So here it is, the pinnacle of my research effort thus far. I'll start by the definitions:
Definition 1. Let and be surfaces so that . The star operator takes two surfaces and creates another in the following way:
with the central transformation being defined by
and the last transformation that takes we will call . Thus
Definition 2. Define a continuous, bounded surface , with , and let be true regardless of the value of . In other words, integrating such surface with respect to yields the uniform probability distribution , . We will call this a strict Pasquali patch, and such is intimately related to probability notions. With , we have a more general definition for a Pasquali patch.
Construction 1. Let , a function which consists of a finite sum of pairs of functions of and . In the spirit of conciseness, we omit the transpose symbology, thusly understanding the first vector listed in the dot product as a row vector and the second vector as a column vector. Then is a Pasquali patch provided
and . Thus, we may choose arbitrary functions of , arbitrary functions of , an th function of so that , and
We may write the normalized version as:
and again observe that the unit contribution to the integral of the Pasquali patch is provided by , so that .
Claim 1. Pasquali patches constructed as by Construction 1 are closed.
Proof. To make the proof clear, let us relabel the normalized version of Construction 1 as
with so as to manipulate the equation more simply, and
with and . Then
with , , and . We're not too concerned of the form of the resultant star product as much as its structure. Observe
can be folded back into function vectors and . Thus the structure of Construction 1 functions is preserved when we multiply one by another, showing closure. Of course the property of Construction 1 being Pasquali patches means is closed under that property, and so is a Pasquali patch also, as can be seen when we integrate across :
and the unit contribution is given by . \qed
Claim 2. Pasquali patches constructed as by Construction 1 have powers:
Proof by Induction. First, using the formula observe
and the second power
which is exactly what we expect from the definition of Construction 1 and Claim 1. Next, let us assume that the formula works and
Let us examine
term by term upon dotting. The first dot the first term is:
The first dot the last term is:
The last dot the first term is:
The last dot the last term is:
The middle dot the first term is:
Finally, the middle dot the last term vanishes:
Putting all this information together we get:
Claim 3. It follows that
and , bounded, both conditions necessary and sufficient to establish that such a limiting surface indeed exists (convergence criterion). Furthermore, we check that this is indeed a Pasquali patch.
Proof. To reach a steady state limit,
Next, the steady state limit must be solely functions of , so the functions of must vanish at the limit. Thus, it follows that . We have now established bounds on , which happen to be exactly the radius of convergence of the geometric series:
gives the desired result:
As a check, we integrate across to corroborate the definition of Pasquali patch:
is the eigenfunction corresponding to eigenvalue of all Construction 1 functions, through each power independently.
Proof. An eigenfunction has the property
where the eigenfunction's corresponding eigenvalue is . The claim is more ambitious, and we will show that for any . The left-hand side is
Observe the first term dotted with the middle and last term produce which annihilates the results, so that the only relevant term is the first dot the first:
The second term dot the first produces:
The second term and the second:
and the second by the last term gives
The parenthetical part of this last formulation is equivalent to
within the bounds already established for , and the result of the star product is
as we wanted to show. \qed
is a constant, is the eigenfunction corresponding to eigenvalue of
Proof. The eigenfunction equation is suggestive of what we must do to prove the claim: . We must show that, starring the eigenfunction with , we obtain times the eigenfunction.