Infinite-Parameter ADHM Transform

ADHM Construction of Noncommutative Instantons

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data, together with an origin of the instanton number for U(1). We also give a derivation of the ADHM construction from the viewpoint of the Nahm transformation of instantons on four-tori. This article is a composite version of [23] and [24].

Infinite-Dimensional Stochastic Transforms and Reproducing Kernel Hilbert space

By way of concrete presentations, we construct two infinite-dimensional transforms at the crossroads of Gaussian fields and reproducing kernel Hilbert spaces (RKHS), thus leading to a new infinite-dimensional Fourier transform in a general setting of Gaussian processes. Our results serve to unify existing tools from infinite-dimensional analysis.

几何细化变换：一种新的不可数无限变换空间

The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space

提出基于分层Voronoi细化的连续变换空间，推广小波与Radon变换，严格证明其完备性、可逆性等性质，并展示在复杂非凸域上的良好表现。