We consider a classical ill-posed problem of reconstruction of continuous functions from their noisy Fourier coefficients. We study the case of functions of two variables that has been much less investigated. The smoothness of reconstructed functions is measured in terms of the Sobolev classes as well as the classes of functions with dominated mixed derivatives. We investigate two summation methods, that are based on ideas of the rectangle and the hyperbolic cross, respectively. For both of these methods we establish the estimates of the accuracy on the classes that are considered as well the estimates of computational costs. Moreover, we made the comparison of their efficiency based on obtained estimates. A somehow surprising outcome of our study is that for both types of the considered smoothness classes one should employ hyperbolic cross approximation that is not typical for the functions under consideration.

我们从其嘈杂的傅立叶系数考虑连续函数重构的经典不适定问题。我们研究了两个变量的函数情况，而这些情况很少进行研究。重建函数的平滑度根据Sobolev类以及具有主导混合导数的函数类进行度量。我们研究了两种求和方法，分别基于矩形和双曲线交叉的思想。对于这两种方法，我们都会在考虑的类上建立准确性的估计以及计算成本的估计。此外，我们根据获得的估计值对它们的效率进行了比较。