Harmonic Functions and the Reflection Principle

2001 ◽  
pp. 274-288
Author(s):  
Theodore W. Gamelin
Keyword(s):  
Harmonic Functions ◽  
Reflection Principle

Remarks on the reflection principle for harmonic functions

1990 ◽  
Vol 54 (1) ◽  
pp. 60-76 ◽  
Author(s):  
Dmitry Khavinson ◽  
Harold S. Shapiro
Keyword(s):  
Harmonic Functions ◽  
Reflection Principle

scholarly journals The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions

2017 ◽  
Vol 60 (1) ◽  
pp. 146-153 ◽  
Author(s):  
Dmitry Khavinson ◽  
Erik Lundberg ◽  
Hermann Render
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Function ◽  
Difference Equation ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Entire Solution ◽  
Reflection Principle ◽  
Harmonic Solution ◽  
Entire Boundary ◽  
Schwarz Reflection Principle

AbstractIt is shown that the Dirichlet problem for the slab (a, b) × ℝd 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 entire harmonic function g, the inhomogeneous difference equation h(t + 1, y) − h(t, y) = g(t, y) has an entire harmonic solution h.

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals NOTES ON BOUNDED INDUCTION FOR THE COMPOSITIONAL TRUTH PREDICATE

2017 ◽  
Vol 10 (3) ◽  
pp. 455-480 ◽  
Author(s):  
BARTOSZ WCISŁO ◽  
MATEUSZ ŁEŁYK
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Truth Predicate ◽  
Base Theory ◽  
Modified Theory ◽  
Induction Scheme

AbstractWe prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ0-induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

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