Abstract

Given function values on a uniform grid in a domain ฮฉ in $\mathbb {R}^{d}$ , one is often interested in extending the values to a larger grid on a box B containing ฮฉ. In particular, we are interested in โ€œperiodic extensions.โ€ For such extensions the discrete Fourier transform (DFT) of the resulting grid values on B is expected to provide good efficient approximation to the underlying function on ฮฉ. This paper presents two different extension algorithms. The first method is a natural approach to this problem, aiming at achieving the fastest decay of the DFT coefficients of the extended data.The second is a fast algorithm which is appropriate for the univariate case and for limited cases of multivariate scenarios. It is shown that if a โ€œgoodโ€ periodic extension exists, the proposed method will find an extension with similar properties.

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