Recostruct a function from the coefficients of its Chebyshev expansion

Question

The Chebyshev polynomials of the first kind are obtained from the recurrence relation $$\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x)~.\end{aligned}$$ I know the numerical result of the integral: $$c_{k}=\int_{-1}^{1} \frac{f(x) T_{k}(x)}{\sqrt{1-x^{2}}} d x$$ for $k=0,1,\dots,n$. From these values is it possible to reconstruct a possible trend of the function?

Answer 1

Sure. The projection coefficient is $\frac{c_k}{(T_k,T_k)}=\frac{2}{\pi} c_k$ and so a Fourier-like approximation of $f$ is

$$\frac{2}{\pi} \sum_{k=0}^n c_k T_k(x)$$

which will converge to $f$ in the $L^2((1-x^2)^{-1/2} \, dx)$ norm as $n \to \infty$.