### Abstract

Consider ℓ _{q}-hulls, 0 < q ≤ 1, from a dictionary of M functions in ^{L p} space for 1 ≤ p < ∞. Their precise metric entropy orders are derived. Sparse linear approximation bounds are obtained to characterize the number of terms needed to achieve accurate approximation of the best function in a ℓ _{q}-hull that is closest to a target function. Furthermore, in the special case of p = 2, it is shown that a weak orthogonal greedy algorithm achieves the optimal approximation under an additional condition.

Original language | English (US) |
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Pages (from-to) | 42-55 |

Number of pages | 14 |

Journal | Journal of Approximation Theory |

Volume | 166 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2013 |

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## Cite this

Gao, F., Ing, C. K., & Yang, Y. (2013). Metric entropy and sparse linear approximation of ℓ

_{q}-hulls for 0 < q ≤ 1.*Journal of Approximation Theory*,*166*(1), 42-55. https://doi.org/10.1016/j.jat.2012.10.002