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Well, I thought I had something pretty slick.

A few months back, I was looking at inner products over one argument function spaces (see here for a snapshot of my thinking at the time). The classic example are the Legendre polynomials.

You can build a function up by looking at its projection along very particular polynomials of increasing degree (constant, linear, quadratic, then degree 3, 4, etc). These polynomials can be thought out as perpendicular projections in the space of functions over the interval [-1, 1] with an inner product (the more abstract version of the vector dot product) being the product of the two polynomials (you can extend to some pretty arbitrary functions) integrated from -1 to 1.

Anyway, long story short, the polynomials for this case have really nice equations to generate the polynomials - the link describes at least four such relations.

So there's a more general case where one integrates the product of two functions of x times a weight function w(x) (which is always nonnegative), integrated again from -1 to 1. In this case, there are no known nice relations unlike the Legendre polynomials above.

You can still construct sequences of increasing degree polynomials just like before, using what's called Gram-Schmidt orthonormalization. You start with x^n as your degree n guess, and then subtract from that the projection of x^n into each of the lower degree polynomials. What's left over is not in the direction of any of the lower degree polynomials you already constructed. Then you divide by the square root of the integral of that residual polynomial squared times that weight function over [-1,1].

Well, I thought I had a trick for fixing that in the case where w(x) is strictly positive and bounded both from below and above by positive numbers (that is, 0 < M1 < w(x) < M2, for all x in [-1, 1]. (Incidentally, the Legendre system is a special case where w(x)=1 for all x in [-1,1].)

The idea is that I would map these polynomials into a new polynomial structure which happens to reduce to the Legendre polynomial system. Then I might be able to pull back those nice Legendre relations to the original non-Legendre situation.

So let's say that P_k(t) is the degree k Legendre polynomial over variable t. Q_l(x) is the degree l polynomial in the weight w system over variable x. I deliberately use two different variables because the polynomial systems will end up being over different multiplication systems which cause all kinds of complications.

Q_l is the usual polynomial over the polynomials ℝ[x], (ℝ being real numbers - this just means real valued polynomials of x) with addition + and multiplication *. I'll write it as (ℝ[x], +, *). P_k(t) will look similar with the same addition, but a completely different and unintuitive multiplication ☆, so we'll write that as (ℝ[t], +, ☆). I'll call the second "star polynomials". Powers of x will be written like x^3 (x to the power 3). Powers of t will be written like t^☆3 to indicate the funky multiplication being used.

The key mapping here between the two flavors of polynomials, is a sequence of polynomials {r_k(x)}, (which will be defined indirectly) where degree r_k = k, k >=0, mapped directly to t^☆k, which is also degree k. It's a basis (maximal set of linearly independent functions) mapped to another basis. This is enough to fully define a one-to-one and onto linear map between (ℝ[x], +, *) and (ℝ[t], +, ☆) with degree and the addition (+) operator being compatible, but not compatibility between the two multiplication operators.

Another important thing that is broken is evaluation of the variable. In the original polynomial series, you can evaluate x and then multiply, or multiply and then evaluate. They give you the same answer. That is (p*q)(3) = p(3)*q(3). It doesn't work for (ℝ[t], +, ☆). Let's say that x -> t, but x^2-3 -> t^☆2. If we naively try to set t = 0, then t^☆2 should be 0. But actually, t=0 implies x=0 and x^2-3 = -3. So t^☆2 = -3. Needless to say, this has ugly repercussions.

Anyway, r_k(x) is chosen so that Q_k(x) maps to P_k(t) for each k >=0. That is, in the ring based on x and normal multiplication, we have the Q_k basis over the weight w system. But we map it to a ring with Legendre polynomials over the t and the star multiplication.

So what can we say about the system. First, it's not elementary to deduce that star multiplication is commutative and associative. You have to define the multiplication so that a☆b = b☆a, and a☆(b☆c) = (a☆b)☆c. It turns out that the definition formula, (t^☆k)☆(t^☆l)=t^☆(k+l) is exactly what you need for both commutativity and associativity. That plus the fact that the star powers of t form a basis, means we get exactly what we need to move on. We dodge our first major problem.

Another feature that comes surprisingly easy is that continuous functions map to continuous functions. Just because we're mapping polynomials to (star) polynomials doesn't mean that we can map the limits of polynomials (such as continuous functions) to limits of polynomials. The somewhat peculiar definition of the weight function w(x) sandwiched between two numbers M1 and M2 is sufficient to get that polynomials that converge to continuous functions map to star polynomials that also converge to continuous functions.

We also can define a derivative, D over our star polynomials. It's linear and maps basis elements, D(t^☆k) = k t^☆(k-1), for k >=0 (equals zero when k=0). We get the usual derivative properties like D(p☆q) = Dp☆q + p☆Dq. I think chain rule works as well since it usually follows from the previous formula.

This allows us to construct an indefinite integral, I. I is what's called the generalized inverse of D. Linear functions need not have inverses, but they always have some sort of generalized inverse. Here, D (I(p))=p and I(D(p)) = p+C, where C is a constant. It's not quite an inverse because D maps constant functions to zero and thus, when you try to invert D with I, you'll be off by some unknown constant function C.

Here's where things went south. I tried to go from the above indefinite integral to a definite integral where I'm integrating from -1 to 1. The idea was that in my original polynomial space Q_k was the orthonormal basis (meaning perpendicular and length 1) of the weight w system. It is mapped directly to P_k, the orthonormal (well when scaled right) basis for the Legendre system. Two different bases and inner products but with the same outcome. (To show the second's inner product would have taken using one or more of the relations that Legendre polynomials have.) To then use the Legendre relations, one would first push the elements of the normal polynomials to the star polynomials, apply the desired Legendre relation, and pull them back to normal polynomial space.

The catch though is that one can't evaluate star polynomials consistently and hence, there is no good boundary conditions [the end points of the title] for which a definite integral can be defined. Perfectly good math ruined by a flaw in the structure.

A final remark is that this sort of thing is a simple version of renormalization from physics. The process is normally used to look at behavior of scale, particularly in attempts to remove infinite blowups that can show up as some aspect of the system is allowed to go to infinity (like perhaps the number of particles or energy of the system). My angle was that renormalization might also shift existing systems into a form that has a lot of nice properties and relations on it, just like my attempt to shift an arbitrary weight w system without known relations to the special Legendre system.