Harmonic function#The mean value property

Mean value property is called in mathematics the property of a function that correspond to each point the function value and the averaged function value in a ball around this point.

A function that satisfies the mean value property is automatically harmonious and smooth, so in.

Definition

Be. A function satisfies the mean value property if and only if for all and all, the meet,

Applies. Stand by and for the volume or the surface of the sphere with radius.

The integrals with prefactor are averaged integrals are often quoted as crossed- integral.

Sufficient conditions

Both requirements, that is the function value equal to the average over the whole of the ball, respectively, on their surface, are equivalent. This follows from the formulas for surface area and volume of the - dimensional sphere, because if the mean value property applies to the surface integral is

Conversely, if the averaging property of the integral of the full sphere is true, in accordance with the law

So it suffices to prove one of the conditions.

Attenuated mean value property

In the study of sub-and super- harmonic functions we used a toned-down formulation of the mean value property, where the equal sign is replaced by smaller - or larger - than:

Discrete mean value property

In the numerical analysis of partial differential equations is called the discrete mean value property associated with the discretization of the Laplace operator: By forming the second centered difference quotient leads to the approximation

The fact that a mean value property applies here, one sees directly by inserting a discrete harmonic function, for which: