About: Non-analytic smooth function

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In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.

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rdf:type	yago:WikicatSmoothFunctions yago:Abstraction100002137 yago:Function113783816 yago:MathematicalRelation113783581 yago:Relation100031921
rdfs:label	Fonction régulière non analytique (fr) 비 해석적 매끄러운 함수 (ko) Non-analytic smooth function (en)
rdfs:comment	수학에서, 매끄러운 함수(무한히 미분가능한 함수)와 해석함수 는 가장 중요한 함수의 유형이다. 어떠한 실수 인자를 가지는 해석함수는 매끄럽다는것은 쉽게 증명된다. 아래의 와 같이 그 역은 참이 아니다 콤팩트 지지 매끄러운 함수 의 중요한 적용 중 하나는 로랑 슈바르츠의 분포이론과 같은 이론에서 중요한 소위 말하는 완화자의 생성의 역할을 하는 것이다. 매끄럽지만 비 해석적인 함수의 존재는 미분기하학과 해석 기하학의 핵심적인 차이점을 나타낸다. 층 이론에서, 이 차이점은 다음과 같이 설명할 수 있다: 해석적인 경우와 비교해서 미분가능한 다양체에서 미분가능한 함수의 층은 단사층이다. 다음 함수는 보통 미분가능한 다양체에서 단위 분할을 만들 때 사용된다. (ko) En mathématiques, les fonctions régulières (i.e. les fonctions indéfiniment dérivables) et les fonctions analytiques sont deux types courants et d'importance parmi les fonctions. Si on peut prouver que toute fonction analytique réelle est régulière, la réciproque est fausse. Une des applications des fonctions régulières à support compact est la construction de fonctions régularisantes, qui sont utilisées dans la théorie des fonctions généralisées, telle la théorie des distributions de Laurent Schwartz. (fr) In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions. (en)
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dcterms:subject	Articles containing proofs Smooth functions
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sameAs	Non-analytic smooth function Non-analytic smooth function Non-analytic smooth function Non-analytic smooth function Non-analytic smooth function Non-analytic smooth function
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