Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain.

Differentiability classes

Differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives.

Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer. The function f is said to be of (differentiability) class C^k if the derivatives f′, f′′, ..., f^(k) exist and are continuous (the continuity is implied by differentiability for all the derivatives except for f^(k)). The function f is said to be of class C^∞, or smooth, if it has derivatives of all orders. The function f is said to be of class C^ω, or analytic, if f is smooth and if it its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. C^ω is thus strictly contained in C^∞. Bump functions are examples of functions in C^∞ but not in C^ω.

Smoothness (probability theory)

In probability theory and statistics, smoothness of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function.

Formally, we call the distribution of a random variable X ordinary smooth of order β if its characteristic function satisfies

for some positive constants d₀, d₁, β. The examples of such distributions are gamma, exponential, uniform, etc.

The distribution is called supersmooth of order β if its characteristic function satisfies

for some positive constants d₀, d₁, β, γ and constants β₀, β₁. Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.

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