Studying Fourier analysis I encountered the following problem:
Consider two periodic functions $f(x) = e^{\sin(4\pi x)} $ and $ g(x) = e^{\cos(5\pi x)} $ with period 1. Let $ \{f_k\}$ and $ \{g_k\} $ denote their respective Fourier coefficients. $$ f_k = \int_{0}^{1} e^{\sin(4\pi x)} e^{-2\pi i k x} \,dx $$ $$g_k = \int_{0}^{1} e^{\cos(5\pi x)} e^{-2\pi i k x} \,dx$$
I am interested in comparing the rate of convergence to zero of the sequences $ \{f_k\} $ and $ \{g_k\} $ as $ k $ tends to infinity (I know this due to the Riemann-Lebesgue lemma). Specifically, I would like to know which sequence converges to zero more rapidly.
Any insights, analytical approaches, or numerical methods for comparing the convergence rates of these Fourier coefficients would be greatly appreciated.
Thank you!