# Differentiability of discrete-time Fourier transform

The discrete-time Fourier transform (DTFT) of real or complex numbers $\{x_n:n\in\mathbb Z\}$ is a Fourier series, which produces a periodic function of a frequency variable and is given by $$X(\omega) = \sum_{n=-\infty}^{\infty} x_n \,e^{-i \omega n},$$ where $i$ is the imaginary unit.

I am interested in the differentiability of the DTFT. Is it a differentiable function? What conditions on $\{x_n:n\in\mathbb Z\}$ are needed for the DTFT to be differentiable? Is there an expression for the derivative? Can the series be differentiated term by term?

In particular, I am interested in the differentiability of the spectral density function $f:[-\pi,\pi]\to[0,\infty)$ defined by $$f(\lambda)=\frac1{2\pi}\sum_{t=-\infty}^\infty\gamma_t e^{-i\lambda t},$$ where $\{\gamma_t:t\in\mathbb Z\}$ are absolutely summable autocovariances of some weakly stationary random variables. Is $f$ a differentiable function?

Any help is much appreciated!