Related papers: Harmonic functions on balls for x-dependent rectil…
In this paper, we study the Dirichlet problem for Laplace's equation in an open disk. The uniqueness of solutions is ensured by the well-known weak maximum principle. We introduce a novel approach to demonstrate the existence of a solution…
Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics \cite{Markovic}. Furthermore, the same result does not hold in higher dimensions…
We consider the operator $\sL$ defined on $C^2(\bR^d)$ functions by \sL f(x)&=&{1/2}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial^2f(x)}{\partial x_i\partial x_j}+\sum_{i=1}^d b_i(x)\frac{\partial f(x)}{\partial x_i}…
Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can…
This note is an addendum to the results of P.O. Frederickson and A.C. Lazer [1], and A.C. Lazer [4] on periodic oscillations, with linear part at resonance. We show that a small modification of the argument in [4] provides a more general…
Based on the Wronski determinant, we propose the construction of linearly independent and orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of functions under consideration, that…
Recent advances in the study of conformally invariant discrete random processes have lead to increasing interest in the study of discrete analogues to holomorphic functions. Of particular interest are results which provide conditions under…
The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the…
Conditioning stable L\'evy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this…
Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\R^n$, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept…
The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order…
This paper developed a systematic strategy establishing RBF on the wavelet analysis, which includes continuous and discrete RBF orthonormal wavelet transforms respectively in terms of singular fundamental solutions and nonsingular general…
We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving…
It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given…
Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we…
This paper continues the studies of symbolic integration by focusing on the stability problems on D-finite functions. We introduce the notion of stability index in order to investigate the order growth of the differential operators…
A real-space formalism for density-functional perturbation theory (DFPT) is derived and applied for the computation of harmonic vibrational properties in molecules and solids. The practical implementation using numeric atom-centered…
In this paper we derive several (and in many cases sharp) estimates for the $\mathrm{L}^2$-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and…
This article contains a self-contained proof of the stability under convolution of the space of resurgent functions associated with a closed discrete subset of the complex plane (the set of possible singularities), under the assumption that…
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…