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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!

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