Orthogonal polynomial approximation

Question

Why we use an orthogonal polynomial (Hermite, Legendre, or Laguerre, etc.) approximation of any function if Taylor series approximation is already there. And what are the criteria to say that which approximation methods are better?

Answer 1

When we study inner product spaces, we usually aim at creating an orthonormal (or orthogonal) basis for a vector space. The main advantage of doing so is that finding the coefficients to form any vector using this basis becomes really easy.

If we have a vector space $V$ with an inner product $\langle \cdot, \cdot \rangle$, then we can always have an orthonormal basis $B$. When we say that $B$ is an orthonormal basis, we automatically mean that for $i \neq j$, $\langle v_i, v_j \rangle = 0$ for basis vectors $v_i$ and $v_j$; and that $\langle v_i, v_i \rangle = 1$.

Hence, if we want to write any vector $v \in V$ as a linear combination of basis vectors, we need to find $\alpha_1, \alpha_2, \cdots, \alpha_n \in \mathbb{F}$ such that

$$v = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n$$

for basis vectors $v_1, v_2, \cdots, v_n$. Then, to find the coefficients, all we have to do is

$$\langle v, v_i \rangle = \alpha_i$$

for $i = 1, 2, \cdots, n$. Since $\langle \cdot, \cdot, \rangle$ is well-defined known function, all the coefficients are automatically known.

This ease is not obtained if we do not have an orthonormal basis. Therefore, even for approximating functions, we use the orthonormal polynomials rather than the usual polynomial basis used in Taylor's expansion. However, since we are approximating, there are a few approximation conditions that need to be added (such as the approximations should pass through certain points, or the error should be minimized, etc.).