A harmonic quadrature formula characterizing open strips

1993 ◽  
Vol 113 (1) ◽  
pp. 147-151 ◽  
Author(s):  
D. H. Armitage ◽  
C. S. Nelson
Keyword(s):  
Harmonic Function ◽  
Lebesgue Measure ◽  
Quadrature Formula ◽  
Open Subset ◽  
Harmonic Functions ◽  
Mean Value ◽  
Open Ball ◽  
Mean Value Property ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, thenA converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such thatfor every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

MEAN VALUE PROPERTY OF HARMONIC FUNCTION ON THE HIGHER-DIMENSIONAL SIERPINSKI GASKET

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050077
Author(s):  
YIPENG WU ◽  
ZHILONG CHEN ◽  
XIA ZHANG ◽  
XUDONG ZHAO
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Sierpinski Gasket ◽  
Mean Value ◽  
Sierpiński Gasket ◽  
Mean Value Property ◽  
Average Value ◽  
Higher Dimensional ◽  
The Mean

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski gasket.

scholarly journals Some Remarks on Harmonic Functions in Minkowski Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 196
Author(s):  
Songting Yin
Keyword(s):  
Harmonic Function ◽  
Minkowski Space ◽  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
Minkowski Spaces ◽  
The Mean ◽  
Mean Value Formula

We prove that in Minkowski spaces, a harmonic function does not necessarily satisfy the mean value formula. Conversely, we also show that a function satisfying the mean value formula is not necessarily a harmonic function. Finally, we conclude that in a Minkowski space, if all harmonic functions have the mean value property or any function satisfying the mean value formula must be a harmonic function, then the Minkowski space is Euclidean.

scholarly journals On the mean value property of superharmonic and subharmonic functions

2006 ◽  
Vol 2006 ◽  
pp. 1-3
Author(s):  
Robert Dalmasso
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Subharmonic Functions ◽  
Mean Value Property ◽  
The Mean

We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer.

On the Inverse mean value Property of Harmonic Functions on Strips

1992 ◽  
Vol 24 (6) ◽  
pp. 559-564 ◽  
Author(s):  
M. Goldstein ◽  
W. Haussmann ◽  
L. Rogge
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

scholarly journals On the mean value property of fractional harmonic functions

2020 ◽  
Vol 201 ◽  
pp. 112112
Author(s):  
Claudia Bucur ◽  
Serena Dipierro ◽  
Enrico Valdinoci
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean

On the mean-value property of harmonic functions

1965 ◽  
Vol 14 (1) ◽  
pp. 109-111 ◽  
Author(s):  
Bernard Epstein ◽  
M. M. Schiffer
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
The Mean

A theorem on positive harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 207-212 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Closed Interval ◽  
Positive Harmonic Function ◽  
Image Position ◽  
Function Conjugate ◽  
Positive Harmonic Functions

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can writewhere α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to writewhere h* is a harmonic function conjugate to h. Then the functionhas the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.

Sign in / Sign up

Export Citation Format

Share Document