Teoría de distribuciones
Teoria dystrybucji
توزيع (رياضيات)
Teoria das distribuições
Distribuzione (matematica)
Обобщённая функция
분포 (해석학)
分布 (数学分析)
Distribution
Distribucio
Distributie (wiskunde)
Узагальнена функція
Distribuce (matematika)
Distribution (mathématiques)
シュワルツ超函数
Distribution (Mathematik)
Distribution (mathematics)
Distribució (matemàtiques)
Sergei Lvovich Sobolev
Vasilii Sergeevich Vladimirov
The vector space is endowed with the locally convex topology induced by any one of the four families of seminorms described above. This topology is also equal to the vector topology induced by of the seminorms in
Definition: Elements of are called ' on and is called the ' on . We will use both and to denote this space.
By definition, a is a continuous linear functional on Said differently, a distribution on is an element of the continuous dual space of when is endowed with its canonical LF topology.
Suppose and is an arbitrary compact subset of Suppose is an integer such that and is a multi-index with length For and define: while for define all the functions above to be the constant map.
is called the and it may also be denoted by Unless indicated otherwise, it is endowed with a topology called , whose definition is given in the article: Spaces of test functions and distributions.
Assumption: For any compact subset we will henceforth assume that is endowed with the subspace topology it inherits from the Fréchet space
true
Theorem
Theorem.
Lemma
Corollary
mathematischer Begriff
دالة رياضية
kontinua funkcionalo sur spaco de testfunkcioj, kiu ĝeneraligas la koncepton de loke integrebla funkcio
un funcional lineal sobre el espacio de funciones infinitamente diferenciables y compactamente soportadas
פונקציונל רציף על מרחב של פונקציות מבחן (מרחב שוורץ), שמכליל את מושג הפונקציות האינטגרביליות באופן מקומי
неперервний функціонал на просторі пробних функцій (простір Шварца), що узагальнює поняття локально інтегровних функцій
un objet qui généralise la notion de fonction et de mesure
функцыя
dział matematyki
istilah analisis matematik
funcțional continuu pe spațiul Schwartz
テスト関数空間上の連続汎関数
a continuous functional on a space of test functions (Schwartz space), which generalizes the concept of locally integrable functions
функция для статистических величин
matematikai fogalom
wiskunde
mga bagay na binibigyan ng tuntuning panlahat ang klasikal na paniniwala ng punsyon sa pagsusuring pangmatematika
Generalized functions, space of
Generalized function algebras
Proof
Generalized functions, product of
Generalized function
Generalized function, derivative of a
V.S.
Michael
Sergei
Oberguggenberger
Sobolev
Vladimirov
The union of all open subsets of in which a distribution vanishes is an open subset of in which vanishes.
Suppose is a distribution on with compact support . There exists a continuous function defined on and a multi-index such that where the derivatives are understood in the sense of distributions. That is, for all test functions on ,
Let be a distribution on . There exists a sequence in such that each has compact support and every compact subset intersects the support of only finitely many and the sequence of partial sums defined by converges in to ; in other words we have: Recall that a sequence converges in if and only if it converges pointwise.
Let be a collection of open subsets of For each let and suppose that for all the restriction of to is equal to the restriction of to . Then there exists a unique such that for all the restriction of to is equal to
Let and If then
Suppose is a distribution on with compact support and let be an open subset of containing . Since every distribution with compact support has finite order, take to be the order of and define There exists a family of continuous functions defined on with support in such that where the derivatives are understood in the sense of distributions. That is, for all test functions on ,
Let be a linear differential operator with smooth coefficients in Then for all we have which is equivalent to:
Suppose has finite order and Given any open subset of containing the support of there is a family of Radon measures in , such that for very and
Let be a collection of open subsets of and let if and only if for each the restriction of to is equal to 0.
Let be a distribution on . For every multi-index there exists a continuous function on such that # any compact subset of intersects the support of only finitely many and # Moreover, if has finite order, then one can choose in such a way that only finitely many of them are non-zero.
Suppose is a Radon measure, where let be a neighborhood of the support of and let There exists a family of locally functions on such that for every and Furthermore, is also equal to a finite sum of derivatives of continuous functions on where each derivative has order