Showing papers on "Entire function published in 1988"
TL;DR: In this article, the decomposition of arbitrary f in L2(R) in terms of the family of functions cp,(x) = w-'/*exp{ - iimnab + iman - i(x - nb)2}, with a, b > 0.
Abstract: We look at the decomposition of arbitrary f in L2(R) in terms of the family of functions cp,,(x) = w-'/*exp{ - iimnab + iman - i(x - nb)2}, with a, b > 0. We derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab = 2w/N, N an integer 2 2. Transported to the Hilbert space F of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the form
188 citations
Journal Article•
TL;DR: In this paper, a generalization of the classical concept of growth order of an entire function is proposed, where the authors define the new parameter ρ_g(f), the relative growth order with respect to ρ(z), and establish a direct comparison between the growth of the moduli of two nonconstant entire functions.
Abstract: In this paper, we essay a generalization of the classical concept of growth order of an entire function. We define the new parameter $\rho_g(f)$, the relative growth order of $f(z)$ with respect to $g(z)$, which establishes a direct comparison between the growth of the moduli of two nonconstant entire functions $f$ and $g$. Diverse properties, relative to sum, product, composition, derivative, real and imaginary parts, Nevanlinna’s characteristic and Taylor’s coefficients are studied.
69 citations
TL;DR: In this paper, the Hellmann-Feynman theorem is applied to derive monotone representations for derivatives with respect to a parameter of the positive zeros of a family of entire functions.
57 citations
TL;DR: In this article, the Nevanlinna charachteristic of f and M (r g) the maximum modulus of g were shown to be the same for a meromorphic function and an entire function.
Abstract: Let f be a meromorphic function and g an entire function. T(r f) the Nevanlinna charachteristic of f and M (r g) the maximum modulus of g. We show that and discuss how far this inequality is best possible.
54 citations
TL;DR: In this paper, an algebraic analysis of the space H of all entire functions spanned by the integer translates of an EB-spline is presented, which is based on the observation that the structure of Z is relatively simple when Z is covered by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process.
Abstract: Exponential box splines (EB-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space Z spanned by exponential polynomials This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients The main part of this paper is devoted to algebraic analysis of the space H of all entire functions spanned by the integer translates of an EB-spline This investigation relies on a detailed description of Z and its discrete analog S° The approach taken here is based on the observation that the structure of Z is relatively simple when Z is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case We also identify and study in detail several types of singularities which occur only for complex EB-splines The first is when the Fourier transform of the EB-spline vanishes at some critical points, the second is when Z cannot be embedded in y and the third is when H is a proper subspace of Z We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh
45 citations
Abstract: If f is a transcendental entire function and D is a non-wandering component of the set of normality of the iterates of f such that fn → ∞ in D then log |fn(z)| = O(n) as n → ∞ for z in D. For a wandering component the convergence of fn to ∞ in D may be arbitrarily fast.
37 citations
19 citations
18 citations
TL;DR: In this article, a solution of second order differential equations with quadratic polynomials as coefficients is studied, and it is shown that there exists a solution which is an entire function of exponential type one and such that this function and all its derivatives are close-to-convex in the unit disc.
18 citations
TL;DR: The usefulness of the method is shown in computer-simulation studies of the reconstruction of the one-dimensional real object from its Hartley intensity, and the theory developed in one dimension is straightforwardly extended to two dimensions.
Abstract: A method for reconstructing a real object function from the intensity of its Hartley transform is proposed. This method is established as a closed-form expression by making use of the mathematical properties of entire functions of the exponential type. The usefulness of the method is shown in computer-simulation studies of the reconstruction of the one-dimensional real object from its Hartley intensity. In addition it is shown that the theory developed in one dimension is straightforwardly extended to two dimensions.
17 citations
TL;DR: In this article, the sum of numbers of finite nonzero deficient values of f(z) and all its primitives does not exceed 21t, based on several lemmas and application of harmonic measure.
Abstract: Let f(z) be an entire function of positive and finite order H1. If f (z) has a finite number of Borel directions of order > ,s, then the sum of numbers of finite nonzero deficient values of f (z) and all its primitives does not exceed 21t. The proof is based on several lemmas and application of harmonic measure.
TL;DR: On donne une bonne approximation d'une fonction C k sur R n et de ses derivees par la restriction d'un Fonction entiere entiere sur C n and de sé derivees respectivement respectivement as discussed by the authors.
Abstract: On donne une bonne approximation d'une fonction C k sur R n et de ses derivees par la restriction d'une fonction entiere sur C n et de ses derivees respectivement
01 Apr 1988
TL;DR: In this article, the authors give a proof for the strong form of Liouville's theorem for the planar case, which is similar to the one given in this paper.
Abstract: We present elementary proofs-shorter than any others that we know for three related theorems. THEOREM I. A function u that is harmonic and positive in the upper half-space {x E R': x?, > O} and zero on the boundary hyperplane must be of the form ax,. When n = 2, for instance, this implies that an entire function which maps the upper half-plane into itself and is real on the real axis is an affine function az + b. Many proofs have been given, for instance [1-7]. Our proof is similar to the one given in [4] for the planar case. THEOREM II. A function u that is harmonic in Rn and bounded from one side by a polynomial must be a polynomial, and of no higher degree. This is a strong form of Liouville's theorem. THEOREM III. A function f, meromorphic in the whole complex plane, real on the real axis, with Im f (z) > 0 when Im z > 0, has the form f(z) = bo + b1z + C +E(Ak Ak) z z -ak ak where bo E R, b1 > 0, c < 0, ak E R \ {0}, Ak < 0, and the series converges uniformly on every compact set that avoids the poles. This is a theorem of Chebotarev and Mefman [1, p. 197]. PROOF OF THEOREM I. By the reflection principle we may assume that u is harmonic in the whole space R', and that u is an odd function of x,. Write 00 00 (1) u(x) =E u (x) = L r'u. (w) ^X=1 ,7=1 where x = rw, r = Ixi, and each u7 is a homogeneous polynomial of degree j. The u, inherit from u the properties of harmonicity and anti-symmetry in the variable x,. In particular ul(x) = axn, and the function lu3(x)/xnl extends continuously to the unit sphere and has some upper bound c3 there. Multiply (1) by g(w) = Ckwn Uk (w) and integrate over the unit sphere. Since spherical harmonics of Received by the editors June 8, 1987. 1980 Mathematics Subject (Cassificatzon (1985 Revzssion). Primary 31B05, Secondary 30D30.
01 Jan 1988
TL;DR: In this paper, the remainder of cardinal spline interpolation (CSI) of the odd degree 2m-1, in its customary Peano form, appears to be endowed with numerous worthwhile properties some of which are described in the first half of this paper.
Abstract: Remainders in terms of high-order derivatives might at times seem rather useless for numerical applications However, they are often effective in theoretical problems of convergence Our present topic is the remainder of cardinal spline interpolation (CSI) of the odd degree 2m-1, in its customary Peano form Its kernel K2m-1(x,t) appears to be endowed with numerous worthwhile properties some of which are described in the first half (Part I) of this paper The remainder of CSI allows us to discuss the behavior of the interpolant, as m → ∞, of entire functions of exponential type (Theorem 6 of §6)
TL;DR: In this article, the authors proved that it is necessary and sufficient that for all, and sufficient for all functions to be a function of completely regular growth in in the sense of weak convergence in, the radial indicator of the entire function is the support function of the compact set.
Abstract: Let be a convex domain and a convex compact set in ; let be the space of analytic functions in , provided with the topology of uniform convergence on compact sets, and the space of germs of analytic functions on with the natural inductive limit topology; and let be the space dual to . Each functional generates a convolution operator , , , which acts continuously from into . Further let be the Fourier-Borel transform of the functional .In this paper the following theorem is proved:Theorem. Let be a bounded convex domain in with boundary of class or , where the are bounded planar convex domains with boundaries of class , and let . In order that it is necessary and sufficient that 1) for all , and 2) be a function of completely regular growth in in the sense of weak convergence in .Here is the regularized radial indicator of the entire function , and is the support function of the compact set .Bibliography: 29 titles.
TL;DR: In this paper, the problem of generalizing Nevanlinna's theory of meromorphic functions has been studied and an elementary and simple method for generalizing Theorem A has been proposed.
Abstract: The method he used in the proof of Theorem A is very complicated and not suitable in generalizing Theorem A. The purpose of this paper is to decide permutable functions of a class of entire functions satisfying certain differential equation and to indicate an elementary and simple method by which we can generalize Theorem A. We assume that the reader is familiar with the fundamental concepts in Nevanlinna's theory of meromorphic functions, in particular, with symbols m{r, /) , T(r, /) and M(r, f) etc. (see [4]).
01 Apr 1988
TL;DR: In this paper, a demonstration de l'independance lineaire de quelques iteres a l'aide d'un resultat classique de Polya concernant la croissance des caracteristiques de Nevanlinna des fronctions composees.
Abstract: Demonstration de l'independance lineaire de quelques iteres a l'aide d'un resultat classique de Polya concernant la croissance des caracteristiques de Nevanlinna des fronctions composees. Etablissement de la non-existence de solutions entieres de l'equation fonctionnelle de Feigenbaum
TL;DR: The Segal-Bargmann space H2(μ) is the space of all entire functions on ℂn that belong to L 2 (μ)-the usual space of Gaussian square-integrable functions as discussed by the authors.
Abstract: Let μ be the Gaussian measure in ℂn given by dμ(z)=(2π)−n exp(−|z|2/2)dV, where dV is the ordinary Lebesgue measure in ℂn. The Segal-Bargmann space H2(μ) is the space of all entire functions on ℂn that belong to L2(μ)-the usual space of Gaussian square-integrable functions. Let P be the orthogonal projection from L2(μ) onto H2(μ). For a measurable function ϕ on ℂn, the multiplication operator Mϕ on L2(μ) is defined by Mϕh =ϕh. The Toeplitz operator Tϕ is defined on H2(μ)by
TL;DR: In this paper, it was proved that for any entire function / of finite nonzero order there is a set S in the plane with density zero and such that almost all the roots of the equation /(ζ) = α belong to S. This conjecture was proved here in a weakened form.
Abstract: It is proved that for any entire function / of finite nonzero order there is a set S in the plane with density zero and such that for any a € C almost all the roots of the equation /(ζ) = α belong to S. This assertion was deduced by Littlewood from an unproved conjecture about an estimate of the spherical derivative of a polynomial. This conjecture is proved here in a weakened form. Bibliography: 11 titles.
TL;DR: In this paper, the continuous Gegenbauer transform is extended to noninteger values of the parameter $2\lambda $ and the transformed function is no longer an entire function but is always holomorphic in a half plane.
Abstract: The continuous Gegenbauer transform is extended to noninteger values of the parameter $2\lambda $. The transformed function is no longer an entire function but is always holomorphic in a half plane. The inverse transform is found for these parameter values. Its kernel is given both by a series and in closed form by an integral.
TL;DR: In this article, a unique analytic generating function pertaining to the two-peaked spectrum characterizing the polydisperse medium was derived based on a formalism given previously, and an example is given to study the spectral features of the poly disperse medium via time interval statistics.
01 Jan 1988
Book•
01 Jan 1988
TL;DR: In this paper, the authors discuss the almost best approximation in the complex plane of the Schwarz-Christoffel integrals and the convergence of these integrals to the Schwarzian derivatives.
Abstract: 1) Cross-ratios and Schwarzian derivatives in Rn.- 2) Remarks on "almost best" approximation in the complex plane.- 3) Conformal mappings onto nonoverlapping regions.- 4) On Wiener conditions for minimally thin and rarefied sets.- 5) The matrix and chordal norms of Mobius transformations.- 6) On meromorphic functions with growth conditions.- 7) A theorem of Wolff-Denjoy type.- 8) Curvature estimates for some minimal surfaces.- 9) On some elementary applications of the reflection principle to Schwarz-Christoffel integrals.- 10) Konforme Verheftung und logarithmisches Potential.- 11) On boundary correspondence for domains on the sphere.- 12) On circulants.- 13) Interpolation by entire functions in ? - another look.- 14) Moglichst konforme Spiegelung an einem Jordanbogen auf der Zahlenkugel.- 15) On BMO and the torsion function.- 16) Subharmonic majorants and some applications.- 17) On weighted extremal length of families of curves.- 18) On approximation by rational functions of class L1.- 19) On fixed points of conformal automorphisms of Riemann surfaces.- 20) The variation of harmonic differentials and their periods.- 21) On the extremality and unique extremality of certain Teichmuller mappings.- 22) Angular distribution of meromorphic functions in the unit disk.
TL;DR: In this article, an integral representation of the periodic function Gyj(a) is given for the general case ρ = O( |JC|~), for some ρ > 0.
Abstract: The function Gyj is a periodic function of a, with period 2. For / = 0 (the general case is obtained by translation) the righthand member of (1) is 2rGyj(l). In the following paper we suppose that / satisfies an additional hypothesis of the form/(x) = O ( |JC|~), for some e > 0, as x —» zboo and we give an integral representation of Gy j(a) which is valid for 0 ^ a ^ 2. More precisely, the Theorems 1, 2 and 3, below, contain formulas giving a representation of Gyj(a) valid respectively for 0 ^ a ê 1, 1 ^ a ^ 3/2 and 3/2 â a ^ 2. Before we examine the method of proof we state explicitly the results in question.
TL;DR: In this article, two particular cases of boundary conditions that are not entire functions and that lead to a relativistic solution of Young's experiment are discussed, and they are discussed in detail.
Abstract: In homogeneous free space the electromagnetic field may be represented by a second rank spinor, each component of which is a solution of the wave equation. This makes it possible to solve the boundary value problem for the electromagnetic field when the data given on the hyperplane z−ct=0 are entire functions. Two particular cases of boundary conditions that are not entire functions and that lead to a relativistic solution of Young’s experiment are discussed.
TL;DR: Arakelyan as discussed by the authors showed that for every countable set A c 9 and every p > i/2 there exists an entire function of order p whose set of defect values contains A [4].
Abstract: It is known that the set of defect values of a function meromorphic in the finite plane 9 is at most countable [1-3]. Arakelyan obtained the following result: for every countable set A c 9 and every p > i/2 there exists an entire function of order p whose set of defect values contains A [4-6]. This theorem of Arakelyan disproved a conjecture of R. Nevanlinna. One is naturally led to asking whether there is an entire function of finite order whose set of defect values coincides with an arbitrarily prescribed countable set.