Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by $$ \hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\ldots, $$ to be monotonic? Given the decay properites of Fourier coefficients, the monotonicity result would translate to $$ |\hat{f}(n)| \geq |\hat{f}(n+1)|, \quad n = 1,2,\ldots. $$ I haven't been able to find any literature regarding this and a result of this nature would be very interesting.

Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by $$ \hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\ldots, $$ to be monotonic? Given the decay properties of Fourier coefficients, the monotonicity result would translate to $$ |\hat{f}(n)| \geq |\hat{f}(n+1)|, \quad n = 1,2,\ldots. $$ I haven't been able to find any literature regarding this and a result of this nature would be very interesting.