相关论文: Connections of the Corona Problem with Operator Th…

In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $(N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients.…

In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of…

The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in $\mathbb{C}^n$.

Let $B$ be a Blaschke product. We prove in several different ways the corona theorem for the algebra $H^\infty_B:=\mC+BH^\infty$. That is, we show the equivalence of the classical {\em corona condition} on data $f_1, ..., f_n \in…

This paper mainly concerns the biholomorphic invariance of $p$-essential normality of Hilbert modules on bounded symmetric domains. By establishing new integral formulas concerning rational function kernels for the Taylor functional…

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following…

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of…

This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the…

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line (measured…

Let $\mathbb{I}$ be a proper ideal of $H^{\infty}(\D)$. We prove the corona theorem for infinitely many generators on the algebra $H^{\infty}_{\mathbb{I}}$ in which the corona theorem for finitely many functions is known to hold. This…

Kernel theorems, in general, provide a convenient representation of bounded linear operators. For the operator acting on a concrete function space, this means that its action on any element of the space can be expressed as a generalised…

Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and…

We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator…

Representing images by compact codes has proven beneficial for many visual recognition tasks. Most existing techniques, however, perform this coding step directly in image feature space, where the distributions of the different classes are…

We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size…

Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and…

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral…

This article studies the problem of approximating functions belonging to a Hilbert space $\mathcal H_d$ with a reproducing kernel of the form $$\tilde K_d(\boldsymbol x,\boldsymbol t):=\prod_{\ell=1}^d…

We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads O. To solve this…

In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in…