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In ], an '''ultradistribution''' (also called an '''ultra-distribution'''<ref>{{cite journal |last=Hasumi |first=Morisuke |title=Note on the n-tempered ultra-distributions |journal=Tohoku Mathematical Journal |volume=13 |issue=1 |year=1961 |pages=94–104 |doi=10.2748/tmj/1178244274}}</ref>) is a ] that extends the concept of a ] by allowing ] whose ]s have ].<ref>{{cite book |last1=Hoskins |first1=R. F. |last2=Sousa Pinto |first2=J. |title=Theories of generalized functions: Distributions, ultradistributions and other generalized functions |edition=2nd |publisher=Woodhead Publishing |location=Philadelphia |year=2011}}</ref> They form an element of the ] 𝒵′, where 𝒵 is the space of test functions whose Fourier transforms belong to 𝒟, the space of ] with compact support.<ref>{{cite journal |last1=Sousa Pinto |first1=J. |last2=Hoskins |first2=R. F. |title=A nonstandard definition of finite order ultradistributions |journal=Proceedings of the Indian Academy of Sciences - Mathematical Sciences |volume=109 |issue=4 |year=1999 |pages=389–395 |doi=10.1007/BF02837074}}</ref>
{{no footnotes|date=December 2024}}
In ], an '''ultradistribution''' is a ] that extends the concept of a ] by allowing ] whose ]s have ]. They form an element of the ] 𝒵′, where 𝒵 is the space of test functions whose Fourier transforms belong to 𝒟, the space of ] with compact support.


== See also == == See also ==
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== References == == References ==
{{reflist}}
*{{cite book |last1=Hoskins |first1=R. F. |last2=Sousa Pinto |first2=J. |title=Theories of generalized functions: Distributions, ultradistributions and other generalized functions |edition=2nd |publisher=Woodhead Publishing |location=Philadelphia |year=2011}}
*{{cite arxiv |last=Vilela Mendes |first=Rui |title=Stochastic solutions of nonlinear PDE's and an extension of superprocesses |eprint=1209.3263 |year=2012}} *{{cite arxiv |last=Vilela Mendes |first=Rui |title=Stochastic solutions of nonlinear PDE's and an extension of superprocesses |eprint=1209.3263 |year=2012}}
*{{cite journal |last=Hasumi |first=Morisuke |title=Note on the n-tempered ultra-distributions |journal=Tohoku Mathematical Journal |volume=13 |issue=1 |year=1961 |pages=94–104 |doi=10.2748/tmj/1178244274}}
*{{cite journal |last1=Sousa Pinto |first1=J. |last2=Hoskins |first2=R. F. |title=A nonstandard definition of finite order ultradistributions |journal=Proceedings of the Indian Academy of Sciences - Mathematical Sciences |volume=109 |issue=4 |year=1999 |pages=389–395 |doi=10.1007/BF02837074}}


] ]
] ]
] ]


{{analysis-stub}} {{analysis-stub}}

Latest revision as of 16:33, 27 December 2024

In functional analysis, an ultradistribution (also called an ultra-distribution) is a generalized function that extends the concept of a distributions by allowing test functions whose Fourier transforms have compact support. They form an element of the dual space 𝒵′, where 𝒵 is the space of test functions whose Fourier transforms belong to 𝒟, the space of infinitely differentiable functions with compact support.

See also

References

  1. Hasumi, Morisuke (1961). "Note on the n-tempered ultra-distributions". Tohoku Mathematical Journal. 13 (1): 94–104. doi:10.2748/tmj/1178244274.
  2. Hoskins, R. F.; Sousa Pinto, J. (2011). Theories of generalized functions: Distributions, ultradistributions and other generalized functions (2nd ed.). Philadelphia: Woodhead Publishing.
  3. Sousa Pinto, J.; Hoskins, R. F. (1999). "A nonstandard definition of finite order ultradistributions". Proceedings of the Indian Academy of Sciences - Mathematical Sciences. 109 (4): 389–395. doi:10.1007/BF02837074.
  • Vilela Mendes, Rui (2012). "Stochastic solutions of nonlinear PDE's and an extension of superprocesses". arXiv:1209.3263.


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