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Due to discontinuity on the boundary, traditional Fourier approximation does not work efficiently for  d −variate functions on [0, 1]  d  . In this paper, we will give a recursive method to reconstruct/approximate functions on [0, 1]  d   well. The main process is as follows: We reconstruct a  d −variate function by using all of its ( d −1)–variate boundary functions and few  d –variate Fourier coefficients. We reconstruct each ( d −1)–variate boundary function given in the preceding reconstruction by using all of its ( d −2)–variate boundary functions and few ( d −1)–variate Fourier coefficients. Continuing this procedure, we finally reconstruct each univariate boundary function in the preceding reconstruction by using values of the function at two ends and few univariate Fourier coefficients. Our recursive method can reconstruct multivariate functions on the unit cube with much smaller error than traditional Fourier methods.

A novel recursive method to reconstruct multivariate functions on the unit cube