Weighted Bergman kernel functions associated to meromorphic functions

We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng Problem.Comment: 11 page

Segal-Bargmann-Fock modules of monogenic functions

In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform for monogenic functionals. This kernel is also the reproducing kernel for the monogenic Bargmann module.Comment: 11 page

A New Algorithm For Difference Image Analysis

In the context of difference image analysis (DIA), we present a new method for determining the convolution kernel matching a pair of images of the same field. Unlike the standard DIA technique which involves modelling the kernel as a linear combination of basis functions, we consider the kernel as a discrete pixel array and solve for the kernel pixel values directly using linear least-squares. The removal of basis functions from the kernel model is advantageous for a number of compelling reasons. Firstly, it removes the need for the user to specify such functions, which makes for a much simpler user application and avoids the risk of an inappropriate choice. Secondly, basis functions are constructed around the origin of the kernel coordinate system, which requires that the two images are perfectly aligned for an optimal result. The pixel kernel model is sufficiently flexible to correct for image misalignments, and in the case of a simple translation between images, image resampling becomes unnecessary. Our new algorithm can be extended to spatially varying kernels by solving for individual pixel kernels in a grid of image sub-regions and interpolating the solutions to obtain the kernel at any one pixel.Comment: MNRAS Letters Accepte

Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems

We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of B\"acklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.Comment: 76 page

Kernel functions based on triplet comparisons

Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similarity triplets, we aim at defining kernel functions that correspond to high-dimensional embeddings. These kernel functions can subsequently be used to apply any kernel method to the data set

Distributions on partitions, point processes, and the hypergeometric kernel

We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed before, see math.RT/9810015, the correlation functions of these limit processes also have determinantal form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a `discrete integrable operator'. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel-Darboux kernel for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel-Darboux kernel for Laguerre polynomials.Comment: AMSTeX, 24 page

Source identities and kernel functions for deformed (quantum) Ruijsenaars models

We consider the relativistic generalization of the quantum $A_{N-1}$ Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh- Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.Comment: 24 page