Stephen Semmes
These notes briefly consider some aspects of the Schwartz class of rapidly decreasing smooth functions, tempered distributions, and harmonic functions of polynomial growth.
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These notes briefly consider convolutions of tempered distributions with functions in the Schwartz class.
These notes are concerned with Abel sums and connections with analytic extensions of Fourier integrals.
These notes deal with a few properties of convolutions in the role of approximations to the identity.
These notes briefly discuss Fourier transforms of finite measures and extensions of Fourier integrals to points in complex domains.
In this brief survey we give an introduction to some aspects of "atoms" on metric spaces and their connection with linear operators.
We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics.
These informal notes briefly discuss some basic topics in harmonic analysis along the lines of convolutions and Fourier transforms.
We generalize recent results of Fleeman and Liaw on the topic of hyponormal Toeplitz operators acting on the Bergman space of the unit disk.
This is a brief survey which reviews some traditional themes in harmonic analysis and some more recent areas of activity, connected to "analysis on fractals" in particular.
The Uncertainty Principle in Harmonic Analysis -- Lecture Notes on Selected Topics
本文通过 Fourier 级数与变换的唯一性和重建问题、频谱间隙影响及对数可积性在逼近与拟解析性中的作用，探讨了不确定性原理的多个经典表现，涵盖 Paley-Wiener 定理等核心结果。
Fourier Transforms is a first in a series of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial differential equations and integration theory and its many and diverse applications in sciences and engineering fields makes it an attractive field of study and research. The purpose of these notes is to introduce the basic ideas and theorems of the subject to students of mathematics, physics or engineering sciences. Our goal is to illustrate the …
Here we look at some situations that are like the unit circle or the real line in some ways, but which can be more complicated or fractal in other ways.
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial differential equations and integration theory and its many and diverse applications in sciences and engineering fields makes it an attractive field of study and research. The purpose of these notes is to introduce the basic ideas and theorems of the subject to students of mathematics, physics, or engineering sciences. Our goal is to illustrate the topics with …
This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifold…
A complex Lie algebra of rotationally symmetric operators and their harmonics
本文研究源于四维复李代数的旋转对称二阶偏微分方程，证明其解具有合流超几何级数表示，并恢复了 Laplace 和 Helmholtz 方程的解。
In this paper, we determine the spherical functions of positive type on the inductive limit space of square complex matrices.
Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions
该论文围绕定量可求长性展开，回顾其与可求长性刻画、里斯变换L²有界性的联系及应用，还探讨了其在粗糙区域中拉普拉斯方程相关问题的L^p可解性方面的关键作用。
For a function algebra A we investigate relations between the following three topics: isomorphisms of singly generated A-modules, Morita equivalence bimodules, and `real harmonic functions' with respect to A. We also consider certain groups which are naturally associated with A. We illustrate the notions considered with several examples.
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