Measurement of the generalized polarizabilities of the proton at intermediate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Q</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>

Background: Generalized polarizabilities (GPs) are important observables to describe the nucleon structure, and measurements of these observables are still scarce.Purpose: This paper presents details of a virtual Compton scattering (VCS) experiment, performed at the A1 setup at the Mainz Microtron by studying the $ep\ensuremath{\rightarrow}ep\ensuremath{\gamma}$ reaction. The article focuses on selected aspects of the analysis.Method: The experiment extracted the ${P}_{\mathrm{LL}}\ensuremath{-}{P}_{\mathrm{TT}}/\ensuremath{\epsilon}$ and ${P}_{LT}$ structure functions, as well as the electric and magnetic GPs of the proton, at three new values of the four-momentum transfer squared ${Q}^{2}$: 0.10, 0.20, and $0.45\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2}$.Results: We emphasize the importance of the calibration of experimental parameters. The behavior of the measured $ep\ensuremath{\rightarrow}ep\ensuremath{\gamma}$ cross section is presented and compared to the theory. A detailed investigation of the polarizability fits reveals part of their complexity, in connection with the higher-order terms of the low-energy expansion.Conclusions: The presented aspects are elements which contribute to minimize the systematic uncertainties and improve the precision of the physics results.