(z) = (Az + B)', A # 0, n a positive integer.

A Dirichlet norm inequality and some inequalities for reproducing kernel spaces

Abstract: Let f be analytic and of finite Dirichlet norm in the unit disk A with f(0) = 0. Then, for any q > 0, IIexpfIIq 1. This also extends with a substantially easier proof, a result of Saitoh concerning the case of q > 1. In addition, a sharp norm inequality, valid for two functional Hilbert spaces whose reproducing kernels are related via an entire function with positive coefficients, is established.

Entire functions on locally convex spaces and convolution operators

Abstract: In this work we generalize the classical results on approximation and existence of solutions of convolution equations in H(Cn). We introduce the spaces HSNb(E) and Nb(E) of the nuclearly Silva entire functions of bounded type and of the nuclearly entire functions of bounded type in a complex locally convex space E. These spaces are endowed with natural locally convex topologies. Convolution equations are considered in these space and results of approximation for solutions of homogeneous convolution equations are proved for any E. Results of existence are demonstrated for a more restrictive class of locally convex spaces which includes the DF-spaces. These results generalize theorems of Gupta and Matos. We also introduce the spaces N(E) of the nuclearly entire functions and SN(E) of the nuclearly Silva entire functions. For these spaces we get results of approximation for solutions of homogeneous convolution equations, thus generalizing, theorems of Gupta-Nachbin.

The minimum of small entire functions

Abstract: It is shown that if f(z) is entire and satisfies lim log M(r, )/(og r)2 = a 1 exp((2k 1)/4a) 2 m(r,)Mr,f)> I+ exp((2k -1)/4o)) + o(1). This proves a long-standing conjecture of P. D. Barry.

A class of Bessel summations

Abstract: These quantities have a host of useful properties which are concisely summarized in [3]. In particular, L(s; x) is an entire function except for the trivial character x(n) -1 for which L(s; 1) = c(s) has a simple pole at s = 1. We shal express S(x) as a power series, indicate various sets of parameters for which it can be summed in closed form, and give some examples. The Mellin transform of S(x) is [2]

Partial inner product spaces of entire functions

Abstract: We investigate various partial inner product space generalizations of Bargmann’s Hilbert space of entire functions as used in the coherent state representation of Quantum Mechanics. In particular, we exhibit a hierarchy of nested Hilbert spaces, the smallest of them being Bargmann’s original scale. RESUME. Nous étudions differents espaces à produit interne partiel qui généralisent l’espace hilbertien de fonctions entières introduit par Bargmann et utilise dans la representation « états cohérents » de la mecanique quantique. Nous obtenons, en particulier, une hiérarchie d’espaces « nestés » (au sens de Grossmann), le plus petit d’entre eux étant l’échelle originelle de Bargmann.

The order of entire functions with radially distributed zeros

Abstract: Let X, IL, p be the order, lower order and the exponent of convergence of the zeros of an entire function f. Whittaker [8, p. 130] has shown that if ,u and p are finite, then X is finite and X = max( IL, p). The finiteness of IL by itself, however, is not enough to make X finite. It is a rather interesting fact, that a radial distribution of the zeros of f makes X finite if ,u is finite. We point out that the theorem whose statement constitutes the title of Whittaker's paper [8], is an immediate corollary of earlier and more informative results of Edrei and Fuchs [2, p. 298], [3, pp. 261, 264]. Using rather difficult estimates of T(r, f), Edrei and Fuchs [2, p. 308] have shown that q 1) having only real negative zeros. Their result implies that q < IL A < q + 1 for such functions provided that X is assumed finite. Years later Shea [6, p. 204], in studying the Valiron deficiencies of meromorphic functions, obtained as a corollary a bound on X in terms of ,u only, for entire functions f having only real negative zeros and finite order X. Our first result (Theorem 1 below) generalizes the above results and the proof extends to subharmonic (and 8-subharmonic) functions in space. In addition, our proof may be of interest because of its simplicity.

Deficient values of entire functions and their derivatives

Abstract: Let f(z) be entire and of finite order, j() be the nth derivative, and An(f) = 28(a, f(n)), the sum of all deficient values of f(n). The authors show that An(f) can be strictly increasing. Let f(z) be entire of order p (y = > 8(a, fj)), lal (f) be strictly increasing. In this paper we give an affirmative answer. More precisely, we have the stronger THEOREM. Let ck (j = 0, 1, 2,. .; k = 1, 2, .. ., K>; 1 < Kj < oo) be finite complex numbers, with cjk # Cjk, (k 7# k'). Given 4 < p < ?, and an increasing sequence {nj) of integers, there exists an entire function f(z) of order p, mean type, such that

A characterization of principal ideal domains

Abstract: This paper explores the concept of a gauge function on an integral domain. The ring of all complex polynomials has a gauge function whereas the bigger ring of entire functions does not have one. A characterization of principal ideal domain explains the reason why T does not possess any gauge function.

Growth problems for a class of entire functions via singular integral estimates

Abstract: for an absolute constant A 0. The upper estimate for C(p) in (2) comes from "Lindel6f functions": these are certain well known entire functions of any finite order with all zeros regularly distributed along a single ray arg z = constant. It has been conjectured that these functions are extremal for this problem but not even the order of magnitude of C(p) is known, for p large. In this direction, we prove:

High Order Continuity Implies Good Approximations to Solutions of Certain Functional Equations.

Abstract: : An entire function of exponential type A 2 Pi is shown that there is a unique entire function f(x) of exponential type A satisfying a certain functional equation. An apparently new principle is described for generating polynomial approximations in 0 = or X or = 1 to the solutions of certain functional equations.

Entire functions that share real zeros and real ones

Abstract: We find all pairs of nonconstant entire functions that have the same zeros and ones (ignoring multiplicities), when the zeros and ones are all real. We say two entire functions f(z) and g(z) share the value c provided thatff(z) = c if and only if g(z) = c. We distinguish between sharing a value CM (counting multiplicities) and IM (ignoring multiplicities). Unless stated otherwise, all functions will be assumed to be nonconstant and entire. It is easy to show that if f and g share 0 and 1 CM and f :# g, then there are entire functions a and ,8 such that f(z) = 2 (z) and g(z) = e2wfl(z) 1 -2rftz 1 e-27ffif(z) -1() Whenf and g have finite order, then C. F. Osgood and C. C. Yang [5, p. 410] have shown that ,B is a polynomial and a is a polynomial in /3 with rational coefficients. Thus all possible pairs are determined. For infinite order, the example a(z) = z sin('lz2) and ,8(z) = z2 shows that a does not even have to be a power series in ,B. When f and g share 0 and 1 IM, the situation is more complicated. One of the authors [3, Theorem 3] has shown that T(r, f) < (3 + o(l)) T(r, g) and T(r, g) < (3 + o(l))T(r, f) as r -* oo outside a set of finite linear measure (T(r, h) is the Nevanlinna characteristic function of h). M. Ozawa [61 has proven some uniqueness theorems when f and g have finite order, by assuming various further hypotheses on the zeros and ones. But in general, no one has come close to finding all possible pairs. Let SC be the class of all nonconstant entire functions which have only real zeros and real ones. We will prove the following: THEOREM. Iff and g are in SC and share 0 and 1 TM, then we necessarily have one of the following six cases where a # 0 and b are real constants: l.f = g, 2. f(z) = sin2(az + b) and g(z) = -i sin(az + b)e'(az +b) 3. f(z) = ; 2(p + b) and g(z) = sin(p(az + b)) ei(p l)(az+b) sin2(az + b) sin(az + b) forp = -2 andforp = -3, Received by the editors March 20, 1980 and, in revised form, July 7, 1980. 1980 Mathematics Subject Classification Primary 30D35.