Assumption 2 below), it is easily seen that, under ${\mathcal H_0}$, $$t_{\mathcal U}  =\frac{\sqrt{b} T \, \mathcal U_T-b^{-1/2} \mu_0}{\theta_0} +o_P(1),$$ i.e., Theorem \[t.asy-test\] implies that the studentized test $t_{\mathcal U} $ is an asymptotically $\alpha$-level test under ${\mathcal H_0}$, for any desired level $\alpha \in (0,1)$.

Bootstrapping The Test Statistic {#sec.boot}
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A problem in implementing the above test occurs from the well-known fact that, even in the finite-dimensional case, the convergence of such $L^2$-norm based test statistics towards their limiting distribution is very slow; see, e.g., Härdle and Mammen (1993), Paparoditis (2000) and Dette and Paparoditis (2009). In this case, bootstrap-based approaches may be very effective. In the following, we develop a bootstrap procedure to estimate the distribution of the test statistic $\mathcal U_T$ defined in (\[eq:test-stat-main\]) (and, hence, of the studentized test $ t_{\mathcal U}$ defined in (\[eq.stud-test\])) under ${\mathcal H_0}$, and we prove its asymptotic validity.

We begin by recalling the fact that for any $k\in \N$ and any set of points $ 0\leq s_1 < s_2 < \ldots < s_k\leq 1$ in the interval $[0,1]$, the corresponding $k$-dimensional vector of finite Fourier transforms $$J_{X,\lambda}=\Big(J_{X,\lambda}(s_j) = (2\pi T)^{-1/2}\sum_{t=1}^{T} X_t(s_j) e^{-i t\lambda},\ j=1,2, \ldots, k\Big),$$ satisfies $$\label{eq.fft1}
\left( 