# smoothness

## English

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### Etymology

From Middle English smethnes, *smothnes, from Old English smēþnes, *smōþnes (smoothness, a smooth place, a level surface), equivalent to smooth +‎ -ness.

### Noun

smoothness (countable and uncountable, plural smoothnesses)

1. The condition of being smooth; the degree or measure of said condition.
• 1998, Vladimir V. Senatov, Normal Approximation: New Results, Methods and Problems, Walter de Gruyter (VSP), page 32,
The ‘smoothness’ of distributions can be understood in various senses, this is why we used quotation marks before; further we will drop them. The smoothness can be understood as the differentiability of the distribution function, boundedness of some of its derivatives, the existence of the absolutely continuous component, the decrease of the characteristic function with a certain rate, the validity of the Cramér condition, the condition ${\displaystyle \sigma (\Phi )\rightarrow 0}$ as ${\displaystyle n\rightarrow \infty }$, etc.
• 2013, Robert Otto Rasmussen, et al., Real-time Smoothness Measurements on Portland Cement Concrete Pavements During Construction, Transportation Research Board, page 3,
With it,[a pavement profile] paving operations can be adjusted "on the fly" to maintain or improve smoothness.
2. (mathematical analysis, of a function) The highest order of derivative (the differentiability class) over a given domain.
Smoothness can vary from 0 (for a nondifferentiable function) to infinity (for a smooth function).
3. (approximation theory, numerical analysis, of a function) The quantity measured by the modulus of smoothness.
• 2013, Feng Dai, Yuan Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer, page 79,
A central problem in approximation theory is to characterize the best approximation of a function by polynomials, or other classes of simple functions, in terms of the smoothness of the function. In this chapter, we study the characterization of the best approximation by polynomials on the sphere. In the classical setting of one variable, the smoothness of a function on ${\displaystyle \mathbb {S} ^{1}}$ is described by the modulus of smoothness, defined by the forward difference.