Appendix A: Construction of polynomial transforms

Let $\{P_j(x)\}$, $j=0,1,\cdots,N$ denote a set of polynomials, which satisfies the orthogonality condition:

$$\sum_{i=0}^{N} P_k(x_i)P_j(x_i) = \delta_{j,k}.$$ (20)

It is known that as polynomials, $P_j(x_i)$ can be expressed

$$P_j(x_i) = \sum_{k=0}^j a_{jk}x_i^k,$$ (21)

$a_{jk}$ denotes polynomial coefficients. It is natural that $x_j$ can be expressed based on superposition of different polynomials:

$$x^j = \sum_{k=0}^j \beta_{jk} P_k(x).$$ (22)

Based on equations 22 and 23, $j$th polynomial can be expressed as lower-order polynomials

$$P_j(x_i) = \left\{ x^j - \sum_{k=0}^{j-1}\beta_{jk}P_k(x_i) \right\}/\beta_{jj},$$ (23)

Get squares of equation 22 and combine with equation 20, we can obtain

$$\beta_{jj} = \sqrt{\sum_{i=0}^{N} x_i^{2j} -\sum_{k=0}^{j-1}\beta_{jk}^2}$$ (24)

and

$$\beta_{jk} = \sum_{i=0}^{N} x_i^j P_k(x_i).$$ (25)

From equations 23 to 25, we can construct the set of polynomials. We first get $\beta_{00}=\sqrt{N}$ based on equation 24, and thus $P_0=1/\beta_{00}$, then compute $\beta_{10}$, $\beta_{11}$ to construct $P_1$. In the same way, we can construct all polynomials.