I have real-world data, which after analysis, produces a series of curves. The data has noise, but each vector $u_1(x), u_2(x), \ldots, u_6(x)$ fits extremely well to a polynomial of that same order, eg. $u_3$ is very well fit by a cubic polynomial.
In addition to this, the polynomials are orthonormal-ish over the valid range of the function. That is
$$ \int_0^{1/2} u_i(x) u_j(x) \ dx = c \delta_{ij} $$
with the same constant $c$ for all values of $i=j$. In addition, the function is not quite zero when $i \neq j$ but small, about $.05c$. This may be due to numerical imprecision or the wrong integrated interval.
See the picture below of the data with the fits superimposed:
Questions: If I have a hypothesis that the polynomials form a orthonormal basis set, how can I show this (other than what I've done)? More importantly, what does this tell me? Is there a "standard-form" that I can convert my polynomials to check if they fall under any of the classic orthogonal polynomials?
Edit: I've checked numerically, between each interval of the roots of $u_i$ lies exactly one root of the polynomial $u_{i+1}$, so this property of orthogonal polynomials seem to hold.