The inversion of the Fourier transform with limited data is a well-known issue that leads to bandlimited extrapolation problems. In some cases, e.g. when the sampling is non-uniform, preliminary interpolation methods are needed to reconstruct the frequency information. In view of this, we drive our attention towards interpolation/extrapolation approaches [2] that consist of the following two steps: 1) Interpolation of the scattered observations of the Fourier transform; 2) Extrapolation and inversion of the so-generated interpolants via Fast Fourier Transform (FFT)-based iterative methods. The interpolant is constructed via Variably Scaled Kernels (VSKs) which are based on the definition of a scaling function. Error estimates on the final inversion step depending on the definition of the scaling function and on the signal formation are provided. We are able to prove that the closer is the scaling function to the sought and unknown function in the frequency domain the smaller is the error in the physical space. As a real case test we consider samples from the Spectrometer/Telescope for Imaging X-rays (STIX), which is a telescope recording X-rays from the Sun with the main purpose of observing solar flares. The the imaging problem can be described as an inversion of the Fourier transform with undersampled data [1]. The results on solar flares reconstructions empirically demonstrate our theoretical claims. [1] E. Perracchione, F. Camattari, A. Volpara, P. Massa, A.M. Massone, M. Piana, Unbiased CLEAN for STIX in Solar Orbiter, Astrophysical Journal, Supplement Series, Vol. 268(2), art. no. 68, 2023. [2] E. Perracchione, A.M. Massone, M. Piana, Feature augmentation for the inversion of the Fourier transform with limited data, Inverse Problems, Vol. 37(10), art. id 105001, 2021.