Given function values on a uniform grid in a domain Ω in \(\mathbb {R}^{d}\), one is often interested in extending the values to a larger grid on a box B containing Ω. In particular, we are interested in “periodic extensions.” 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 Ω. 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 “good” periodic extension exists, the proposed method will find an extension with similar properties.

给定\（\ mathbb {R} ^ {d} \）中域Ω上统一网格上的函数值，人们通常会对将值扩展到包含Ω的框B上的更大网格感兴趣。我们尤其对“定期扩展”感兴趣。对于此类扩展，B上生成的网格值的离散傅立叶变换（DFT）预期可以为Ω的基础函数提供良好的有效近似。本文提出了两种不同的扩展算法。第一种方法是解决此问题的自然方法，旨在实现扩展数据的DFT系数的最快衰减。第二种方法是一种适用于单变量情况和有限情况的多变量情况的快速算法。结果表明，如果存在“良好的”周期性扩展，则所提出的方法将找到具有相似特性的扩展。