SG2

Splines

Home
Up
Pluribiharmonic Functions
Splines
Decomposition
Finite Element Method
Green's Function
Poisson Kernel
Differential Operators
Eigenvalues

 A spline is built up out of a basis of piecewise pluri-n-harmonic functions that satisfy certain matching conditions. We first need some terminology.

A junction point is a point where two cells of level m intersect at a point x. If x is a junction point, then there will be at least two words, |w1| = |w2| = m such that, x = Fw1¢¢(V0)×Fw2¢¢¢¢(V0) ÇFw2¢¢(V0) ×Fw2¢¢¢¢(V0). For example:

F¢0 (V0¢) ×F¢¢0 (V0¢¢) ÇF¢1 (V0¢) ×F¢¢2 (V0¢¢) = (F¢0(v1),F¢¢0(v2))

The intersection of the two contractions is one point. The set of all junction points is Jm.

Similarly, a junction cell,cm is a level m cell that is the intersection of two contractions, |w1| = |w2| = m such that cm = Fw1¢¢(V0)×Fw2¢¢¢¢(V0) ÇFw2¢¢(V0) ×Fw2¢¢¢¢(V0). As an example,

F¢0 (V0¢) ×F¢¢0 (V0¢¢) ÇF¢0 (V0¢) ×F¢¢1 (V0¢¢) = F0¢(V0) ×F¢¢0 (v1)

This intersection is a cell of level 1.

In order to define splines on the product, we need to examine what our matching conditions should be on the product. Remember that on the p.c.f. fractal that for each junction point, we require:

 
å
(w,k) Î Jm(x) 		 rw-1(rw mw)-l n Dl (f °Fw)(vk) = 0

In other words, the normal derivatives of each different representation sum to zero at the junction point. This is a very natural definition on SG. However, on SG2, we have a different situation because the intersection of two contractions can result in one point (as is the case on SG), or two cells. We are prompted to define two different matching conditions:

Matching Condition If f °Fw and g °Fw¢ are in Hj, and the support of f and g are in cell w and w¢, respectively, which intersect at one point x, then the matching conditions are satisfied if f(x) = g(x) and n f(x) = n g(x) (with the normal derivative taken with respect to w and w¢).

We are now ready to define the spline on the product of two Sierpinski Gaskets:

Definitions
 

bulletS(H0, G) (G is a set of points) is defined to be the space of all continuous functions f such that D(f °Fw) = 0 for all words w for which there exists an i such that Fw(Vi) Î G.

 

bulletS(H1, G) is defined to be the space of all continuous functions f such that D(f ° Fw) = 0 for all words w for which there exists an i such that Fw(Vi) Î G and the matching conditions hold at all points in G.

 

bulletS0(Hj, G) is the subspace of S(Hj, G) of functions vanishing on SG2