A basis in the space of all tempered distributions over R^n
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What is a(n uncountable) basis in the topological vector space $mathcal{S}' left(mathbb{R}^nright)$ ? How can any tempered distribution be expanded in terms of such a basis?
functional-analysis distribution-theory
asked Nov 23 at 18:19
DanielC
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What is a(n uncountable) basis in the topological vector space $mathcal{S}' left(mathbb{R}^nright)$ ? How can any tempered distribution be expanded in terms of such a basis?
functional-analysis distribution-theory
asked Nov 23 at 18:19
DanielC
429316
Any compactly supported distribution is of the form $sum_{|alpha|le N}partial_alpha f_alpha$ with $f_alpha$ continuous. Also $e^{-pi |x|^2/k^2} ( T ast k^n e^{-pi k^2 |x|^2}) in S$ and $to T$.
– reuns
Nov 23 at 18:27
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up vote
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favorite
up vote
1
down vote
favorite
What is a(n uncountable) basis in the topological vector space $mathcal{S}' left(mathbb{R}^nright)$ ? How can any tempered distribution be expanded in terms of such a basis?
functional-analysis distribution-theory
asked Nov 23 at 18:19
DanielC
429316
What is a(n uncountable) basis in the topological vector space $mathcal{S}' left(mathbb{R}^nright)$ ? How can any tempered distribution be expanded in terms of such a basis?
functional-analysis distribution-theory
functional-analysis distribution-theory
asked Nov 23 at 18:19
DanielC
429316
asked Nov 23 at 18:19
DanielC
429316
asked Nov 23 at 18:19
DanielC
429316
asked Nov 23 at 18:19
DanielC
429316
asked Nov 23 at 18:19
DanielC
429316
429316
Any compactly supported distribution is of the form $sum_{|alpha|le N}partial_alpha f_alpha$ with $f_alpha$ continuous. Also $e^{-pi |x|^2/k^2} ( T ast k^n e^{-pi k^2 |x|^2}) in S$ and $to T$.
– reuns
Nov 23 at 18:27
add a comment |
Any compactly supported distribution is of the form $sum_{|alpha|le N}partial_alpha f_alpha$ with $f_alpha$ continuous. Also $e^{-pi |x|^2/k^2} ( T ast k^n e^{-pi k^2 |x|^2}) in S$ and $to T$.
– reuns
Nov 23 at 18:27
Any compactly supported distribution is of the form $sum_{|alpha|le N}partial_alpha f_alpha$ with $f_alpha$ continuous. Also $e^{-pi |x|^2/k^2} ( T ast k^n e^{-pi k^2 |x|^2}) in S$ and $to T$.
– reuns
Nov 23 at 18:27
Any compactly supported distribution is of the form $sum_{|alpha|le N}partial_alpha f_alpha$ with $f_alpha$ continuous. Also $e^{-pi |x|^2/k^2} ( T ast k^n e^{-pi k^2 |x|^2}) in S$ and $to T$.
– reuns
Nov 23 at 18:27
add a comment |
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You said "uncountable" which suggests you are talking about a Hamel basis (only allowed finite linear combinations to get all vectors). This is a useless notion in the present context. What you might need rather is a Schauder basis (where you are allowed infinite sums, with suitable notion of convergence). There is an countable Schauder basis given by Hermite functions. See this article by B. Simon.
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
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1 Answer
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1 Answer
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active
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active
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votes
up vote
0
down vote
You said "uncountable" which suggests you are talking about a Hamel basis (only allowed finite linear combinations to get all vectors). This is a useless notion in the present context. What you might need rather is a Schauder basis (where you are allowed infinite sums, with suitable notion of convergence). There is an countable Schauder basis given by Hermite functions. See this article by B. Simon.
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
add a comment |
up vote
0
down vote
You said "uncountable" which suggests you are talking about a Hamel basis (only allowed finite linear combinations to get all vectors). This is a useless notion in the present context. What you might need rather is a Schauder basis (where you are allowed infinite sums, with suitable notion of convergence). There is an countable Schauder basis given by Hermite functions. See this article by B. Simon.
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
add a comment |
up vote
0
down vote
up vote
0
down vote
You said "uncountable" which suggests you are talking about a Hamel basis (only allowed finite linear combinations to get all vectors). This is a useless notion in the present context. What you might need rather is a Schauder basis (where you are allowed infinite sums, with suitable notion of convergence). There is an countable Schauder basis given by Hermite functions. See this article by B. Simon.
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
You said "uncountable" which suggests you are talking about a Hamel basis (only allowed finite linear combinations to get all vectors). This is a useless notion in the present context. What you might need rather is a Schauder basis (where you are allowed infinite sums, with suitable notion of convergence). There is an countable Schauder basis given by Hermite functions. See this article by B. Simon.
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
answered Nov 28 at 18:51
Abdelmalek Abdesselam
396110
396110
add a comment |
add a comment |
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Any compactly supported distribution is of the form $sum_{|alpha|le N}partial_alpha f_alpha$ with $f_alpha$ continuous. Also $e^{-pi |x|^2/k^2} ( T ast k^n e^{-pi k^2 |x|^2}) in S$ and $to T$.
– reuns
Nov 23 at 18:27