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Is it possible to define orthogonal polynomials on the interval $[0, +\infty[$ ? Maybe using the Gram-Schmidt process from the monomial basis $(1, x, x^2, ...)$?

My problem is that I have some data for which I defined the polynomial model $f(x) = c_0+c_1 x + c_2 x^2 + c_3 x^3$. I use afterwards the parameters $c_0, c_1, c_2, c_3$, I estimate, to describe my data (regression). I found that there are some correlations between these coefficients so I wondered if there is any transformation that can give me another (orthogonal) basis $\phi_k(x), k = 0, \dots, 3$ such that $f(x) = d_0\phi_0(x)+d_1\phi_1(x)+d_2\phi_2(x)+d_3\phi_3(x)$ that guaranties the independence of parameters $d_0, d_1, d_2, d_3$.

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The answer to the questions in your first paragraph should be that it is possible and it has been done since long ago. The Laguerre polynomials are orthogonal in $[0, +\infty[$ using an exponential measure.

Anyway, you probably don't need the polynomials to be orthogonal in the whole set of positive reals. You need them to be orthogonal in your data set (that is, your $x$), and that is easier. For example, poly function in R can compute them.

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  • $\begingroup$ I don't use R, but I might check the references related to the function. I already found a way to do it by using the Legendre polynomials orthogonal in $[-1, 1]$ for which I modified the describing polynomials (stretching the interval). $\endgroup$ – Learn_and_Share Dec 25 '16 at 12:25

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