smoothness

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.

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.

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).

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