analysis, these finite-dimensional vectors can be transformed to functional objects using a basis in $L^2$, for instance, the Fourier basis. In this case, the bootstrap approximation of the test statistic $ \mathcal U_T$ defined in (\[eq:test-stat-main\]) will then be given by $$\mathcal U^\ast_T=\frac{2\pi}{T}\sum_{\substack{l=-N\\l\neq0}}^{N}\int_0^1\int_0^1 \Big| \widehat{f}^\ast_{X,\lambda_l}(\tau,\sigma)
-\widehat{f}^\ast_{Y,\lambda_l}(\tau,\sigma) \Big|^2d\tau d\sigma. 
\label{eq:boot-test-stat}$$ From an asymptotic point of view, and as an inspection of the proof of Theorem \[t.boo-test\] below shows, both bootstrap approximations, $ \mathcal U^\ast_{T,k}$ and $ \mathcal U^\ast_{T} $, will lead to the same result, provided that for $ \mathcal U^\ast_{T,k}$ the number of points $ k$ increases to infinity as the sample size $T$ increases to infinity.]{}

Following the bootstrap procedure described in Steps 1-4, a bootstrap-based test then rejects ${\mathcal H_0}$ if $$t_{\mathcal U} \geq t^\ast_{\mathcal U,1-\alpha},$$ where $ t^\ast_{\mathcal U, 1-\alpha}$ denotes the upper $1-\alpha$ percentage point of the distribution of the bootstrap studentized test $$\label{tst-fanis}
t^\ast_{\mathcal U}=(\sqrt{b} T\,\mathcal U^\ast_T  -b^{-1/2}\widehat{\mu}^\ast_0)/\widehat{\theta}^\ast_0,$$ where $\mathcal U^\ast_T$ is defined in (\[eq:boot-test-stat\]) and $\widehat{\mu}_0^\ast$ and $ \widehat{\theta}^\ast_0$ are obtained by replacing the unknown spectral density kernel $ f_{X,\lambda}$ in the expressions for $\mu_0$ and $\theta_0$ given in Theorem \[t.asy-test\] by its pooled estimator $ \widehat{f}^\ast_{\lambda}(\sigma, \tau) = \widehat{f}^\ast_{X,\lambda}(\sigma, \tau)/2+ \widehat{f}^\ast_{Y,\lambda}(\sigma, \tau)/2 $, 