Notes on commutative algebra and harmonic analysis

A note on the Neuman-Sándor Mean

In this article, we present the best possible upper and lower bounds for the Neuman-Sándor mean in terms of the geometric combinations of harmonic and quadratic means, geometric and quadratic means, harmonic and contra-harmonic means, and geometric and contra-harmonic means.

Harmonic Analysis Lecture Notes

These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove L^p convergence of Fourier series. Maximal functions and Calderon--Zygmund decompositions are treated in R^d, so that they can be applied again in the second part of the course, where the Fourier transform is studied. The final part of the course treats band limited functions, Poisson summation and uncertainty principle…

A note on harmonic continuation of characteristic function

We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic functions.

$\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$ 的表示理论与非交换谐振子的推广

Representation theory of $\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$ and a generalization of non-commutative harmonic oscillators

本文利用李代数表示理论，将非交换谐振子推广至高维旋转不变情形，证明其特征值问题可化为单位圆盘上的富克斯型全纯微分方程。

Notes on Stein-Sahi representations and some problems of non $L^2$ harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.

A Note on the Weighted Harmonic-Geometric-Arithmetic Means Inequalities

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.

A Note on Gradually Varied Functions and Harmonic Functions

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $σ-$-net might be very small. Keeping the original discretization (square or triangulation), can we get some interesting properties related to gradual variation? In this note, we try to prove that many harmonic functions are gradually varied or near gradually varied; this means that the value of the center point differs from that of its neighbor at most by 2. It is obvious that most of the gradually varied functions are not harmonic.This note discusses some of…

A note on mean-value properties of harmonic functions on the hypercube

For functions defined on the $n$-dimensional hypercube $I_n (r) = \{\bm{x} \in \mathbb{R}^n ~\vert~ \vert x_i \vert \le r,~ i = 1, 2, \ldots , n\}$ and harmonic therein, we establish certain analogues of Gauss surface and volume mean-value formulas for harmonic functions on the ball in $\mathbb{R}^n$ and their extensions for polyharmonic functions. In particular, our results contribute to the best one-sided $L^1$-approximation by harmonic functions on $I_n (r)$.

Covering models of the asymmetric quantum Rabi model: $η$-shifted non-commutative harmonic oscillators

The non-commutative harmonic oscillator (NCHO) is a matrix valued differential operator originally introduced as a generalization of the quantum harmonic oscillator having a weaker $\mathfrak{sl}_2(\mathbb{R})$-symmetry. The spectrum of the NCHO has remarkable properties, including the presence of number theoretical structures such as modular forms, elliptic curves and Eichler cohomology observed in the special values of the associated spectral zeta function. In addition, the Heun ODE picture of the eigenvalue problem of the NCHO reveals a connection with the quantum Rabi model (QRM), a fundam…

关于Sun型二项 harmonic 和的若干注记

Notes on certain binomial harmonic sums of Sun's type

本文证明并推广了Sun关于包含调和数和二项式系数乘积的无穷级数的猜想，通过自守对象方法求得闭合形式。