Chebyshev polynomials and the proper decomposition of functions

We study the equivalence property of scalar products, based on which we can find the rows of the Chebyshev polynomial sets. For each function in the space $\mathcal L^2_{\mathfrak g}$, the approximation by a row of Chebyshev polynomials is characterized by the standard deviation. In the case of simple algebras, the sets of standard Chebyshev polynomials ensure rapid convergence of the rows. The presented calculation algorithm produces correct results for the algebras $B_3$, $C_3$, and $D_3$.