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

Open Mathematics (2017)

• Volume: 15, Issue: 1, page 1568-1577
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Zhihua Zhang. "A novel recursive method to reconstruct multivariate functions on the unit cube." Open Mathematics 15.1 (2017): 1568-1577. <http://eudml.org/doc/288456>.

@article{ZhihuaZhang2017,
abstract = {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.},
author = {Zhihua Zhang},
journal = {Open Mathematics},
keywords = {Recursive method; Hyperbolic cross truncation; Multivariate functions; Fourier series},
language = {eng},
number = {1},
pages = {1568-1577},
title = {A novel recursive method to reconstruct multivariate functions on the unit cube},
url = {http://eudml.org/doc/288456},
volume = {15},
year = {2017},
}

TY - JOUR
AU - Zhihua Zhang
TI - A novel recursive method to reconstruct multivariate functions on the unit cube
JO - Open Mathematics
PY - 2017
VL - 15
IS - 1
SP - 1568
EP - 1577
AB - 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.
LA - eng
KW - Recursive method; Hyperbolic cross truncation; Multivariate functions; Fourier series
UR - http://eudml.org/doc/288456
ER -

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