Showing papers on "Entire function published in 1979"
TL;DR: The existence of solutions of the inhomogeneous Cauchy-Riemann equations as a powerful tool in the study of analytic functions of several complex variables is well demonstrated in this article.
129 citations
TL;DR: In this paper, it was shown that the most general integrable evolution equations in one spatial dimension which is solvable using the inverse scattering transform (i. s. t.) associated with the n th order eigenvalue problem V x = ( ξR 0 + P ( x, t )) V has the simple and elegant form G ( D R, t ) P t + F ( DR, t ) x [ R 0, P ] = Ω( D R, t ] [ C, P ], where G, F and Ω are entire functions
Abstract: This paper presents some new results in connection with the structure of integrable evolution equations. It is found that the most general integrable evolution equations in one spatial dimension which is solvable using the inverse scattering transform (i. s. t.) associated with the n th order eigenvalue problem V x = ( ξR 0 + P ( x , t )) V has the simple and elegant form G ( D R , t ) P t – F ( D R , t ) x [ R 0 , P ] = Ω ( D R , t ) [ C , P ], where G , F and Ω are entire functions of an integro-differential operatos D R and the bracket refers to the commutator. The list provided by this form is not exhaustive but contains most of the known integrable equations and many new ones of both mathematical and physical significance. The simple structure allows the identification in a straightforward manner of the equation in this class which is closest to a given equation of interest. The x dependent coefficients enable the inclusion of the effects of field gradients. Furthermore when the partial derivative with respect to t is zero, the remaining equation class contains many nonlinear ordinary differential equation of importance, such as the Painleve equations of the second and third kind. The properties of the scattering matrix A( ξ , t ) corresponding to the potential P( x , t ) are investigated and in particular the time evolution of A ( ξ , t ) is found to be G ( ξ , t ) A t + F ( ξ , t ) A ξ = Ω ( ξ , t )[ C , A ], The role of the diagonal entries and the principal corner minors in providing the Hamiltonian structure and constants of the motion is discussed. The central role that certain quadratic products of the eigenfunctions play in the theory is briefly described and the necessary groundwork from a singular perturbation theory is given when n = 2 or 3.
118 citations
TL;DR: In this paper, a complex Banach space is characterized in terms of compact holomorphic mappings and equicontinuity conditions on the spaces of n-homogeneous Taylor polynomial coefficients.
Abstract: Let E be an infinite dimensional complex Banach space. Let Hwu(e) be the space of entire complexvalued functions on E which are weakly uniformly continuous when restricted to any bounded subset of E, and let Hwsc(E) be the space of entire complex-valued functions on E which map weakly convergent sequences in E to convergent sequences. These and intermediate spaces are characterized in terms of compact holomorphic mappings and in terms of equicontinuity conditions on the spaces of n-homogeneous Taylor polynomial coefficients.
33 citations
TL;DR: In this article, a theorem on the asymptotic behavior of meromorphic functions of completely regular growth (as previously defined by the author) as outside a set of zero linear density is proved.
Abstract: A theorem is proved on the asymptotic behavior of meromorphic functions of completely regular growth (as previously defined by the author) as outside a set of zero linear density.For entire functions of completely regular growth a uniformity property is established, and some of its applications are presented. An upper bound for the number of deficient values (in the sense of R. Nevanlinna) of such functions is also obtained.Bibliography: 11 titles.
19 citations
16 citations
TL;DR: In this paper, translation-invariant locally-convex algebras of entire functions on a connected, simply-connected complex nilpotent Lie group were constructed.
14 citations
TL;DR: In this paper, the authors considered the problem of estimating the value of g(x 0) of a spectral density function and showed that the risk has an asymptotic lower bound similar to those previously obtained by Farrell [6] for density function estimation.
Abstract: Discrete parameter stationary processes with joint Gaussian distributions are considered. Loss is measured by squared error. It is shown that when estimating the value of g(x
0) of a spectral density function g the risk has an asymptotic lower bound similar to those previously obtained by Farrell [6] for the problem of density function estimation. A similar result is obtained for estimates of an entire function when risk is measured by integrated square error.
13 citations
TL;DR: The analytic nature of the scattered field in the Fresnel region is explored in this paper, where it is shown that in the near field, as well as in the far field, the zeros encode the information about the object wave.
Abstract: The analytic nature of the scattered field in the Fresnel region is explored. It is found that in the near field, as well as in the far field, the zeros encode the information about the object wave. The scattered field is essentially an entire function of exponential type, although in certain circumstances a multiplicative factor may impose a higher-order envelope.
12 citations
TL;DR: In this article, the authors studied trace formulas for a class of operators of the form ΠT-GΠT in which G designates multiplication by a suitable restricted d × d matrix valued function G(γ) of γ ϵ R1 and T stands for the diagonal d ×d matrix (δijPT) of orthogonal projections PT of L2(R1, dδ) onto the space IT(dδ), of entire functions of exponential type ⩽T which are square summable on the line relative to the
10 citations
TL;DR: In this paper, the Paley-Wiener-Schwartz theorem in infinite dimensions was shown to hold for a complex valued holomorphic function of exponential type on (E') c and slowly increasing on E' (when E belongs to a wide class of separable Banach spaces).
Abstract: The Paley-Wiener-Schwartz theorem characterizes the Fourier transforms of distributions with bounded (compact) support as being exactly the entire functions of exponential type which are slowly increasing (cf.[4], [18], [20], [2l]). Nachbin and Dineen [9] defined the Frechet space ɛNbc (E;F) of infinitely nuclearly differentiable mappings of boundedcompact type from E valued in F, when E is a real Banach space and F is any Banach space (§1). When E is finite dimensional and F = C, the space ɛNbc (E;C) = = ɛNbc (E) is the space ɛ(E) endowed with the Schwartz topology [20]. For this reason and on account of theorem 3, ɛ'Nbc (E), the dual space to ɛNbc (E), is called the space of distribution with bounded support in infinite dimensions. In contrast with the finite dimensional case, if E is infinite dimensional, then there exist complex valued holomorphic functions of exponential type on (E')c, bounded on E' (and hence slowly increasing) which are not the Fourier transform of any distributions with bounded support (cf. [9]). Here I establish, as a main result of this work, a necessary and sufficient condition for a complex valued holomorphic function of exponential type on (E') c and slowly increasing on E' (when E belongs to a wide class of separable Banach spaces) to be the Fourier transform of a distribution with bounded support: the Paley-Wiener-Schwartz theorem in infinite dimensions.
10 citations
TL;DR: In this paper, it was shown that the classical scattering operator S is analytic in the manifold D defined by the inequality |λ||ϕin|2Y<ƞ] and that S cannot be an entire function in the coupling constant and initial data.
01 Jan 1979
TL;DR: In this paper, the authors studied simultaneous interpolation and approximation of a function f on the whole real line by entire functions of exponential type, where the function f is supposed to be uniformly continuous and bounded on (-∞,∞).
Abstract: Given the values of a function and possibly the values of some of its derivatives, at certain points, a practical problem of numerical analysis is to use this information to construct other functions which approximate it. Simultaneous interpolation and approximation of continuous functions on a compact interval, by polynomials, has been extensively studied by Runge, Bernstein, Faber, Fejer, Turan and others. Here we study simultaneous interpolation and approximation of a function f on the whole real line by entire functions of exponential type. The function f is supposed to be uniformly continuous and bounded on (-∞,∞).
TL;DR: In this paper, a general method of Wiman-Valiron type for dealing with entire functions of finite lower growth is presented and used to obtain the lower-order version of a result of W. K. Hayman on the real part of a function of small lower growth.
Abstract: A general method of Wiman-Valiron type for dealing with entire functions of finite lower growth is presented and used to obtain the lower-order version of a result of W. K. Hayman on the real part of entire functions of small lower growth.
01 Jan 1979
TL;DR: In this paper, a function of finite lower order μ is shown to be convergent in the series (δ(a, f~((j))} is convergent.
Abstract: 1. Let f(z) be an entire function of finite lower order μ. If(?)δ(a,f~((j)))=1, then (?)p_j≤μ, where P_j denotes the number of finite and non-zero deficient values of f~((j))(z). More over, every deficient value of f~((j))(z) (j=0, ±1, ±2, …; f~((0))≡f) is an asymptotic value of f~((j))(z) and every deficiency is a multiple of 1/μ.2. If f(z) is an entire function of finite lower order μ, then the series (?){δ(a, f~((j))} is convergent.
TL;DR: In this article, the authors give a minimum modulus theorem which enables them to prove the invertibility of a large class of ultradifferential operators in the ultradistributions space.
Abstract: In this work we give a minimum modulus theorem which enables us to prove the invertibility of a large class of ultradifferential operators. It is known that the invertibility of convolution operators defined by ultradistributions S with compact support is equivalent to the existence of a certain lower estimation for the modulus of the Fourier transform of S (see [1], [3], [8], [9]). While usual differential operators with constant coefficients are all invertible even in the space of Schwartz's distributions, the following problem is still open: Is every ultradifferential operator invertible in the corresponding ultradistributions space or at least in the \"union\" of all ultradistributions? In [2] Ch. Ch. Chou positively solved this problem for elliptic ultradifferential operators. For the general case some results are given by the same author in [1]; unfortunately, the invertibility is proved under very restrictive conditions on the considered ultradistributions space. The aim of this work is to give a general minimum modulus theorem, improving the well-known theorem of L. Ehrenpreis [7] and which yields to an invertibility result in ultradistributions spaces satisfying less restrictive conditions then those of Ch. Ch. Chou. In particular we prove that all ultradifferential operators of class {k! (//~=2 In j)~} with ~ ~ 1, are invertible, while Chou's result works only for ~ >2.
TL;DR: In this paper, the authors present criteria for the solvability of the general moment problem in countably Hilbert spaces and the relationship between the general interpolation problem in reflexive spaces and corresponding moment problem is also established.
Abstract: The article contains criteria for the solvability of the general moment problem in countably Hilbert spaces. The relationship between the general interpolation problem in reflexive spaces and the corresponding moment problem is also established. The results lead on the one hand to criteria for solvability of the general interpolation problem (and, in particular, of the Hermite problem) in various function spaces, and, on the other hand, to necessary and sufficient conditions for a system , where is an entire function, to form a basis in its linear hull.Bibliography: 27 titles.
TL;DR: In this article, a series is assigned (according to a specific rule) to an arbitrary entire function of order and necessary and sufficient conditions on are found under which this series always converges to in some topology.
Abstract: Let be an entire function of order and an entire function of order with simple zeros . A series is assigned (according to a specific rule) to an arbitrary entire function of order . Necessary and sufficient conditions on are found under which this series always converges to in some topology.Bibliography: 5 titles.
TL;DR: In this paper, Treves et al. showed that the strong duals (H(CN),τ)′ are spaces of analytic functions on the countable product of complex lines.
TL;DR: In this article, it was shown that the Volterra integral equation of convolution type w − w⊗g = f has a continuous solution w when f, g are continuous functions on Rx and ⊗ denotes a truncated convolution product.
Abstract: In this article, it is shown that the Volterra integral equation of convolution type w − w⊗g = f has a continuous solution w when f, g are continuous functions on Rx and ⊗ denotes a truncated convolution product. A similar result holds when f, g are entire functions of several complex variables. Also simple proofs are given to show when f, g are entire, f⊗g is entire, and, if f⊗g=0, then f = 0 or g = 0. Finally, the set of exponential polynomials and the set of all solutions to linear partial differential equations are considered in relation to this convolution product.
01 Jan 1979
TL;DR: A survey of known results on functions of bounded index was recently given by S. M. Shah [4] as mentioned in this paper, where the sinus function is a function of bounded indices and the famous Euler-formula e = cos z + i sin z.
Abstract: A survey of known results on functions of bounded index was recently given by S. M. Shah [4]. Let us now look at the sinus-function a function of bounded index iz 1 and the famous Euler-formula e = cos z + i sin z. If we set f :-sin z , it can be expressed as follows: There are entire functions gl' go of exponential type such that 1 = gl f' + g0 f. This result is a special case of the following theorem.
TL;DR: In this article, the authors used Cauchy's integral formula and the Bore1 transformation to derive the integral representation of Tricomi's equation, and then they applied it to the theory of transonic gas dynamics.
TL;DR: In this article, it was shown that the restriction to R of an entire function of exponential type < s is the restriction of a cardinal Hermite spline of degree n having knots of multiplicity s at the integers.
Abstract: Let Sns denote the class of cardinal Hermite splines of degree n having knots of multiplicity s at the integers. In this paper we show that if fn -f uniformly on R, where fn E i,s in oo as n oo, andf is bounded, then f is the restriction to R of an entire function of exponential type < s. In proving this result, we need to derive some extremal properties of certain splines n(n'11.
01 Jan 1979
TL;DR: In this article, the authors prove existence of an infinite product representation of an entire function with zero having positive deficiency to meet the requirements of lower order, finite lower order and having a finite deficient value.
Abstract: A. A. Goldberg in "The Possible Magnitude of the Lower Order of an Entire Function with a Finite Deficient Value" [4] poses the question of existence of entire functions of infinite order, finite lower order, and having a finite deficient value. The answer to both questions is affirmative. We prove existence by constructing an explicit infinite product representation of an entire function with zero having positive deficiency to meet the requirements. Our methods include generalizing a result of B. Ja. Levin concerning particular entire functions with zeros evenly distributed on two rays. Next we exhibit a polynomial substitute for the exponential convergence factor which appears in the standard Weierstrass primary factor. Then we partition the complex plane into annular regions which are appropriate for our purposes of interpolating through a family of entire functions. Finally we take a comparison function by D. Drasin and generalize it to obtain a counting function for zeros which we use to construct our entire function.