by using results for uniform consistency of spectral density estimators of univariate time series, since $$\begin{aligned}
\int_0^1\int_0^1 \widehat{f}_{X,\lambda} (\sigma,\tau)d\sigma d\tau & = \frac{1}{Tb}\sum_{t=-N}^N W\Big(\frac{\lambda-\lambda_t}{b}\Big) \int_0^1 \int_0^1 \widehat{p}_{X,\lambda_t}(\sigma,\tau) d\sigma d\tau\end{aligned}$$ can be interpreted as a kernel estimator of the spectral density of the univariate time series $ \int_0^1 X_t(s)ds$, $ t=1,2, \ldots, n$, the periodogram of which at frequency $ \lambda_t$ equals $\int_0^1 \int_0^1 \widehat{p}_{X,\lambda_t}(\sigma,\tau) d\sigma d\tau$. For instance, for the linear functional process $\{X_t, t\in\Z\}$ considered in this paper, $\int_0^1 X_t(s)ds$ is a univariate linear process as well and, under certain conditions, Assumption 2 is satisfied; see Franke and Härdle (1992). However, the same assumption can be fulfilled under different conditions on the integrated process $\int_0^1 X_t(s)ds$; see Wu and Zaffaroni (2015) for a discussion.\
The following theorem establishes the asymptotic validity of the suggested bootstrap procedure.\

\[t.boo-test\] Suppose that the Assumptions 1 and 2 as well as the conditions (i) and (ii) of Theorem \[t.asy-test\] are satisfied. Then, conditional on $X_1,\ldots,X_T,Y_1,\ldots, Y_T$, as $T\rightarrow \infty$, $$\sqrt{b} T \, \mathcal U^\ast_T-b^{-1/2} \widetilde{\mu}_0  \stackrel{d}{\rightarrow}  \mathcal N(0,\widetilde{\theta}_0),$$ in probability, where $$\widetilde{\mu}_0 = \frac{1}{\pi}\, \int_{-\pi}^\pi\left\{\int_0^1 f_{\lambda}(\tau,\tau) \,d\tau\,\right\}^2 d\lambda\int_{-\pi}^\pi W^2\left(u\right)\, du,$$ $$\begin{aligned}
\widetilde{\theta}_0^2&=\frac{2}{\pi^2}\int_{-2\pi}^{2\pi}\left\{\int_{-\pi}^\pi W(u)W(u-x)\, du\right\}^2dx\, \int_{-\pi}^\pi \left\{\int_0^1\int_0^1f_{\lambda}^2(\sigma, \tau)\,\,d\sigma\, d\tau\right\}^2\, d\lambda.\end{aligned}$$

Notice that, under $\mathcal 