## Abstract

It is well known that ω^{r} (f, t)_{p} ≤ t ω^{r-1} f′, t)_{p} ≤ t^{2} ω^{r-2} f″, t)_{p} ≤ ⋯ for functions f ∈ W_{p}^{r}, 1 ≤ p ≤ ∞ For general functions f ∈ L_{p}, it does not hold for 0 < p < 1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p = ∞ Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_{p} space for 0 < p ≤ ∞ We then show its analogues for the Ditzian-Totik modulus of smoothness ω_{φ} ^{r} (f, t)_{p} and the weighted Ditzian-Totik modulus of smoothness ω_{φ}^{r} (f, t)_{w,p} for polynomials with φ (x) = √1 - x^{2}.

Original language | English |
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Pages (from-to) | 182-197 |

Number of pages | 16 |

Journal | Journal of Approximation Theory |

Volume | 136 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2005 |

## Keywords

- Equivalence
- Moduli of smoothness
- Polynomials
- Weighted moduli of smoothness

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