Potential theory and approximation: highlights from the scientific work of Stephen Gardiner

Abstract

This survey is dedicated to the scientific achievements of Stephen Gardiner in honour of his 60th birthday.

Poisson-Jensen Formulas and Balayage of Measures

“…Balayage of charges and measures with a non-compact support is also occur frequently and are used in Analysis. So, a bounded domain D ⊂ R d is called a quadrature domain (for harmonic functions) if there is a charge ∆ ∈ Meas cmp (D) such that the restriction of the Lebesgue measure λ to D is a balayage of ∆ with respect to the class of all harmonic λ-integrable functions on D. In connection with the quadrature domains, see very informative overview [19,3] and bibliography in it.…”

Affine Balayage of Measures and Distribution of Riesz Measures of Subharmonic Functions

“…So, a bounded domain D ⊂ R d is called a quadrature domain (for harmonic functions) if there is a charge µ ∈ Meas cmp (D) such that the restriction λ d D is a balayage of µ for the class har(D) ∩ L 1 (D). In connection with the quadrature domains, see very informative overview [22,3] and bibliography in it. Example 5.1 ([18, 3], [51], [52], [14], [15], [11], [23], [27]- [43], [47], [6]).…”