English Dictionary

smoothness

名詞

可算名詞 不可算名詞 of句

The condition of being smooth; the degree or measure of said condition. / (mathematical analysis, of a function) The highest order of derivative (the differentiability class) over a given domain.

日本語の意味

なめらかさ、滑らかさ – 物体や表面が滑らかである性質やその程度 / (数学的文脈) 関数の定義域において、存在する最高次数の導関数に基づく微分可能性の度合い

The smoothness of the silk surprised everyone at the market.

その絹の滑らかさは市場にいた誰もを驚かせた。

意味（1）

The condition of being smooth; the degree or measure of said condition.

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.

意味（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).

意味（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.

smoothnesses （ plural ）