How to express cosine of Fourier series as Fourier series again

Question

I have the following Fourier series exapansion: \begin{equation} \phi(t) = a_0 + \Sigma_{n=1}^\infty (a_n\cos pnt + b_n\sin pnt). \end{equation} I want to express $\cos(\phi(t))$ as Fourier series again. More precisely, I want to analytically find the coefficients $\left\{c_n \right\}_{n=0}^\infty$ and $\left\{d_n \right\}_{n=1}^\infty$ which satisfy the following: \begin{equation} \cos(\phi(t)) = c_0 + \Sigma_{n=1}^\infty (c_n\cos pnt + d_n\sin pnt). \end{equation} Is there a way to do this, maybe for example by using some special functions such as Bessel's one?

Answer 1

We can use the generalized Bessel functions with an infinite number of variables, (see for example here). These functions are defined by \begin{equation} J_n\left( \left\lbrace \alpha_m\right\rbrace \right)=\frac{1}{\pi}\int_0^\pi \cos\left(n\theta-\sum_{m=1}^\infty \alpha_m\sin m\theta \right)\,d\theta \end{equation} where $\left\lbrace \alpha_m\right\rbrace $ are real coefficients such that the series $\sum_{m}m\left|\alpha_m\right|$ is convergent. They verify a Anger-Jacobi-like expansion \begin{equation} \exp\left(i\sum_{m=1}^\infty\alpha_m\sin m\theta \right)=\sum_{n=-\infty}^\infty e^{in\theta}J_n\left( \left\lbrace \alpha_m\right\rbrace \right) \end{equation} Then, one can obtain the desired result for a Fourier sine expansion. Similarly, for a Fourier cosine, \begin{align} I_n\left( \left\lbrace \alpha_m\right\rbrace \right)&=\frac{1}{\pi}\int_0^\pi \cos\left(n\theta-\sum_{m=1}^\infty \alpha_m\cos m\theta \right)\,d\theta\\ \exp\left(i\sum_{m=1}^\infty\alpha_m\cos m\theta \right)=&\sum_{n=-\infty}^\infty e^{in\theta}I_n\left( \left\lbrace \alpha_m\right\rbrace \right) \end{align} Many properties of these infinite variable Bessel functions were derived and methods of calculation were also given (see the works of Lorenzutta, Dattoli...).