A C*-function on an n-dimensional torus is approximated by its Fourier polynomial of degree N with the accuracy of order C*k . However, for piecewise C* - functions the approximation accuracy drops to the order of √ CN. The following conjecture by Eckhoff (1995) has been intensively studied: There is a non-linear algebraic procedure reconstructing any piecewise C*-function (of one or several variables) from its first N Fourier coefficients, with the overall accuracy of order C*k . This includes the discontinuities’ positions, as well as the smooth pieces over the continuity domains. Recently we have shown that for functions in one variable at least half the conjectured accuracy (i.e. an order of C/N k 2 ) can be achieved. However, the multidimensional question remains open, as well as the question of maximal possible reconstruction accuracy. In this talk we present some recent results towards this reconstruction problem. We also discuss the case of “mixed measurements” where a part of the samples is taken in the geometric domain, and a part in the Fourier one. In this last setting the Whitney C*- extension problem naturally enters the considerations. This is a joint work with D. Batenkov.