Smoothness

Definitions

1. The condition of being smooth; the degree or measure of said condition.
   countable, uncountable
2. The highest order of derivative (the differentiability class) over a given domain.
   countable, uncountable
3. The quantity measured by the modulus of smoothness.
   countable, uncountable

Equivalents

Examples

“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⟳.”

“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 #92;sigma(#92;Phi)#92;rightarrow 0 as n#92;rightarrow#92;infty, etc.”

“With it,^([a pavement profile]) paving operations can be adjusted "on the fly⟳" to maintain⟳ or improve⟳ smoothness.”

“Smoothness can vary⟳ from 0 (for a nondifferentiable function) to infinity (for a smooth function).”

“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 #92;mathbbS¹ is described by the modulus of smoothness, defined by the forward⟳ difference.”

CEFR level