Distributions and analytic continuation of Dirichlet series

This is the second in a series of three papers; the other two are “Summation Formulas, from Poisson and Voronoi to the Present” [Progr. Math. 220 (2004) 419-440] and “Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)” (preprint). The first paper is primarily expository, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on . The present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3)×GL(1) converse theorem given in the third paper. The primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold. Applications are given to the analytic continuation of Riemann's zeta function, degree 1 and degree 2 L-functions, the converse theorem for GL(2), and a characterization of the classical Mellin transform/inversion relations on functions with specified singularities.     

Theorem concerning analytic continuation

A. I. Markushevich obtained the following representation of a function in its holomorphicity star with a sequence {m _v }, for which m _v+1/m _v: Here it is proved that this condition is necessary; more precisely, . This result is derived from certain properties of over-convergent power series.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 57–62, July, 1971.The author wishes to thank his scientific director A. I. Markushevich for his advice concerning this work.     

A remark on analytic continuation

A simple example is given to show that the space of germs obtained by analytic continuation of a given germ need not be a covering space in the topological sense.

     

On the continuation of analytic sets

Mathematische Annalen -     

The analytic continuation of hyperbolic space

We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction inspires a new concrete insight for the study of the hyperbolic geometry and Lorentzian geometry as a unified object. We also discuss the advantages of this new geometric model as well as some of its applications.     

Numerical methods for analytic continuation and mesh generation

The problem of computing the analytic continuation of a holomorphic function known on a circle is considered. Several fast numerical schemes based on solving an initial-value problem for the Cauchy-Riemann equations are analyzed. To avoid instability problems, some of the schemes consist of two parts: one for integrating the Cauchy-Riemann equations, and one for smoothing the function values so obtained. We show that with appropriate integration and smoothing methods, the stability and accuracy of such schemes is sufficient for many applications. The schemes are well-suited for generating level curves and stream lines of conformal mappings. Computed examples are presented. We also indicate how the schemes can be used to generate near-orthogonal boundary-fitted grids with given mesh sizes along the boundary.     

Unique continuation for operatorswith partially analytic coefficients

A general unique continuation result for partial differential operators with partially analytic coefficients was obtained in Tataru (1995); however, certain technical assumptions were used there. A part of these assumptions were elliminated independently by Hörmander (1985), and Robbiano and Zuily (preprint). The aim of this note is to remove the remaining technical restriction and, at the same time, to provide a simple proof for the entire result.     

Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

On the analytic continuation of the Poisson kernel

We give a “heat equation” proof of a theorem which says that for all ε sufficiently small, the map ${S_{\epsilon} \colon f \mapsto \exp(- \epsilon \sqrt{\Delta})f}$ extends to an isomorphism from H ^s (X) to ${\mathcal{O}^{s + (n-1)/4}(\partial M_{\epsilon})}$ . This result was announced by Boutet de Monvel (C R Acad Sci Paris Sér A-B 287(13):A855–A856, 1978) but only recently has a proof, due to Zelditch (Spectral geometry, volume 84 of proceedings of the symposium in pure mathematics, pp 299–339. American Mathematical Society, Providence, RI, 2012), appeared in the literature. The main tools in our proof are the subordination formula relating the Poisson kernel to the heat kernel, and an expression for the singularity of the Poisson kernel in the complex domain in terms of the Laplace transform variable ${s =d^{2}(z, y) + \epsilon^{2}}$ where d ² is the analytic continuation of the distance function squared on ${X,\,z \in M_{\epsilon}}$ , and ${y \in X}$ .