polynomials

Using Horner's method to evaluate the polynomials is probably suboptimal because it introduces dependencies and eliminates instruction-level parallelism.

The terms in the mean vector would then be polynomials of the window members.

His polynomials give inifintely huge L^1 and L^2 errors near origin; the minimax is finite (1) but it doesn't make an infinitely big error OK.

Which brings me to the other unspelled problem in some of his polynomials; the derivative is very bumpy and discontinuous, which spells out disaster in general math applications.

Moreover, extremely similar problems are studied in automated geometric theorem proving, and they also get pulled into using polynomials.

For example, differentiation and integration of polynomials can be expressed as a linear algebra problem, etc.

Some polynomials of arbitrary degree are indeed solvable with a finite number of such operations.

A better way to do this is to use a polynomial basis that's more stable, such as Legendre, Chebyshev, or Bernstein polynomials.

For example polynomials of the form c+bx are equivalent to the complex numbers (over the reals) which are equivalent to two-vectors (x,y).

Finding exact roots of high-order (>4) polynomials isn't just difficult; it's provably impossible[1].

Recently, the polynomials defined by Bernstein have become again of interest to mathematicians.

I stand by what I said earlier: I don't recommend using the polynomials in the write up.

Just like you could create a field by considering a set of elements modulo some prime, if you pick a "prime" polynomial and take the set of polynomials modulo your "prime" polynomial, then you get another field.

"Euler's original derivation of the value π2/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series." Sorry, not very convincing or geometric, and I'm sure someone else can provide a better answer, but that's how I visualize it.