Holomorphic Continuation Research Articles

Boundary value problems and best approximation by holomorphic functions

A classical problem in complex analysis consists in finding the distance of a functionfe L” on the unit circle U to H”, the space of functions which extend to a bounded holomorphic function in the unit disk D. It is closely related to some other questions, such as the Pick-Nevanlinna problem of minimizing the supremum norm over the set of bounded holomorphic functions in D, subject to a finite or infinite set of interpolation conditions [S-lo] or the problem of seeking the largest circular domain of a positive harmonic function whose first Taylor coefficients are given [2]. The problem has a remarkable variety of applications, especially in systems engineering. Recent heightening of theoretical interest was brought about by results of Adamyan, Arov, and Krejn [l] on an equivalent problem in operator theory. Nowadays a series of related interpolation and approximation problems can be handled by several alternative mathemati- cal approaches in a unified treatment (problems with matrix-valued functions included, cf. [7, 141, introduction, for instance). Much less is known about a far-reaching generalization of the above problem, which was brought into discussion in the recent paper [6] by J. W. Helton and R. E. Howe (Unfortunately, this paper is not available to me at present, therefore I refer to [S]). The authors study the following optimization problem: Given a function F T x CN + Iw, find inf sup F( t, w(t)), WEE tt~ (1.1) where E= (H” n C)” denotes the space of all continuous cCN-valued func- tions on U with holomorphic continuation into D. Assuming the existence

Dirac operators, Schrödinger type operators in [formula omitted]n, and Huygens principle

A detailed study of the holomorphic continuations of primitives of iterates of the Dirac operator D in C n is given. This analysis is used to study holomorphic continuations of solutions to Shrödinger type equations. Also analytic and continuous extensions of solutions to homogeneous Dirac equations are studied, and it is shown that for each cell of harmonicity considered here there exist solutions to each iterate of the Dirac operator which may not be extended beyond any point of the closure of the cell.

Extending proper holomorphic mappings of positive codimension

In this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real-analytic boundaries in complex spaces of different dimensions. Equivalently, we obtain results concerning the analyticity of Cauchy-Riemann mappings between real-analytic pseudoconvex hypersurfaces in complex spaces of different dimensions. To begin with, we recall the corresponding results for mappings of equidimensional domains. Let D and D' be bounded pseudoconvex domains with smooth boundaries in C". If the boundaries of D and D' are strictly pseudoconvex or, more generally, of finite type in the sense of D'Angelo [15], then every proper holomorphic map of D onto D' extends smoothly t o / ) according to the results of Bell and Catlin [7, 8] and Diederich and Forn~ess [19]. For mappings between strictly pseudoconvex domains this was proved by Fefferman [26] and Nirenberg, Webster, and Yang [37]. If the boundaries of the pseudoconvex domains D, D ' c C" are real-analytic, then every proper holomorphic mapping of D onto D' extends holomorphically to a neighborhood of /3 according to Baouendi and Rothschild [2] and Diederich and Forn~ess [21]. This 'reflection principle' was first discovered by Lewy [34] and Pin6uk [38] for mappings between strictly pseudoconvex domains. In the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of Baouendi, Jacobowitz, and Treves [5]. Results in this direction were obtained in recent years by several authors; see the papers [4, 6, 18, 22, 33, 47, 49]. In this paper we are treating the case when the domains D and D' have different dimensions. To be specific, we assume that D e C " and D ' c C N are bounded pseudoconvex domains with real-analytic boundaries and N > n > l . In this situation a proper holomorphic map f : D , D ' need not be regular at the boundary. For instance, there exist proper holomorphic maps of balls of different dimensions

Holomorphic continuation in several complex variables

Holomorphic continuation in several complex variables

Uniformly holomorphic continuation in locally convex spaces

We investigate the simultaneous uniformly holomorphic continuation of the uniformly holomorphic functions defined in a domain spread of uniform type, ( X, ϑ), over a locally convex Hausdorff space E. We construct the envelope of uniform holomorphy of ( X, ϑ) with an analogous method of the results of M. Schottenloher ( Portugal. Math. 33 (1974)) . Finally, we use this construction to the problem of extending uniformly holomorphic maps f: ( X, ϑ) → F, with values in a complete locally convex space to the envelope of uniform holomorphy of X.

Congruences for eigenvalues of Hecke operators on Siegel modular forms of degree two

where ( , ) is the normalized Petersson inner product (cf. Notation 2 ~ below). By Shimura [19], Zagier [22], and the author [17], L2(s,f) has a holomorphic continuation to the whole s-plane. By Sturm [20], Zagier [22], and [17], L'~(m,f) for each even integer m with k = 1). Now, let [ f ] be the Eisenstein series associated with f with respect to Sp(2, Z) in the sense of Langlands [13] and Klingen [6]. This I f ] is an eigenform of weight k satisfying q~([ f ] )=f where 9 is the Siegel operator; conversely, [ f ] is characterized by these properties [11]. A Siegel modular form is called an cigenform if it is a non-zero common eigenfunction of all Hecke operators. In [9], Kurokawa raised the following conjecture (after some numerical data): For a certain prime I in Q dividing the numerator of L~(2k2,f), there will exist an eigen cusp form F of weight k and degree 2 satisfying

Holomorphic continuation of solutions of partial differential equations across the multiple characteristic surface

Holomorphic continuation of solutions of partial differential equations across the multiple characteristic surface

Localization of differential operators and holomorphic continuation of the solutions

Localization of differential operators and holomorphic continuation of the solutions

ANALYTIC CONTINUATION OF SYMMETRIC SQUARES

In this paper the author constructs a holomorphic analytic continuation onto the whole complex plane of special Euler products-symmetric squares-corresponding to Siegel modular forms for congruence-subgroups of .The proof of this theorem is based on the analytic properties of mixed Eisenstein series for arithmetic congruence-subgroups of with character . The paper contains a proof that holomorphic analytic continuation onto the whole complex plane is possible for these series, and a derivation of their functional equation in the case of primitive .Bibliography: 13 titles.

A system of quaternionic equations in four-dimensional complex space

The boundary properties are studied of a class of quaternionic functions containing the class of holomorphic functions of four variables. A condition is found in order for a function defined on some five- or six-dimensional part of the whole topological boundary ∂D of a domain d⊂4 of a special type to have a holomorphic continuation. The results obtained are used to solve singular integral equations in C4.

Holomorphic continuation of solutions of degenerate partial differential equations in two variables

Holomorphic continuation of solutions of degenerate partial differential equations in two variables

Holomorphic continuation at a totally real edge

Holomorphic continuation at a totally real edge

Holomorphic Continuation of Smooth Functions Over Levi-Flat Hypersurfaces

Here we consider the singularities of a Levi-flat real hypersurface S in C that lie in an analytic variety of codimension 2. It is shown that, from the geometric point of view, there are two kinds of singularities, and the type of singularity determines whether S bounds a domain of holomorphy of type ${A^\infty }$.

Holomorphic continuation of smooth functions over Levi-flat hypersurfaces

Here we consider the singularities of a Levi-flat real hypersurface S in C that lie in an analytic variety of codimension 2. It is shown that, from the geometric point of view, there are two kinds of singularities, and the type of singularity determines whether S bounds a domain of holomorphy of type A ∞ {A^\infty } .

On vector-valued versus scalar-valued holomorphic continuation

The purpose of this note is to show that, under mild assumptions, obstacles to holomorphic continuation of vector-valued mappings are the same as in the case of scalar-valued functions.

A theorem of continuation for functions of several complex variables

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.