This paper serves as an extension of known 1D results of B. Adcock et al. [Appl. Comput. Harmon. Anal. 36, No. 3, 387–415 (2014; Zbl 1293.42037)], in this way it provides a necessary first step in the study of reconstructions from Fourier samples within the context of generalized sampling in higher-dimensional settings. In [loc cit.] the respective authors proved the linearity of the stable sampling rate for one-dimensional compactly supported wavelets based on finitely many Fourier samples. This means, up to a constant, one needs the same number of samples as reconstruction elements. In the paper under review the authors extend the previous method from one to two dimensions. Most of the applications involve two- or three-dimensional images, for this reason this is an important non-trivial extension, because non-diagonal scaling matrices are allowed neglecting straightforward arguments for separable two dimensional wavelets from 1D to 2D. The authors not only prove the linearity for standard two-dimensional separable wavelets, but also for two-dimensional boundary wavelets which are of particular interest for smooth images. Only uniform samples are considered but in the 2D setting. It is shown that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions. This means that one can reconstruct a function from its Fourier coefficients and yet get error bounds on the reconstruction (up to a constant) in terms of the decay properties of the wavelet coefficients.
Introduced are also the stable sampling rate, the wavelet reconstruction systems and the Fourier sampling systems. The main results are presented in Section 4. The efficiency of the theoretical results in application is presented by some numerical experiments. They illustrate that the generalized sampling reconstruction provides a substantial gain over the classical Fourier reconstruction.