Showing papers on "Entire function published in 1991"
TL;DR: The Julia set as discussed by the authors is the closure of the set of all preimages of poles in the plane, and it is defined as a subset of a straight line and general classes of functions for which this is the case can be determined.
Abstract: For functions meromorphic in the plane, apart from an exceptional case, the Julia set J is the closure of the set of all preimages of poles. The repelling periodic cycles are dense in J. In contrast with the case of transcendental entire functions, J may be a subset of a straight line and general classes of functions for which this is the case can be determined. J may also lie on a quasicircle through infinity which is not a straight line.
131 citations
Book•
01 Jan 1991
TL;DR: Agarwal and G.Leviatan as discussed by the authors gave a characterization of the classical orthogonal poly-nominals, R.Kahn best L-approximations, K.Rohwer entire functions associated with eq where q is of slower than polynomial growth, MAtteia on the norm of the best approximation operator.
Abstract: A characterization of the classical orthogonal poly-nominals, R.P.Agarwal and G.V.Milovanovic monotone approximation by pseudopolynomicals, G.A.Anastassiou complex-valued spline functions defined on an open complex set, bounded, and multiply connected, MAtteia on the norm of the best approximation operator, M.W.Bartlett and J.J.Swetits remainders for boolean interpolation, G.Baszenski and F.J.Delvos some spectral approximations of mono-dimensional fourth-order problems, C.Bernardi and Y.Manday able summability of entire functions of exponential type, W.T.Butterworth mean and uniform convergence of quadrature rules for evaluating the finite Hilbert Transform, G.Criscuolo and G.Mastroianni some properties of a rational operator of Bernstein-type, B.D.Vecchia anti-periodic interpolation of uniform meshes, F.J.Delvos a nodal spline generalization of the Lagrange Interpolant, J.M.De Villiers and C.H.Rohwer entire functions associated with eq where q is of slower than polynomial growth, K.A.Driver quadrature formulae with Birkhoff-type data on equidistant nodes for 2-periodic functions, D.P.Dryanov et al rate of convergence of moving least squares interpolation methods - the univariate case, R.Farwig on proximinal subspaces of continuous function spaces, M.Feder discretization of convolution and reconstruction of band-limited functions from irregular sampling, H.G.Feichtinger Chebyshev Polynomials are not always optimal, B.Fischer and R.Freund Banach Spaces with a basic inequality property and the best compact approximation property, R.J.Fleming and J.E.Jamison M-ideals and a basic inequality in Banach Spaces, R.J.Fleming and J.E.Jamison strong asymptotics and the limit distribution of the zeros of Jacobi Polynomials P, W.Gawronski and B.Shawyer rational approximation of a class of infinite dimensional systems, K.Glover et al the Posse-Markov-Stieltjies Inequality for Turan type quadratic sums, L Gori on the convergence of Quasi-Gaussian Functionals, L.Gori and E.Santi equivalence of a weighted modulus of smoothness and a modified weighted K-functional, M.Heilman and M.W.Muller best L-approximation by convex functions of several variables, S.Legg and D.Townsend approximation properties of beta operators, M.K.Khan some properties of a Bernstein-type operator of Blimann, Butzer and Hahn, R.A.Kahn best L-approximations, K.Kitahara generalized series expansions of a function, A.Knopfmacher the dimension of the space of periodic splines on the regular hexagonal lattice, F.Krebs on Fejer Means with respect to orthogonal polynomials - a hypergroup theoretic approach, R.Lasser and J.Obermaier degree of approximation by polynomials with restricted co-efficients, D.Leviatan. (Part contents) ...
120 citations
TL;DR: In this article, a logarithmic Sobolev inequality was deduced for all 0 < p ⩽ q < t8 where the Lp norms are taken with respect to the measure dμh above.
101 citations
TL;DR: In this paper, a q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons, which are represented as multiplication by z and q-differentiation with respect to z.
Abstract: A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmann's scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.
81 citations
TL;DR: In this paper, it was shown that every (necessarily entire) nonconstant solution / of the differential equation f + gf + hf = 0 has infinite order.
Abstract: Suppose g and h are entire functions with the order of h less than the order of g . If the order of g does not exceed j , it is shown that every (necessarily entire) nonconstant solution / of the differential equation f\" + gf + hf = 0 has infinite order. This result extends previous work of Ozawa and Gundersen.
56 citations
TL;DR: In this article, it was shown that if n ≥ 2, then there are infinitely many fixpoints of an entire transcendental function which are not fixpoints for any k satisfying 1 ≤ k ≤ n.
Abstract: Let f be an entire transcendental function and denote the nth iterate off by f n. Our main result is that if n ≥ 2, then there are infinitely many fixpoints of f n which are not fixpoints of fk for any k satisfying 1 ≤ k ≤ n. This had been conjectured by I. N. Baker in 1967. Actually, we prove that there are even infinitely many repelling fixpoints with this property. We also give a new proof of a conjecture of E Gross from 1966 which says that if h and g are entire transcendental functions, then the composite function hog has infinitely many fixpoints. We show that h∘g. has even infinitely many repelling fixpoints.
45 citations
TL;DR: In this paper, the electric surface current is modeled as a finite series of sinusoids whose domain consists of the entire generating curve, which results in a matrix size of less than 5% of that produced with subdomain basis functions.
Abstract: The moment-method technique utilizing entire domain basis functions is applied to the analysis of large, axially symmetric reflector antennas. The electric surface current is modeled as a finite series of sinusoids whose domain consists of the entire generating curve. This expansion results in a matrix size of less than 5% of that produced with subdomain basis functions. Only a slight increase in the CPU requirements occurs from this analysis. The results from this technique show good agreement when compared to both physical optics and a subdomain-based moment-method formulation on small, axially fed paraboloidal and hyperboloidal reflector antennas. Extension to a large 100- lambda paraboloidal reflector with f/D=0.4 produces results comparable to that obtained using physical optics. Convergence is obtained with as few as two expansion terms per wavelength. Discretization of the generating curve with four points per wavelength leads to results which agree within 0.5 dB over data from a more densely defined curve. >
35 citations
TL;DR: In this class of impulse responses calledpseudorational, it is shown that the difficulty is related to classical complex analysis, especially that of entire functions of exponential type, and the infinite-product representation makes it possible to prove that stability is indeed determined by the location of spectrum or by a modified H∞ condition.
Abstract: It is well known that for infinite-dimensional systems, exponential stability is not necessarily determined by the location of spectrum. Similarly, transfer functions in theH∞ space need not possess an exponentially stable realization. This paper addresses this problem for a class of impulse responses calledpseudorational. In this class, it is shown that the difficulty is related to classical complex analysis, especially that of entire functions of exponential type. The infinite-product representation for such entire functions makes it possible to prove that stability is indeed determined by the location of spectrum or by a modifiedH∞ condition. Examples are given to illustrate the theory.
35 citations
TL;DR: In this paper, the authors introduce a family of operators called C-groups and apply them to the first and second order abstract Cauchy problem, for a large class of linear operators on a Banach space.
Abstract: We introduce a family of operators that we will callentire C-groups, and apply them to the first and second order abstract Cauchy problem, for a large class of linear operators on a Banach space. This produces unique solutions, for all initial data in a large (often dense) set, eachof which extends to an entire function, with continuous dependence on the initial data.
29 citations
TL;DR: It is shown that complex Kergin interpolation may be defind in any domain that is C -convex, whereas the original definition required ordinary, real convexity.
28 citations
TL;DR: In this article, various forms of the Whittaker-Kotelnikov -Shannon sampling theorem are derived by viewing them as sums of residues, and the contour integral method provides a powerful way of generating the correct form of the series and of obtaining the uniform convergence.
Abstract: Various forms of the Whittaker-Kotelnikov -Shannon sampling theorem, which allow certain entire functions to be represented by interpolatory series, are derived by viewing them as sums of residues. Series in which the sample points are distributed in one and in two dimensions are considered.This contour integral method provides a powerful way of generating the correct form of the series, and of obtaining the uniform convergence.Particular emphasis is placed upon series representations involving derivative samples as well as samples of the function itself. The general derivative sampling series is treated, as well as derivative sampling at points which are slightly perturbed from uniform spacing. Finally, derivative sampling at lattice points in the complex plane is considered
TL;DR: It was shown in this article that for any function on C n, a semigroup of operators T(t), t ⩾ 0, generated by the number operator in the complex wave representation is contractive if and only if p ϵ [1, re2t], and if p > re 2t.
01 Jan 1991
TL;DR: In this paper, the test function space H∞(SR) is introduced, which is the basic local element of the duality H ∞(G),H-∞ (G).
Abstract: Here we introduce the test-function space H∞(SR), which is the basic local element of the duality {H∞(G),H-∞(G)}.
Book•
24 Sep 1991
TL;DR: A selection of some important topics in complex analysis, intended as a sequel to the author's classical complex analysis (see preceding entry), can be found in this article, where five chapters are devoted to analytic continuation; conformal mappings, univalent functions, and nonconformal mapping; entire function; meromorphic fu
Abstract: A selection of some important topics in complex analysis, intended as a sequel to the author's Classical complex analysis (see preceding entry). The five chapters are devoted to analytic continuation; conformal mappings, univalent functions, and nonconformal mappings; entire function; meromorphic fu
TL;DR: In this paper, a Banach space for 1 0 consists of all entire functions of exponential type (not exceeding) that belong to HP(C+NOH) that are defined almost everywhere on the line.
Abstract: = K~{c+)d~---etHP(C+)NOH~(C+), I ~p ~+oo (Now, 8 and elements of classes HP are functions defined almost everywhere on the line R=FrC+). Endowed with the LP-norm, KOP becomes a Banach space for 1 0 consists of all entire functions of exponential type (not exceeding) a that belong to HP(C+). The papers [5-13] are devoted to a study of spaces KsP. The following basic facts will be useful for our purposes.
TL;DR: In this article, the authors studied the fine geometric structure of strongly continuous semigroups that satisfy the following property: the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type.
Abstract: In this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property: the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type. This class includes the solution map for functional differential equations and certain partial differential equations. In particular, we present necessary and sufficient conditions for one-to-oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator.
TL;DR: In this paper, a limiting equality between the best approximations in L ∞ of functions of several variables by algebraic polynomials and entire functions of exponential type was established.
Abstract: A limiting equality is established between the best approximations in L∞ of functions of several variables by algebraic polynomials and entire functions of exponential type.
TL;DR: Three elementary approximations for computing Dawson's integral are developed that are convenient to implement in those situations where high accuracy is not required and can be used to compute Dawson’s integral with a relative error.
Abstract: We develop three elementary approximations for computing Dawson's integral. These approximations are convenient to implement in those situations where high accuracy is not required. The most accurate of the three approximations can be used to compute Dawson's integral with a relative error
TL;DR: In this article, it was shown that if the singularity lies at 0, then at least lim inf n⇒∞ 1 n ∑ j=1 n g({jβ})= ʃ 0 1 g(t)dt, for each bounded and Riemann integrable g.
01 Apr 1991
TL;DR: In this article, the inverse Laplace transform of the frequency-domain solution of the transient scattered field from a lossless dielectric sphere is evaluated by the inverse Langevin transform, and the numerical results obtained from both methods are in complete agreement.
Abstract: The transient scattered field from a lossless dielectric sphere is evaluated by the inverse Laplace transform of the frequency-domain solution. Two methods of obtaining the inverse transform are employed. In the first approach, direct numerical evaluation of the Bromwich integral is carried out. To overcome the slow convergence of the Mie series at high frequencies (HF), asymptotic expressions are used in the large | s | portion of a properly chosen Bromwich contour, allowing a significant reduction of the computation time. In the second approach, the asymptotically compensated singularity expansion method is employed. In this method, a multiplicative function is chosen to compensate for the asymptotic growth of the field in the right half of the s–plane, leading to a singularity expansion without entire functions. The transient scattered field is then recovered. The numerical results obtained from both methods are in complete agreement. Reference to the physical origins of the various structures appearing in the time-domain response is made by considering the dependence of their magnitude and time of arrival on the permittivity. Verification of the various predictions previously given by frequency-domain analysis can be carried out in a different perspective with new physical insight.
01 May 1991
TL;DR: Valiron[3] showed that if f(z) is an entire function for which where A(r) = log(sup{|f(z)|:|z| = r}) , then there is a function x(R) = r ρ( r) satisfying both
Abstract: Valiron[3] showed that if f(z) is an entire function for whichwhere A(r) = log(sup{|f(z)|:|z| = r}), then there is a function x(r) = rρ(r) satisfying both
TL;DR: In this paper, it was shown that for any subharmonic on a plane function u of finite order p and any ~ > p, there exists an entire function f, a constant C(~) dependent only on ~, and a set E(a) such that the inequality ]u(z) l n ] / ( z ) ] ] < C(a ln ]z I.
Abstract: Results on approximation of a subharmonic function by the logarithm of the modulus of an analytic function are employed in construction of analytic functions with prescribed asymptotic properties. This kind of approximation has been considered in a series of works by Yulmukhametov, in which it has been shown [I, p. 278], among other things, that for any subharmonic on a plane function u of finite order p and any ~ > p there exist an entire function f, a constant C(~) dependent only on ~, and a set E(a) such that the inequality ]u(z) l n ] / ( z ) ] ] < C(a) ln ]z I. h o l d s f o r a l l z q/~E(a). The e x c e p t i o n a l s e t E(~) i s c o n t a i n e d in t h e u n i o n of d i s k s {z: Iz zil < t i} (i = i, 2 .... ) such that
TL;DR: The notion of linearly invariant families of functions defined in the unit disk D was introduced by Pommerenke as mentioned in this paper, where the family of all Koebe transforms of a function has a finite linear invariant order.
Abstract: Pommerenke initiated the study of linearly invariant families of functions defined in the unit disk D. A holomorphic function f on D is called linearly invariant if the family of all Koebe transforms of f has finite linear invariant order. A function f is linearly invariant on D if and only if f is uniformly locally univalent in the hyperbolic sense; that is, there is an r > 0 such that f is univalent in every hyperbolic disk of radius r. We present two extensions of the notion of linear invariance to general planar regions, one involves the hyperbolic metric and the other the quasihyperbolic metric. We relate these two concepts of linear invariance to uniform local univalence relative to each of these metrics. For uniformly perfect regions all of these concepts coincide; we obtain various inclusion relations for non-uniformly perfect regions. Finally, we characterize entire functions which are uniformly locally univalent relative to the euclidean metric and establish a curious connection between function...
01 Apr 1991
TL;DR: In this article, it was shown that the set {hn : n = 0, 1,... } is a linearly independent sequence of entire functions, where h0 = 1, hx = gx, h2 = S\ ° Sj.. hx o g2o g?, gx is a nonconstant entire function and gn {n > 2} are entire functions which are not polynomials of degree < 1.
Abstract: We prove the following result: The set {hn : n = 0, 1, ... } is a linearly independent sequence of entire functions, where h0 = 1 , hx = gx , h2 = S\ ° Sj.. hi = gx o g2o g?, ... , gx is a nonconstant entire function and gn {n > 2) are entire functions which are not polynomials of degree < 1 . Our theorem generalizes a previous one about linear independence of iterates.
TL;DR: For the case of a simply connected domain in the plane, the authors proves necessary and sufficient conditions for the representation of functions of the Hardy class H1 by an integral with respect to the harmonic measure of its boundary values.
Abstract: For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H1 by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.