Saulius Norvidas
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.
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On the harmonic continuation of the Riemann xi function
本文将黎曼xi函数的调和延拓推广到n维，解决了$\mathbb{R}_{+}^{n+1}$上的Dirichlet问题，并基于Duffin展开给出了该延拓的新展开式。
We derive necessary and sufficient conditions for a continuous bounded function $f: R\to C$ to be a characteristic function of a probability measure. The Cauchy transform $K_f$ of $f$ is used as analytic continuation of $f$ to the upper and lower half-planes in $C$. The conditions depend on the behavior of $K_f(z)$ and its derivatives on the imaginary axis in $C$. The main results are given in terms of completely monotonic and absolutely monotonic functions.
Let $Ω\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $Ω$ and continuous in $\overline Ω$ which vanishes in a relatively open subset $Σ\subset\partialΩ$ and moreover the normal derivative $\partial_νu$ vanishes in a subset of $Σ$ with positive surface measure, then $u$ is identically $0$.
Harmonic analysis of oscillators through standard numerical continuation tools
提出一种基于边界值问题的数值延拓方法，利用标准延拓包（如AUTO）对非线性周期振荡器进行谐波分析，并通过Colpitts振荡器和非线性阻尼振荡器两个案例验证了其在分析与设计中的有效性。
These notes are concerned with harmonic and holomorphic functions on Euclidean spaces, using quaternions and Clifford algebras in higher dimensions. The main themes are weak solutions, the mean-value property, and subharmonicity.
Fourier series with absolutely summable coefficients provide a classical example of a commutative Banach algebra, and these notes are concerned with this and related matters.
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.
This paper is the continuation of the study on discrete harmonic analysis related to Jacobi expansions initiated in [1]. Considering the operator $\mathcal{J}^{(α,β)}=J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we focus on the study of weighted inequalities for the Riesz transform associated with it.
We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.
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.
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…
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 $I_n (r)$.
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…
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 …
These informal notes briefly discuss some basic topics in harmonic analysis along the lines of convolutions and Fourier transforms.
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。
These informal notes concern some basic themes of harmonic analysis related to representations of groups.
These informal notes briefly discuss Fourier inversion in terms of Gauss--Weierstrass kernels and summability.
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。
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 are concerned with the Schwartz class of rapidly decreasing smooth functions on R^n, Fourier transforms, etc.
These notes briefly discuss basic notions concerning locally compact abelian topological groups and Fourier transforms of functions on them.
These notes briefly consider some aspects of the Schwartz class of rapidly decreasing smooth functions, tempered distributions, and harmonic functions of polynomial growth.
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 …
These notes deal with a few properties of convolutions in the role of approximations to the identity.
These informal notes deal with Fourier series in one or more variables, Fourier transforms in one variable, and related matters.
These notes briefly discuss Fourier transforms of finite measures and extensions of Fourier integrals to points in complex domains.
These informal notes consider Fourier transforms on a simple class of nice functions and some basic properties of the Fourier transform.
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