### Abstract

Original language | English |
---|---|

Pages (from-to) | A1075-A1099 |

Journal | SIAM Journal on Scientific Computing |

Volume | 38 |

Issue number | 2 |

Early online date | 12 Apr 2016 |

DOIs | |

Publication status | E-pub ahead of print - 12 Apr 2016 |

### Cite this

*SIAM Journal on Scientific Computing*,

*38*(2), A1075-A1099. https://doi.org/10.1137/15M1018630

**A practical guide to the recovery of wavelet coefficients from Fourier measurements.** / Gataric, Milana; Poon, Clarice.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 38, no. 2, pp. A1075-A1099. https://doi.org/10.1137/15M1018630

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TY - JOUR

T1 - A practical guide to the recovery of wavelet coefficients from Fourier measurements

AU - Gataric, Milana

AU - Poon, Clarice

PY - 2016/4/12

Y1 - 2016/4/12

N2 - In a series of recent papers [B. Adcock, A. C. Hansen, and C. Poon, Appl. Comput. Harmon. Anal., 36 (2014), pp. 387--415; B. Adcock, M. Gataric, and A. C. Hansen, SIAM J. Imaging Sci., 7 (2014), pp. 1690--1723; Adcock et al., SIAM J. Math. Anal., 47 (2015), pp. 1196--1233], it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its Fourier transform via the method of generalized sampling. While these papers focused on the optimality of generalized sampling in terms of its stability and error bounds, the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm. In particular, we show that generalized sampling has a computational complexity of $\mathcal{O}\left(M(N)\log N\right)$ when recovering the first $N$ boundary-corrected wavelet coefficients of an unknown compactly supported function from $M(N)$ Fourier samples. Therefore, due to the linear correspondences between the number of samples $M$ and the number of coefficients $N$ shown previously, generalized sampling offers a computationally optimal way of recovering wavelet coefficients from Fourier data.

AB - In a series of recent papers [B. Adcock, A. C. Hansen, and C. Poon, Appl. Comput. Harmon. Anal., 36 (2014), pp. 387--415; B. Adcock, M. Gataric, and A. C. Hansen, SIAM J. Imaging Sci., 7 (2014), pp. 1690--1723; Adcock et al., SIAM J. Math. Anal., 47 (2015), pp. 1196--1233], it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its Fourier transform via the method of generalized sampling. While these papers focused on the optimality of generalized sampling in terms of its stability and error bounds, the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm. In particular, we show that generalized sampling has a computational complexity of $\mathcal{O}\left(M(N)\log N\right)$ when recovering the first $N$ boundary-corrected wavelet coefficients of an unknown compactly supported function from $M(N)$ Fourier samples. Therefore, due to the linear correspondences between the number of samples $M$ and the number of coefficients $N$ shown previously, generalized sampling offers a computationally optimal way of recovering wavelet coefficients from Fourier data.

U2 - 10.1137/15M1018630

DO - 10.1137/15M1018630

M3 - Article

VL - 38

SP - A1075-A1099

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 2

ER -