smoothness

Etymology

From Middle English smothenesse; equivalent to smooth + -ness. Compare Old English smēþnes (“smoothness, a smooth place, a level surface”).

noun

1. The condition of being smooth; the degree or measure of said condition.
   The admirable smoothness of the riding also reflected the greatest credit on those who, despite the difficulties caused by the shortage of men and materials, have succeeded in maintaining the track in such first-class order. 1946 July and August, Cecil J. Allen, “British Locomotive Practice and Performance”, in Railway Magazine, page 213
   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 σ(Φ)→0 as n→∞, etc. 1998, Vladimir V. Senatov, Normal Approximation: New Results, Methods and Problems, Walter de Gruyter (VSP), page 32
   With it, paving operations can be adjusted "on the fly" to maintain or improve smoothness. 2013, Robert Otto Rasmussen, “et al.”, in Real-time Smoothness Measurements on Portland Cement Concrete Pavements During Construction, Transportation Research Board, page 3
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.
   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 𝕊¹ is described by the modulus of smoothness, defined by the forward difference. 2013, Feng Dai, Yuan Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer, page 79