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In some applications, one is interested in reconstructing a function $f$ fromits Fourier series coefficients. The problem is that the Fourier series isslowly convergent if the function is non-periodic, or is non-smooth. In thispaper, we suggest a method for deriving high order approximation to $f$ using aPad\'e-like method. Namely, by fitting some Fourier coefficients of theapproximant to the given Fourier coefficients of $f$. Given the Fourier seriescoefficients of a function on a rectangular domain in $\mathbb{R}^d$, assumingthe function is piecewise smooth, we approximate the function by piecewise highorder spline functions. First, the singularity structure of the function isidentified. For example in the 2-D case, we find high accuracy approximation tothe curves separating between smooth segments of $f$. Secondly, simultaneouslywe find the approximations of all the different segments of $f$. We start bydeveloping and demonstrating a high accuracy algorithm for the 1-D case, and weuse this algorithm to step up to the multidimensional case.

Reconstruction of piecewise-smooth multivariate functions from Fourier data