In last week’s post on polynomial approximations for sine, I showed that the polynomial based on Chebyshev series was much better than a couple alternatives. I calculated a few terms of the Chebyshev series for sin(πx) but didn’t include the calculations in that blog post. I calculated the series coefficients numerically, but this post will show how to calculate the coefficients analytically.
The Chebyshev series for a function f(x) on [-1, 1] is given by
where Tn(x) is the nth Chebyshev polynomial of the first kind. The coefficients are given by
One way of defining the polynomials Tn(x) is
and so the change of variables x = cos θ lets us conclude
Series for sin(πx)
Now for our particular function, f(x) = sin(πx), we know by symmetry that the coefficients with even subscripts will be zero. This is because sine is an odd function, and Tn is an even function when n is even,
Using equation 10.9.2 here we can prove that if n = 2k+1 then
where Jn is the nth Bessel function of the first kind.
(The preceding sentence was the conclusion to a fair amount of fumbling around on my part. As is often the case in mathematics, the length of the write-up is unrelated to the length of the discovery process.)