scholarly journals Asymptotic Power Series of Field Correlators

10.14311/1199 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
I. Caprini ◽  
J. Fischer ◽  
I. Vrkoč
Keyword(s):  
Power Series ◽  
Large Class ◽  
Complex Plane ◽  
Perturbation Expansion ◽  
Asymptotic Power ◽  
General Contour ◽  
Asymptotic Perturbation ◽  
Class Of Functions ◽  
Borel Type

We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified form of theWatson lemma recently proved elsewhere, we discuss a large class of functions determined by the same asymptotic power expansion and represented by various forms of integrals of the Laplace-Borel type along a general contour in the Borel complex plane. Some remarks on possible applications in QCD are made.

scholarly journals Some Algebraic Properties Of Asymptotic Power Series

1956 ◽  
Vol 8 ◽  
pp. 220-224
Author(s):  
T. E. Hull
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Power ◽  
Algebraic Properties ◽  
Image Position ◽  
The Right ◽  
The Given ◽  
Class Of Functions

1. Introduction. Let us consider all power series of the formIt was shown first by Borel (1) that to each such series there corresponds a non-empty class of functions such that each function in the class has the given series as its asymptotic expansion about z = 0, the expansion being valid in a sector of the right half z-plane with vertex at the origin.

A Short Proof of an Interpolation Theorem

1974 ◽  
Vol 17 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Simple Proof ◽  
Real Line ◽  
Short Proof ◽  
Interpolation Theorem ◽  
Operator Interpolation ◽  
Upper Half Plane ◽  
Positive Functions ◽  
Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

Partial Hölder continuity of minimizers of functionals satisfying a VMO condition

2017 ◽  
Vol 10 (1) ◽  
pp. 83-110 ◽  
Author(s):  
Christopher S. Goodrich
Keyword(s):  
Large Class ◽  
Partial Regularity ◽  
Hölder Continuity ◽  
Full Measure ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Open Set ◽  
Principal Assumption ◽  
Class Of Functions

AbstractFor a bounded, open set${\Omega\hskip-0.569055pt\subseteq\hskip-0.569055pt\mathbb{R}^{n}}$we consider the partial regularity of vectorial minimizers${u\hskip-0.853583pt:\hskip-0.853583pt\Omega\hskip-0.853583pt\rightarrow\hskip-% 0.853583pt\mathbb{R}^{N}}$of the functional$u\mapsto\int_{\Omega}f(x,u,Du)\,dx,$where${f:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times n}\rightarrow\mathbb{R}}$. The principal assumption we make is thatfis asymptotically related to a function of the form${(x,u,\xi)\mapsto a(x,u)F(\xi)}$, whereFpossessesp-Uhlenbeck structure and the partial maps${x\mapsto a(x,\cdot\,)}$and${u\mapsto a(\,\cdot\,,u)}$are, respectively, of class VMO and${\mathcal{C}^{0}}$. We demonstrate that any minimizer${u\in W^{1,p}(\Omega)}$of this functional is Hölder continuous on an open set${\Omega_{0}}$of full measure. Finally, we show by means of an example that our asymptotic relatedness condition is very general and permits a large class of functions.

scholarly journals Polynomially bounded multisequences and analytic continuation

1982 ◽  
Vol 23 (1) ◽  
pp. 41-52
Author(s):  
Daniel J. Troy
Keyword(s):  
Power Series ◽  
Analytic Continuation ◽  
Linear Functional ◽  
Necessary And Sufficient Condition ◽  
Convergence Criteria ◽  
Dimensional Torus ◽  
Unit Factor ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Class Of Functions

Given a polynomially bounded multisequence {fm}, where m = (m1, …, mk) ∈ ℤk, we will consider 2k power series in exp(iz1), …, exp(izk), each representing a holomorphic function within its domain of convergence. We will consider this same multisequence as a linear functional on a class of functions defined on the k-dimensional torus by a Fourier series, , with the proper convergence criteria. We shall discuss the relationships that exist between the linear functional properties of the multisequence and the analytic continuation of the holomorphic functions. With this approach we show that a necessary and sufficient condition that the multisequence be given by a polynomial is that each of the power series represents, up to a unit factor, the same function that is entire in the variables

Inverse Problems for Partition Functions

2001 ◽  
Vol 53 (4) ◽  
pp. 866-896
Author(s):  
Yifan Yang
Keyword(s):  
Asymptotic Behavior ◽  
Partition Function ◽  
Inverse Problems ◽  
Large Class ◽  
Generating Function ◽  
Riemann Hypothesis ◽  
Arithmetic Function ◽  
Partition Functions ◽  
Class Of Functions ◽  
Weighted Partition

AbstractLet pw(n) be the weighted partition function defined by the generating function , where w(m) is a non-negative arithmetic function. Let be the summatory functions for pw(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Φ(u) and λ(u), an estimate for Pw(u) of the formlog Pw(u) = Φ(u){1 + Ou(1/λ(u))} (u→∞) implies an estimate forNw(u) of the formNw(u) = Φ*(u){1+O(1/ log ƛ(u))} (u→∞) with a suitable function Φ*(u) defined in terms of Φ(u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

The Wavefield Radiated Into an Elastic Half-Space by a Transducer of Large Aperture

1988 ◽  
Vol 55 (2) ◽  
pp. 398-404 ◽  
Author(s):  
John G. Harris
Keyword(s):  
Power Series ◽  
Plane Wave ◽  
Half Space ◽  
Wave Equations ◽  
Ultrasonic Transducer ◽  
Elastic Half Space ◽  
The Other ◽  
Circular Region ◽  
Asymptotic Power ◽  
Normal Traction

The wavefield radiated into an elastic half-space by an ultrasonic transducer, as well as the radiation admittance of the transducer coupled to the half-space, are studied. Two models for the transducer are used. In one an axisymmetric, Gaussian distribution of normal traction is imposed upon the surface, while in the other a uniform distribution of normal traction is imposed upon a circular region of the surface, leaving the remainder free of traction. To calculate the wavefield, each wave emitted by the transducer is expressed as a plane wave multiplied by an asymptotic power series in inverse powers of the aperture’s (scaled) radius. This reduces the wave equations satisfied by the compressional and shear potentials to their parabolic approximations. The approximations to the radiated waves are accurate at a depth where the wavefield remains well collimated.

Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

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