Some topics in complex and harmonic analysis

Elements of harmonic analysis

These informal notes briefly discuss some basic topics in harmonic analysis along the lines of convolutions and Fourier transforms.

Hyponormal Toeplitz Operators with Non-Harmonic Algebraic Symbol

We generalize recent results of Fleeman and Liaw on the topic of hyponormal Toeplitz operators acting on the Bergman space of the unit disk.

Some topics concerning analysis on metric spaces and semigroups of operators

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.

调和分析中的不确定性原理——精选主题讲义

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。

Notes on Harmonic Analysis Part I: The Fourier Transform

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 …

Some topics related to real, complex, and Fourier analysis

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.

Notes on harmonic analysis Part II: the Fourier Series

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 …

Plane wave formulas for spherical, complex and symplectic harmonics

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…

旋转对称算子及其调和函数的复李代数

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。

Rectifiability; a survey

This is a survey on rectifiability. I discuss basic properties of rectifiable sets, measures, currents and varifolds and their role in complex and harmonic analysis, potential theory, calculus of variations, PDEs and some other topics.

Asymptotic Harmonic Analysis on the Space of Square Complex Matrices

In this paper, we determine the spherical functions of positive type on the inductive limit space of square complex matrices.