Daniel Blasi, Artur Nicolau
A natural interpolation problem in the cone of positive harmonic functions is considered and the corresponding interpolating sequences are geometrically described.
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We consider a free interpolation problem in Nevanlinna and Smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions. We also consider the related problem of characterizing positive functions in the disc having a harmonic majorant. An answer is given in terms of a dual relation which involves positive measures in the disc with bounded Poisson balayage. We deduce necessary and sufficient geometric conditions, both expressed in terms of certain maximal functions.
This expository thesis contains a study of four interpolation theorems, the requisite background material, and a few applications. The materials introduced in the first three sections of Chapter 1 are used to motivate and prove the Riesz-Thorin interpolation theorem and its extension by Stein, both of which are presented in the fourth section. Chapter 2 revolves around Calderón's complex method of interpolation and the interpolation theorem of Fefferman and Stein, with the material in between providing the necessary examples and tools. The two theorems are then applied to a brief study of line…
This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical semi-infinite lines.) It characterizes, in terms of the spacing between the hyperplanes, those domains for which there exist minimal harmonic functions with a certain exponential growth.
In this paper we present the necessary modifications of the proof by Mozolyako and Havin in to work for Lipschitz domains instead of $\mathcal{C}^2$ domains.
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…
Harmonic Analysis on the Space of M-positive Vectors
本文在由伸缩矩阵M定义的M-正向量空间上建立调和分析框架，引入Walsh函数类似物和Fourier变换，证明Poisson求和公式、Plancherel定理等，并刻画了函数及其Fourier变换均具紧支撑的Schwartz类类似物。
Fatou theorem and its converse for positive eigenfunctions of the Laplace-Beltrami operator on Harmonic $NA$ groups
本文研究调和 $NA$ 群上正本征函数的边界收敛，证明容许极限存在当且仅当对应边界测度的强导数存在且二者相等，推广了欧氏上半空间结果。
A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.
Any three circles theorem for discrete harmonic functions must contain an inherent error term. In this paper we find the sharp error term in an $L^2$-three circles theorem for harmonic functions defined in $\Zb^2$. The proof is highly indirect due to combinatorial obstacles and cancellations phenomena. We exploit Newton interpolation methods and recursive arguments.
We give a local characterization of the class of functions having positive distributional derivative with respect to $\bar{z}$ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild non-degeneracy assumptions. As a consequence, we give conditions that guarantee that any subharmonic piecewise harmonic function coincides locally with the maximum of finitely many harmonic functions and we describe the topology of their level curves. These results are valid in a quite general setting as they assume no {\em à priori} conditions on the differentiable struct…
For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h\mathbb{Z}^d$ via the Matérn kernel $Φ_{m,d}$---the fundamental solution of $(1-Δ)^m$ in $\mathbb{R}^d$. We prove that the Lebesgue constants of the corresponding interpolation operators are uniformly bounded as $h\to0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Matérn and related compactly supported polyharmonic kernels.
What can be said about the domain $\Om$ in $\bR^n$ for which its Green's function $G(z)$ satisfies $G(z)\asymp \dist (z, \pd\Om)^δ$? What can we say about $\Om$ if the Boundary Harnack Principle holds in the form $u/v=\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are positive harmonic functions on $\Om$ vanishing on $E$. Is this part of the boundary also nice? We discuss these questions below and give answers in very special cases.
Vertex-removal stability and the least positive value of harmonic measures
该研究证明$\text{Z}^d$中调和测度的顶点移除稳定性仅在d=2时成立，借助离散克莱因瓶完成几何论证，同时验证了$\text{Z}^2$上调和测度最小正值指数衰减的猜想，并证明了d≥3时的类似猜想。
We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, $k \ge 1$) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as $e^{-cn^α}$ strictly inside E, where c and $α$ are positive constants independent of the degree n of the approximating polynomial.
Boundary exceptional sets for radial limits of superharmonic functions on non-positively curved Harmonic manifolds of purely exponential volume growth
本文研究了纯指数体积增长非正曲率Harmonic流形上，函数沿径向测地射线发散的边界例外集大小，利用Poisson积分和Green位势获得了Hausdorff维数的尖锐界限与上界，并结合Riesz分解定理得到了正上调和函数例外集维数的主要结果。
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.
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