Numerical Integration of Harmonic Functions with Restricted Sampling Data

A Numerical Integration Method by Using Generalized Series of Functions

Numerical indefinite integration of functions with singularities

We derive an indefinite quadrature formula, based on a theorem of Ganelius, for Hp functions, for p > 1, over the interval (−1, 1). The main factor in the error of our indefinite quadrature formula is O(e−π √ ), with 2N nodes and 1 p + 1 q = 1. The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of √ 2 in the constant of the exponential. We conjectur...

Harmonic functions via restricted mean-value theorems

Let f be a function on a bounded domain Ω ⊆ R and δ be a positive function on Ω such that B(x, δ(x)) ⊆ Ω. Let σ(f)(x) be the average of f over the ball B(x, δ(x)). The restricted mean-value theorems discuss the conditions on f, δ, and Ω under which σ(f) = f implies that f is harmonic. In this paper, we study the stability of harmonic functions with respect to the map σ. One expects that, in gen...

a numerical integration method by using generalized series of functions

A Remark on Disk Packings and Numerical Integration of Harmonic Functions

X iv :1 40 3. 80 02 v1 [ m at h. N A ] 3 1 M ar 2 01 4 A REMARK ON DISK PACKINGS AND NUMERICAL INTEGRATION OF HARMONIC FUNCTIONS STEFAN STEINERBERGER Abstract. We are interested in the following problem: given an open, bounded domain Ω ⊂ R2, what is the largest constant α = α(Ω) > 0 such that there exist an infinite sequence of disks B1, B2, . . . , BN , · · · ⊂ R 2 and a sequence (ni) with ni ...