  We consider a model where a signal (discrete or continuous) is observed with
an additive Gaussian noise process. The signal is issued from a linear
combination of a finite but increasing number of translated features. The
features are continuously parameterized by their location and depend on some
scale parameter. First, we extend previous prediction results for off-the-grid
estimators by taking into account here that the scale parameter may vary. The
prediction bounds are analogous, but we improve the minimal distance between
two consecutive features locations in order to achieve these bounds. Next, we
propose a goodness-of-fit test for the model and give non-asymptotic upper
bounds of the testing risk and of the minimax separation rate between two
distinguishable signals. In particular, our test encompasses the signal
detection framework. We deduce upper bounds on the minimal energy,expressed as
the $\ell_2$-norm of the linear coefficients, to successfully detect a signal
in presence of noise. The general model considered in this paper is a
non-linear extension of the classical high-dimensional regression model. It
turns out that,in this framework, our upper bound on the minimax separation
rate matches (up to a logarithmic factor) the lower bound on the minimax
separation rate for signal detection in the high-dimensional linear model
associated to a fixed dictionary of features. We also propose a procedure to
test whether the features of the observed signal belong to a given finite
collection under the assumption that the linear coefficients may vary, but have
prescribed signs under the null hypothesis. A non-asymptotic upper bound on the
testing risk is given.We illustrate our results on the spikes deconvolution
model with Gaussian features on the real line and with the Dirichlet kernel,
frequently used in the compressed sensing literature, on the torus.
