Chebyshev series for sine

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(x) 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.

Generalities

The Chebyshev series for a function f(x) on [-1, 1] is given by

where T_n(x) is the nth Chebyshev polynomial of the first kind. The coefficients are given by

One way of defining the polynomials T_n(x) is

and so the change of variables x = cos lets us conclude

Series for sin(x)

Now for our particular function, f(x) = sin(x), we know by symmetry that the coefficients with even subscripts will be zero. This is because sine is an odd function, and T_n is an even function when n is even,

Using equation 10.9.2 here we can prove that if n = 2k+1 then

where J_n 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.)

Related posts

• Chebyshev approximation
• Products of Chebyshev polynomials
• Fourier-Bessel series

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