---
tag: 868V3H5VX8
kind: definition
name: Chebyshev series
msc: [41A10, 42A10, 33C45]
defines: GP98BJP4JH
uses: [M6EAR1JHPP, G12KXDHPPS, 8TXASARQN1]
sources:
  - url: https://en.wikipedia.org/wiki/Chebyshev_polynomials
---

## Definition

Let $T_k$ be the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind.
Let $f \colon [-1, 1] \to \R$ be measurable with $\int_{-1}^{1} \abs{f(x)} (1 - x^2)^{-1/2} \, dx < \infty$.
The *Chebyshev coefficients* of $f$ are
$$a_k = \frac{2}{\pi} \int_{-1}^{1} \frac{f(x) \, T_k(x)}{\sqrt{1 - x^2}} \, dx, \qquad k = 0, 1, 2, \dots$$
The *Chebyshev series* of $f$ is the formal series
$$\frac{a_0}{2} + \sum_{k=1}^{\infty} a_k T_k(x).$$

## Remarks

Example: for $f(x) = x^2$, one finds $a_0 = 1$, $a_2 = 1/2$ and $a_k = 0$ otherwise.
Check: $\frac{2}{\pi} \int_{-1}^{1} x^2 (1 - x^2)^{-1/2} \, dx = \frac{2}{\pi} \cdot \frac{\pi}{2} = 1$, and $\frac{1}{2} + \frac{1}{2}(2x^2 - 1) = x^2$.

Non-example: $f(x) = (1 - x^2)^{-1/2}$ has no Chebyshev series, since $\int_{-1}^{1} (1 - x^2)^{-1} \, dx$ diverges.

With $x = \cos \theta$ and $g(\theta) = f(\cos \theta)$, one gets $a_k = \frac{2}{\pi} \int_0^{\pi} g(\theta) \cos(k\theta) \, d\theta$.
So the Chebyshev series of $f$ is the [[Fourier series|G12KXDHPPS]] of the even function $g$, and convergence results transfer.
For example, if $f$ is Lipschitz on $[-1, 1]$, the series converges uniformly to $f$.
The $T_k$ are [[orthogonal polynomials|8TXASARQN1]] for the weight $(1 - x^2)^{-1/2}$, and, when $\int_{-1}^{1} f(x)^2 (1 - x^2)^{-1/2} \, dx < \infty$, the partial sums are the least squares approximations of $f$ for this weight.