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- In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the canonical LF-topology, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called the space of distributions on and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology. There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If then the use of Schwartz functions as test functions gives rise to a certain subspace of whose elements are called tempered distributions. These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of the space of distributions and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support. Use of analytic test functions leads to Sato's theory of hyperfunctions. (en)
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- 108121 (xsd:nonNegativeInteger)
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- Vasilii Sergeevich Vladimirov (en)
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- G/g043810 (en)
- G/g043820 (en)
- G/g043830 (en)
- G/g043840 (en)
- G/g130030 (en)
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- Vladimirov (en)
- Oberguggenberger (en)
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- Let and If then (en)
- 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 (en)
- Each of the canonical maps below are TVS isomorphisms:
Here represents the completion of the injective tensor product and has the topology of uniform convergence on bounded subsets. (en)
- 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 (en)
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- Theorem (en)
- Theorem. (en)
- Fubini's theorem for distributions (en)
- Schwartz kernel theorem (en)
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- Assumption: For any compact subset we will henceforth assume that is endowed with the subspace topology it inherits from the Fréchet space (en)
- Definition: Elements of are called ' on and is called the ' on . We will use both and to denote this space. (en)
- 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 (en)
- Suppose and is an arbitrary compact subset of Suppose an integer such that and is a multi-index with length For define:
while for define all the functions above to be the constant map. (en)
- Definition and notation: , denoted by is the continuous dual space of endowed with the topology of uniform convergence on bounded subsets of More succinctly, the space of distributions on is (en)
- is called the and it may also be denoted by (en)
- The is the finest locally convex topology on making all of the inclusion maps continuous . (en)
- As is common in mathematics literature, the space is henceforth assumed to be endowed with its canonical LF topology . (en)
- By definition, a is defined to be 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. (en)
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- Generalized function (en)
- Generalized function algebras (en)
- Generalized function, derivative of a (en)
- Generalized functions, product of (en)
- Generalized functions, space of (en)
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- In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the canonical LF-topology, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called the space of distributions on and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology. (en)
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- Spaces of test functions and distributions (en)
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