Intuition for polynomial bases

Question

In my linear algebra course I stumbled upon the following observations.

We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$.

$f(x)$ may be composed of elementary functions or not, but in either case, we can express $f(x)$ in any basis we want.

If we express $f(x)$ in the following basis:

$$\{1, x, x^2, x^3, \ldots\}$$

We get the coefficients of the Taylor series of $f(x)$.

If we express $f(x)$ in the following basis:

$$\{1, \sin x, \cos x, \sin {2x}, \cos {2x}, \ldots\}$$

We get the coefficients of the Fourier transform of $f(x)$.

I have two questions.

1

If we express $f(x)$ in the following basis:

$$\{1, x, \frac12(3x^2 - 1), \frac12(5x^3 - 3x), \ldots\} =\ \text{Legendre polynomials}$$

Do the coefficients tell us anything useful?

2

Other than the Taylor series basis and the Fourier transform basis, have there been other famous polynomial bases in mathematics? What were they used for?

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I tried looking online and on this site but couldn't find any good information.

Answer 1

As a remark, these are not algebraic bases. But you can talk about orthogonal bases in a Hilbert space, as you alluded to.

For example, if you look at weighted inner products, you get some $L^2$ spaces and "orthogonal polynomials" (like Legendre's polynomials). They can be used for Gaussian quadrature to compute numerical approximations of integrals.

Answer 2

Answering question 2, Using orthonormal polynomial bases turns out to be very efficient for approximating functions.

Suppose we want a degree-$n$ polynomial approximation $p(x)$ for $f(x)$ on some finite interval $T$.

Let $F$ be the space of continuous functions on $T$. Let $P$ be a subspace of $F$, consisting of n-degree polynomials. We need to minimize the distance from $P$ to $f$.

We're dealing with functions so we can't use the euclidean notion of distance. Let $\langle f,g\rangle = \int_T f(x)g(x) dx $

Now the problem can be phrased to find $p \in P$ that minimizes $||f-p|| = \langle f-p,f-p\rangle = \int_T|f(x) - p(x)|^2 dx $

Consider $f = p+w$ where $p \in P$ and $w \in W$ where W is the orthogonal complement of $P$. Turns out $p$ is unique, and it is the desired polynomial, AND it can be expressed explicitly by $\langle f, p_1\rangle p_1 + ... + \langle f, p_m\rangle p_m$ for any orthonormal basis $(p_1, ..., p_m)$ of $P$.

Applying the Gram-schmidt algorithm to $(1, x, x^2, ... x^n)$ will yield an orthonormal basis of P. Applying it to $\langle f, p_1\rangle p_1 + ... + \langle f, p_m\rangle p_m$ gives the desired approximation.

The error term is hundreds of times smaller than the corresponding taylor series!

I discovered this from the book "Linear algebra done right" 3rd ed. page 199.