## Real Analysis Exchange

- Real Anal. Exchange
- Volume 37, Number 1 (2011), 189-202.

### Least Squares and Approximate Differentiation

#### Abstract

The least squares derivative and the approximate derivative are both generalizations of the ordinary derivative. The existence of either of these generalized derivatives does not guarantee the existence of the other and it is even possible for both generalized derivatives to exist at a point but have different values. Several examples of such functions are presented in this paper. In addition, conditions for which the existence of the approximate derivative implies the existence (and equality) of the least squares derivative are stated and proved. These conditions involve the notion of Hölder continuity and a stronger version of approximate differentiability.

#### Article information

**Source**

Real Anal. Exchange, Volume 37, Number 1 (2011), 189-202.

**Dates**

First available in Project Euclid: 30 April 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1335806771

**Mathematical Reviews number (MathSciNet)**

MR3016859

**Zentralblatt MATH identifier**

1247.26009

**Subjects**

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A16: Lipschitz (Hölder) classes

Secondary: 26A05

**Keywords**

approximate derivative least squares derivative Hölder continuity

#### Citation

Gordon, Russell A. Least Squares and Approximate Differentiation. Real Anal. Exchange 37 (2011), no. 1, 189--202. https://projecteuclid.org/euclid.rae/1335806771