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there is such a vector space it's too bad it has an uncountable basis: https://en.wikipedia.org/wiki/Laplace_transform https://en.wikipedia.org/wiki/Laplace_tra
by mlevental 7y ago
there is such a vector space it's too bad it has an uncountable basis: https://en.wikipedia.org/wiki/Laplace_transform https://en.wikipedia.org/wiki/Laplace_transform
- MauiWarrior 7y agoI searched the article, but could not find basis or uncountable. Could you point where exactly I should look?
- mlevental 7y agoequation 1 (in formal definition). the basis is e^(-st). if you don't know how that's a basis you need to read a little bit about functional analysis but just look at the integral as a continuous sum and f(t) as the basis coefficients and e^(-st) starts to look like a vector space basis (hilbert space) basis.
- MauiWarrior 7y agoI am afraid it is a bit over my head. Would you be able to point me to the source?
- peterhj 7y agoNot the most explanatory sources but function spaces essentially are vector spaces: https://en.wikipedia.org/wiki/Vector_space#Function_spaces https://en.wikipedia.org/wiki/Vector_space#Function_spaces http://mathworld.wolfram.com/HilbertSpace.html http://mathworld.wolfram.com/HilbertSpace.html
- mlevental 7y agohttps://www.youtube.com/watch?v=7ICYxBuS2iw https://www.youtube.com/watch?v=7ICYxBuS2iw looks like a pretty elementary introduction
- lapink 7y agoThere exists many such bases, you could also take all the monomes and approximate a function by its Taylor development. However, it does not mean that such bases can be efficiently approximated in a reasonable dimension, nor that conversion from/to textual representation is easy. A deep learning network would achieve those points.
- mlevental 7y agowhat is a monome? https://en.wikipedia.org/wiki/Monome https://en.wikipedia.org/wiki/Monome ?
- MauiWarrior 7y agoI think he meant monomial.
- lapink 7y agoIndeed...