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**Least squares and approximate differentiation.**
*(English)*
Zbl 1247.26009

This paper deals with the notions of least squares derivative and approximate derivative, which are both generalizations of the usual derivative. The existence of either of these generalized derivatives does not guarantee the existence of the other. This paper provides several examples of such functions. In addition, conditions for which the existence of the approximate derivative implies the existence of the least squares derivative are stated and proved. These conditions involve the notion of Hölder continuity and a stronger version of approximate differentiability.

Reviewer: Teodora-Liliana Rădulescu (Craiova)

### MSC:

26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |

26A16 | Lipschitz (Hölder) classes |

26A06 | One-variable calculus |

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\textit{R. A. Gordon}, Real Anal. Exch. 37, No. 1, 189--202 (2012; Zbl 1247.26009)

### References:

[1] | N. Burch, P. Fishback, R. Gordon, The least squares property of the Lanczos derivative , Math. Magazine 75 (2005) 368-378. · Zbl 1086.65015 |

[2] | R. Gordon, A least squares approach to differentiation , Real Anal. Exchange 35 (2010) 205–228. · Zbl 1200.26003 |

[3] | R. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock , Graduate Studies in Mathematics, Vol. 4, American Mathematical Society, Providence, 1994. · Zbl 0807.26004 |

[4] | C. Groetsch, Lanczos’ generalized derivative , Amer. Math. Monthly 105 (1998) 320-326. · Zbl 0927.26003 |

[5] | D. Hicks and L. Liebrock, Lanczos’ generalized derivative: insights and applications , Appl. Math. and Comp. 112 (2000) 63-73. · Zbl 1023.65016 |

[6] | C. Lanczos, Applied Analysis , Prentice Hall, 1960. · Zbl 0111.12403 |

[7] | B. Thomson, Differentiation , Handbook of measure theory, Vol. I, II, 179-247, North-Holland, Amsterdam, 2002. · Zbl 1028.28001 |

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