Relationship between kernel function and basis











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2
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Is there any relationship between a kernel function and the concept of basis in a vector space?



For example does an equality like this



$$
g(x) = int_{Omega} alpha(y) K(x,y)dy
$$



have anything to do with something like



$$
g(x) = sum_{i}alpha_iK_i(x)
$$

?



Is there any concept of "uncountable" basis maybe?










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  • 1




    see here: en.wikipedia.org/wiki/Mercer%27s_theorem
    – Hamed
    Nov 15 at 15:07












  • Given it's a soft question, can't you just summarize few keypoints in an answer?
    – user8469759
    Nov 15 at 15:52










  • The thing is my knowledge of this area is pretty shallow. Even if it's a soft question, I don't wanna possibly write nonsense as an answer. Hopefully, someone more knowledgeable will come along and provide those key points.
    – Hamed
    Nov 15 at 18:58















up vote
2
down vote

favorite












Is there any relationship between a kernel function and the concept of basis in a vector space?



For example does an equality like this



$$
g(x) = int_{Omega} alpha(y) K(x,y)dy
$$



have anything to do with something like



$$
g(x) = sum_{i}alpha_iK_i(x)
$$

?



Is there any concept of "uncountable" basis maybe?










share|cite|improve this question


















  • 1




    see here: en.wikipedia.org/wiki/Mercer%27s_theorem
    – Hamed
    Nov 15 at 15:07












  • Given it's a soft question, can't you just summarize few keypoints in an answer?
    – user8469759
    Nov 15 at 15:52










  • The thing is my knowledge of this area is pretty shallow. Even if it's a soft question, I don't wanna possibly write nonsense as an answer. Hopefully, someone more knowledgeable will come along and provide those key points.
    – Hamed
    Nov 15 at 18:58













up vote
2
down vote

favorite









up vote
2
down vote

favorite











Is there any relationship between a kernel function and the concept of basis in a vector space?



For example does an equality like this



$$
g(x) = int_{Omega} alpha(y) K(x,y)dy
$$



have anything to do with something like



$$
g(x) = sum_{i}alpha_iK_i(x)
$$

?



Is there any concept of "uncountable" basis maybe?










share|cite|improve this question













Is there any relationship between a kernel function and the concept of basis in a vector space?



For example does an equality like this



$$
g(x) = int_{Omega} alpha(y) K(x,y)dy
$$



have anything to do with something like



$$
g(x) = sum_{i}alpha_iK_i(x)
$$

?



Is there any concept of "uncountable" basis maybe?







linear-algebra functional-analysis soft-question






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 15 at 14:19









user8469759

1,2741516




1,2741516








  • 1




    see here: en.wikipedia.org/wiki/Mercer%27s_theorem
    – Hamed
    Nov 15 at 15:07












  • Given it's a soft question, can't you just summarize few keypoints in an answer?
    – user8469759
    Nov 15 at 15:52










  • The thing is my knowledge of this area is pretty shallow. Even if it's a soft question, I don't wanna possibly write nonsense as an answer. Hopefully, someone more knowledgeable will come along and provide those key points.
    – Hamed
    Nov 15 at 18:58














  • 1




    see here: en.wikipedia.org/wiki/Mercer%27s_theorem
    – Hamed
    Nov 15 at 15:07












  • Given it's a soft question, can't you just summarize few keypoints in an answer?
    – user8469759
    Nov 15 at 15:52










  • The thing is my knowledge of this area is pretty shallow. Even if it's a soft question, I don't wanna possibly write nonsense as an answer. Hopefully, someone more knowledgeable will come along and provide those key points.
    – Hamed
    Nov 15 at 18:58








1




1




see here: en.wikipedia.org/wiki/Mercer%27s_theorem
– Hamed
Nov 15 at 15:07






see here: en.wikipedia.org/wiki/Mercer%27s_theorem
– Hamed
Nov 15 at 15:07














Given it's a soft question, can't you just summarize few keypoints in an answer?
– user8469759
Nov 15 at 15:52




Given it's a soft question, can't you just summarize few keypoints in an answer?
– user8469759
Nov 15 at 15:52












The thing is my knowledge of this area is pretty shallow. Even if it's a soft question, I don't wanna possibly write nonsense as an answer. Hopefully, someone more knowledgeable will come along and provide those key points.
– Hamed
Nov 15 at 18:58




The thing is my knowledge of this area is pretty shallow. Even if it's a soft question, I don't wanna possibly write nonsense as an answer. Hopefully, someone more knowledgeable will come along and provide those key points.
– Hamed
Nov 15 at 18:58















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