The union of all open subsets of in which a distribution vanishes is an open subset of in which vanishes.
mathematischer Begriff
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
Theorem
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
V.S.
פונקציונל רציף על מרחב של פונקציות מבחן (מרחב שוורץ), שמכליל את מושג הפונקציות האינטגרביליות באופן מקומי
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.
Generalized functions, space of
Theorem.
неперервний функціонал на просторі пробних функцій (простір Шварца), що узагальнює поняття локально інтегровних функцій
Generalized function algebras
Sergei Lvovich Sobolev
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
Michael
Lemma
Oberguggenberger
un objet qui généralise la notion de fonction et de mesure
Let and If then
Proof
Vasilii Sergeevich Vladimirov
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 ,
функцыя
dział matematyki
istilah analisis matematik
Corollary
funcțional continuu pe spațiul Schwartz
Definition: Elements of are called ' on and is called the ' on . We will use both and to denote this space.
テスト関数空間上の連続汎関数
Generalized functions, product of
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.
a continuous functional on a space of test functions (Schwartz space), which generalizes the concept of locally integrable functions
Sergei
Sobolev
Generalized function
функция для статистических величин
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.
matematikai fogalom
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.
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.
Assumption: For any compact subset we will henceforth assume that is endowed with the subspace topology it inherits from the Fréchet space
Generalized function, derivative of a
wiskunde
true
Vladimirov
mga bagay na binibigyan ng tuntuning panlahat ang klasikal na paniniwala ng punsyon sa pagsusuring pangmatematika
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
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)