Szego Kernel Research Articles

Products of Nevanlinna-Pick kernels and operator colligations

The Schur classSd of analytic functionsf ofd complex variables for whichf(T1,T2,...,Td) has norm at most one for everyd-tuple of commuting strict contractions is characterized by J. Agler [1] in terms of a Szego kernel factorization property and in terms of a transfer function of a certain type ofd-variable unitary colligation. Replacing the Szego kernel by positive kernels whose reciprocal has one positive square we can define a new Schur class in terms of a kernel factorization property. By using, in part, the approach found in Ball and Trent [3], we characterize this Schur class in terms of a transfer function of a certain type of unitary colligation. Further, with these results we establish a Nevanlinna-Pick interpolation theorem for our Schur class. Such interpolation theorems already exist in the literature for two special cases where [1] all kernels are taken to be the Szego kernel, and [2] d=1.

Powers of the Szegő kernel and Hankel operators on Hardy spaces.

In this paper we study the action of certain integral operators on spaces of holomorphic functions on some domains in Cn: These integral operators are defined by using powers of the Szego kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily integral order. These operators may be used to give equivalent norms for the Besov spaces Bp of holomorphic functions. As a consequence we prove that, when 1 p < 1; the small Hankel operators hf on Hardy and weighted Bergman spaces are in the Schatten class Sp if and only if the symbol f belongs to Bp: The type of domains we deal with are the smoothly bounded strictly pseudoconvex domains in Cn and a class of complex ellipsoids in Cn: Our results for strictly pseudo-convex domains depend on Fefferman's expansion of the Szego kernel. In this case, its powers act like a power of the derivation in the normal direction. The ellipsoids we consider are the simplest examples of domains of finite type. In this case, the symmetries of the domains can be exploited to use methods of harmonic analysis and describe the pseudo-differential operators involved.

On the Thomae Formula for $\mathbf Z_N$ Curves

We shall give an elementary and rigorous proof of the Thomae formula for \mathbf Z_N curves which was discovered by Bershadsky and Radul [1, 2]. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szego kernels and prove the vanishing of zero values of derivatives of theta functions with \mathbf Z_N invariant 1/2N characteristics.

The geometry of flag manifold and holomorphic extension of Szegö kernels for SU(p,q)

Let G0 = SU(p, q), K0 = S(U(p) × U(q)) a maximal compact subgroup, and let G,K be their complexifications. Finally, let B be a Borel subgroup of G. We define a number of algebraic functions on G/B × G/K and use them to construct a Stein extension of the Riemannian symmetric space G0/K0. These functions capture the singularities that can occur in the meromorphic extensions of the Knapp-Wallach Szego kernels. These facts imply that all solutions of the Schmid equations extend holomorphically to the space of linear cycles.

Recipes for the classical kernel functions associated to a multiply connected domain in the plane

I have recently shown that the Bergman kernel associated to a finitely connected domain in the plane is given as an explicit rational combination of finitely many basic functions of one complex variable. In this paper, it is proved that all the basic functions and constants in the new formula for the Bergman kernel can be evaluated using onedimensional integrals and simple linear algebra. In fact, all integrals used in the computations are line integrals over boundary curves; at no point is an integral with respect to area measure required. From a theoretical perspective, these results lead to an understanding of the complexity of the Bergman kernel. From a practical point of view, they give an efficient method to numerically compute the Bergman kernel. Similar results are also proved for the Szego kernel function, the Poisson kernel, and the classical Green's function.

Conformal Mapping-Based Image Processing: Theory and Applications

In this paper several techniques for numerical conformal mapping are surveyed and their applications to the development of novel methods in shape analysis and image classification are discussed. One of these techniques, based on the Szegö kernel, is illustrated by examples comprising distance transform and face representation and recognition. A conformal mapping-based face representation is presented. This face representation technique combined with an eigenface-based method extends and improves the results obtained with other eigenface algorithms.

Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains

The objective of this paper is to determine the Szegő kernel of the domain D = { ( z , ζ , w ) ∈ C n + m + 1 ; ℑ m w &gt; ‖ z ‖ 2 + ‖ ζ ‖ 2 p } \mathcal {D} = \{ (z,\zeta ,w) \in {\mathbb {C}^{n + m + 1}};\Im {\text {m}}w &gt; {\left \| z \right \|^2} + {\left \| \zeta \right \|^{2p}}\} explicitly in closed form.

The szegö kernel and the rational proper mappings between planar domains

Suppose D 1is a smoothly bounded domain in whose associated Szego kernel is a rational function and suppose D 2 is the unit disc in . Then any proper holomorphic mapping f: D1— D 2 is rational.

On a new hyperelliptic formula

In this Letter, the correlation functions of the free fermions with odd spin structures are derived on a general hyperelliptic curve. In order to express the correlators explicitly in terms of the branch points, we construct the analogous of the Szego kernel for the odd spin structures.

Les noyaux de Bergman et Szegö pour des domaines strictement pseudo-convexes qui généralisent la boule

Let G be a complex semi-simple group with a compact maximal group K and an irreducible holomorphic representation ? on a finite dimensional space V. There exists on V a K-invariant Hermitian scalar product. Let O be the intersection of the unit ball of V with the G-orbit of a dominant vector. O is a generalization of the unit ball (case obtained for G = SL(n,C) and ? the natural representation on Cn). We prove that for such manifolds, the Bergman and Szego kernels as for the ball are rational fractions of the scalar products and these fractions can be computed explicitely, using invariants of ?. To compute this kernels, one uses a good orthonormal basis related to ?, and then proves that one has a rational fraction, using Schur's orthogonality relations and Weyl's dimensional formula for V.

Lie group structures and reproducing kernels on homogeneous siegel domains

We introduce a continuous family of Lie group structures on a homogeneous Siegel domain. We associate to each one a Hilbert space of holomorphic functions getting an interpolation between the Bergman and the Szego kernels.

The Szego Kernel as a Singular Integral Kernel on a Family of Weakly Pseudoconvex Domains

The Szego Kernel as a Singular Integral Kernel on a Family of Weakly Pseudoconvex Domains

The Cauchy and the Szegö kernels on multiply connected regions

LetD be a bounded plane region whose boundary ∂D is Dini-smooth. An operatorB:L 2(∂D)→L 2(∂D) that has been considered by Kerzman and Stein is investigated. This operator is a compact self-adjoint integral operator whose kernel β (z, ζ) is bounded on ∂Dx∂D and has a geometric description involving chords inD. With the aid ofB the Szego kernel can be expressed in terms of the Cauchy kernel. Here, the operatorB is recovered from the classical theory of kernel functions resulting in extension of the work of Kerzman and Stein. In particular, the eigenvalue problem associated with the operatorB is studied and its complete equivalence with a very classical problem due to Bergman, Schiffer and Singh is established.

The use of the hypercircle inequality in deriving a class of numerical approximation rules for analytic functions

a Hilbert space having as its reproducing kernel an elementary form of the Szego kernel function. For approximation in terms of a given set of point functionals, the simplicity of this kernel yields equations defining the various terms of the hypercircle approximation which may be solved in closed form. In Section 2 we review the hypercircle inequality and the various equations relevant to its use in constructing approximation rules. These results are specialized in the following section to a particular space of analytic functions, while Section 4 contains examples to illustrate their use in specific cases. Presented as examples are an elementary interpolation rule and an integration formula for complex functions closely analogous to the 5-point rule of Birkhoff and Young.

Schwarz’s lemma and the Szegö kernel function

Introduction. This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus f cIf Ids of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis [8](2) for the investigation of schlicht functions in a multiply-connected domain. Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformal mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3). We shall find in addition to this that, while the recent papers of Bergman and Schiffer [5, 16] have developed a close relationship between the theory of schlicht canonical mappings and the Bergman kernel function [3], we are able to develop here a relationship between the theory of bounded functions and the Szego kernel function [19]. Thus the Szego kernel function does for the theory of distortion of bounded functions what the Bergman kernel function does for the theory of distortion of schlicht functions. We point out that both these kernel functions are actually differentials, and that in the Szego case one is dealing with length and in the Bergman case one is dealing with area. Thus the mean modulus fc f I ds, or ffD IfI 2dxdy, should be thought of as a length, or area, and not as a mean modulus. All these remarkable relationships are brought to light by using the simple boundary relations satisfied by the classical domain functions, such as