We assume our timeseries data are discrete observations from a diffusion process $x$ following the Ito stochastic differential equation $$dx(t)= \sigma(t) \ dW(t) + b(t) \ dt,$$ where $W$ is a Brownian motion on a filtered probability space. Let $\sigma$ and $b$ be random processes, adapted to the Brownian filtration.

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The integrated variance of the process over the time interval $[0,T]$ is defined as $$\int_0^T \sigma^2(t) dt.$$ For any positive integer $n$, let ${\cal S}_{n}:=\{ 0=t_{0}\leq \cdots \leq t_{n}=T \}$ be the observation times. Moreover, let $\delta_i(x):= x(t_{i+1})-x(t_i)$ be the increments of $x$.

The Fourier estimator of the integrated variance over $[0,T]$, is defined as $$\widehat\sigma^{2}_{n,N}:= {T^2 \over {2N+1}}\sum_{|s|\leq N} c_s(dx_n) c_{-s}(dx_n),$$ where for any integer $k$, $|k|\leq N$, the discretized Fourier coefficients of the increments are $$c_k(dx_{n}):= {1\over {T}} \sum_{i=0}^{n-1} e^{-{\rm i} {{2\pi}\over {T}} kt_i}\delta_i(x).$$ The cutting frequency $N$ is a scalar integer. If not specified, $N$ is set equal to $n/2$ (Nyquist frequency).