Holomorphic Continuation Research Articles

Holomorphic Continuation of Functions along a Fixed Direction (Survey)

Holomorphic Continuation of Functions along a Fixed Direction (Survey)

On some sets sufficient for holomorphic continuation of functions with generalized boundary Morera property

This article considers continuous functions defined on the boundary of a bounded domain $D$ in $\mathbb C^n$, $n>1$, and having a generalized boundary Morera property. The question of the existence of a holomorphic continuation of such functions into the domain $D$ for some sufficient sets $\Gamma$ of complex lines intersecting the germ of the generating manifold lying inside the domain is investigated.

Holomorphic Continuation of Functions Along a Fixed Direction (Survey)

In this article, we give an overview of the most significant and important results on holomorphic extensions of functions along a fixed direction. We discuss the following geometric questions of multidimensional complex analysis: holomorphic extension along a bundle of complex straight line, the Forelly theorem; holomorphic continuation of functions with thin singularities along a fixed direction; holomorphic continuation of functions along a family of analytic curves.

Holomorphic continuation into a matrix ball of functions defined on a piece of its skeleton

The question of the possibility of holomorphic continuation into some domain of functions defined on the entire boundary of this domain has been well studied. The problem of describing functions defined on a part of the boundary that can be extended holomorphically into a fixed domain is attracting more interest. In this article, we reformulate the problem under consideration: Under what conditions can we extend holomorphically to a matrix ball the functions given on a part of its skeleton? We describe the domains into which the integral of the Bochner—Hua Luogeng type for a matrix ball can be extended holomorphically. As the main result, we present the criterion of holomorphic continuation into a matrix ball of functions defined on a part of the skeleton of this matrix ball. The proofs of several results are briefly presented. Some recent advances are highlighted. The results obtained in this article generalize the results of L.A. Aizenberg, A.M. Kytmanov and G. Khudayberganov.

Standard model O(α) renormalization of gA and its impact on new physics searches

We present an $\mathcal{O}(\alpha)$ Standard Model calculation of the inner radiative corrections to Gamow-Teller $\beta$ decays. We find that \textit{a priori} contributions arise from the photonic vertex correction and $\gamma W$ box diagram. Upon evaluation most elastic contributions vanish due to crossing symmetry or cancellation between isoscalar and isovector photonic contributions, leaving only the polarized parity-odd contribution, i.e., the Gamow-Teller equivalent of the well-known axial $\gamma W$ box contribution for Fermi decays. We show that weak magnetism contributes significantly to the Born amplitude, and consider additional hadronic contributions at low energy using a holomorphic continuation of the polarized Bjorken sum rule constrained by experimental data. We perform the same procedure for the Fermi inner radiative correction through a combination of the running of Bjorken and Gross-Llewellyn Smith sum rules. We discuss heavy flavor, higher-twist, and target mass corrections and find a significant increase at low momentum from the latter. We find $\Delta_R^A = 0.02532(22)$ and $\Delta_R^V = 0.02473(27)$ for axial and vector inner radiative corrections, respectively, resulting in $\Delta_R^A-\Delta_R^V=0.60(5) \times 10^{-3}$, which allows us to extract $g_A^0$ for the first time to our knowledge. We discuss consequences for comparing experimental data to lattice calculations in beyond Standard Model fits. Further, we show how some traditional $\beta$ decay calculations contain part of this effect but fail to account for cancellations in the full $\mathcal{O}(\alpha)$result. Finally, we correct for a double-counting instance in the isospin $T=1/2$ mirror decay extraction of $|V_{ud}|$, the up-down matrix element of the Cabibbo-Kobayashi-Maskawa matrix, resolving a long-standing tension and leading to increased precision.

Holomorphic continuation of a formal series along analytic curves

ABSTRACT The work is devoted to the holomorphy of a formal homogeneous series whose restrictions to analytic curves passing through 0 are holomorphic. In content, the main result of the paper is a variation of the well-known Forelli theorem on the holomorphic continuation of formal series along with a set of complex lines.

Boundary Behavior of Functions Representable by Weighted Koppelman Type Integral and Related Hartogs Phenomenon

In the present paper we study the boundary behavior of a weighted Koppelman type integral with a specific choice of weight for a function $$\phi $$ that is integrable on a bounded domain $$D\subset \mathbb {C}^n$$ and is continuous on its $$\mathcal {C}^1$$-boundary. Applying the above results, we derive a variation of Hartogs phenomena about the holomorphicity of a function $$\phi $$ which is integrable in a D and continuous on $$\partial D$$, provided that it satisfies, in some sense, a stronger version of “one-dimensional holomorphic continuation property” along any complex line meeting the domain.

On family of complex straight lines sufficient for existence of holomorphic continuation of continuous functions on boundary of domain

On family of complex straight lines sufficient for existence of holomorphic continuation of continuous functions on boundary of domain

Rapid polynomial approximation on Stein manifolds

We generalize to a certain class of Stein manifolds the Bernstein–Walsh–Siciak theorem which describes the equivalence between possible holomorphic continuation of a function $f$ defined on a compact set $K$ in $\mathbb {C}^N$ and the rapidity of the best

On Holomorphic Continuation Integrable Functions of Along Finite Families of Complex Lines in n-circular Domain

On Holomorphic Continuation Integrable Functions of Along Finite Families of Complex Lines in n-circular Domain

Holomorphic Extension of Continuous Functions Along Finite Families of Complex Lines in a Ball

This paper presents some results related to the holomorphic extension of functions f , defined on the boundary of a ball B ⊂ C, n > 1, in this ball. We deal with a functions with the one-dimensional holomorphic extension property along the complex lines. The first result related to our subject was received by M.L.Agranovsky and R.E.Valsky in [1], who studied the functions with a one-dimensional holomorphic continuation property on the boundary of a ball. The proof was based on the automorphism group properties of a sphere. E.L.Stout in [2] used complex Radon transformation to generalize the Agranovsky and Valsky theorem for an arbitrary bounded domain with a smooth boundary. An alternative proof of the Stout theorem was obtained by A.M.Kytmanov in [3] by applying the Bochner–Martinelli integral. The idea of using the integral representations (Bochner–Martinelli, Cauchy–Fantappie, multidimensional logarithmic residue) has been useful in the study the functions with onedimensional holomorphic continuation property (see review [4]). The problem of finding the different families of complex lines, sufficient for holomorphic extension was posed in [5]. Clearly, the family of complex lines passing through one point is not enough. As shown in [6], the family of complex lines passing through a finite number of points also, generally speaking, is not sufficient. In [6] we proved that the family of complex lines crossing the germ of generic manifold γ, is sufficient for the holomorphic extension. In [7] we consider a family of complex lines passing through the germ of a complex hypersurface. In particular, C, it can be any real analytic curve. Various other families are given in [8–11]. We note here the work [9, 11], where it is shown that a family of complex lines passing through somehow located a finite number of points is sufficient for holomorphic extension. But it is approved only for real-analytic or infinitely differentiable functions defined on the boundary. Then in C Agranovsky and Globevnik for real-analytic

Interfacial line inclusion between two dissimilar thermo-electro-magneto-elastic solids

In this study, a rigid line inclusion on the interface between two dissimilar thermo-electro-magneto-elastic solids is considered. Based on the Muskhelishvili’s theory and extended Stroh formalism, it is reduced to the non-homogeneous Hilbert equation. According to the holomorphic continuation technology and Muskhelishvili’s theory, the exact solutions for both thermal and electro-magneto-elastic fields are derived. The explicit expressions of the multiple coupling fields are presented, and the generalized stress intensity factors are illustrated with the closed form. From the results, it can be observed the coupling of the thermo-electro-magneto-elastic effects. Moreover, the generalized stress has singularities and oscillatory properties near the inclusion tip.

On the GL(r)×GL(r+s)×GL(s) convolution

Abstract In this paper, we show a new kind of Rankin–Selberg functions, and we prove the functional equation and holomorphic continuation. As an application, we give a new Voronoi formula.

On the rapid decay of cuspidal automorphic forms

Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets, and furthermore state and prove the decay for smooth but not necessarily K-finite cusp forms. We also state a general theorem about the convergence of Rankin–Selberg integrals involving unipotent periods, closing a gap in the literature on L-functions. These properties serve as the analytic basis (Miller and Schmid, 2009 [15]) of a new method to establish holomorphic continuations of Langlands L-functions, in particular the exterior square L-functions (Miller and Schmid, 2009 [16]) on GL(n).

Space–time analyticity of weak solutions to linear parabolic systems with variable coefficients

Analytic smoothing properties of a general, strongly parabolic linear Cauchy problem of second order in RN×(0,T) with analytic coefficients (in space and time variables) are investigated. They are expressed in terms of holomorphic continuation of global (weak) L2-type solutions to the system. Given 0<T′<T⩽∞, it is proved that any L2-type solution u:RN×(0,T)→RM possesses a bounded holomorphic continuation u(x+iy,σ+iτ) into a complex domain in CN×C defined by (x,σ)∈RN×(T′,T), |y|<A′ and |τ|<B′, where A′,B′>0 are constants depending upon T′. The proof uses the extension of a solution to an L2-type solution in a domain in CN×C, such that this extension satisfies the Cauchy–Riemann equations. The holomorphic extension is thus obtained in a Hardy space H2. Applications include market completion by European options in Finance.

Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors

In this article, we study local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.

Holomorphic continuation of functions from closed hypersurfaces with singular edges

Consider a bounded domain D in ℂn such that its boundary contains a finite set of singular edges. We study the boundary behavior of the Bochner-Martinelli integral over such domains. As an application, we obtain generalizations of Hartogs-Bochner and Aizenberg-Kytmanov theorems on the holomorphic continuation of functions from the boundary into the domain.

Holomorphic Hermite polynomials and a non-commutative plane

One more coherent state quantization of a complex plane is presented. Although the complex plane is equipped with a non-rotationally invariant measure, we still obtain a canonical commutation rule (up to a simple rescaling). We explain how the involved coherent states, built from holomorphic continuations of Hermite polynomials, are related to the non-commutative plane.

Bessel models for general admissible induced representations: the compact stabilizer case

A holomorphic continuation of Jacquet type integrals for parabolic subgroups with abelian nilradical is studied. Complete results are given for generic characters with compact stabilizer and arbitrary representations induced from admissible representations. A description of all pertinent examples is given. These results give a complete description of the Bessel models corresponding to compact stabilizer.

Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula

Abstract We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions fromwhich some generalizations of the classical sum formula can be deduced.