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Dose there exist a map with high capacity(i.e. can be modeled by a lot of parameters) that maps a function to another?

0

$begingroup$

Note: The following two paragraph have nothing to do with the question I ask, by they explain my motivation.

Let $S = sum_idelta (t-t_i) $, where $delta$ is the Dirac function. My main purpose is to find a map defined on S which can be modeled by a lot of parameters, so that I can adjust these parameters to approximate a given function (which is implicit defined by solving some PDEs).

To find such a map directly is very hard. I make it easier by choosing a smooth function $h$ and trying to find a map defined on $T = h*S:= t_i in[0,T]$

Now I only need to find a map between functions. Such a map must can be modeling by a lot of parameters which will be adjusted to approximate another implicit-defined function. (e.g. A map defined by $R^nrightarrow R^n,xrightarrow Wx$ certainly have lots of parameters——$W$.)

I have think up some ideas, but none of they satisfies me. For example, you can find a basis of the space of real-valued function(using RKHS or Fourier expansion), and then a function can by represented by an infinite dimensional vector. The desired map can be modeled by the map between the vectors,but you must cut short the vector first to get a finite dimensional one, since we cannot handle infinite dimensional vectors using computers. This will induce extra inaccuracy because of the vector cut-off.

Thanks for reading my question, I will appreciate you a lot if you can help me.

linear-algebra functional-analysis signal-processing

share|cite|improve this question

asked Apr 1 at 5:41

AlbertCityAlbertCity

11

$endgroup$

add a comment | 

0

$begingroup$

Note: The following two paragraph have nothing to do with the question I ask, by they explain my motivation.

Let $S = sum_idelta (t-t_i) $, where $delta$ is the Dirac function. My main purpose is to find a map defined on S which can be modeled by a lot of parameters, so that I can adjust these parameters to approximate a given function (which is implicit defined by solving some PDEs).

To find such a map directly is very hard. I make it easier by choosing a smooth function $h$ and trying to find a map defined on $T = h*S:= t_i in[0,T]$

Now I only need to find a map between functions. Such a map must can be modeling by a lot of parameters which will be adjusted to approximate another implicit-defined function. (e.g. A map defined by $R^nrightarrow R^n,xrightarrow Wx$ certainly have lots of parameters——$W$.)

I have think up some ideas, but none of they satisfies me. For example, you can find a basis of the space of real-valued function(using RKHS or Fourier expansion), and then a function can by represented by an infinite dimensional vector. The desired map can be modeled by the map between the vectors,but you must cut short the vector first to get a finite dimensional one, since we cannot handle infinite dimensional vectors using computers. This will induce extra inaccuracy because of the vector cut-off.

Thanks for reading my question, I will appreciate you a lot if you can help me.

linear-algebra functional-analysis signal-processing

share|cite|improve this question

asked Apr 1 at 5:41

AlbertCityAlbertCity

11

$endgroup$

add a comment | 

$begingroup$

Note: The following two paragraph have nothing to do with the question I ask, by they explain my motivation.

Let $S = sum_idelta (t-t_i) $, where $delta$ is the Dirac function. My main purpose is to find a map defined on S which can be modeled by a lot of parameters, so that I can adjust these parameters to approximate a given function (which is implicit defined by solving some PDEs).

To find such a map directly is very hard. I make it easier by choosing a smooth function $h$ and trying to find a map defined on $T = h*S:= t_i in[0,T]$

Now I only need to find a map between functions. Such a map must can be modeling by a lot of parameters which will be adjusted to approximate another implicit-defined function. (e.g. A map defined by $R^nrightarrow R^n,xrightarrow Wx$ certainly have lots of parameters——$W$.)

I have think up some ideas, but none of they satisfies me. For example, you can find a basis of the space of real-valued function(using RKHS or Fourier expansion), and then a function can by represented by an infinite dimensional vector. The desired map can be modeled by the map between the vectors,but you must cut short the vector first to get a finite dimensional one, since we cannot handle infinite dimensional vectors using computers. This will induce extra inaccuracy because of the vector cut-off.

Thanks for reading my question, I will appreciate you a lot if you can help me.

linear-algebra functional-analysis signal-processing

share|cite|improve this question

asked Apr 1 at 5:41

AlbertCityAlbertCity

11

$endgroup$

share|cite|improve this question

asked Apr 1 at 5:41

AlbertCityAlbertCity

11

share|cite|improve this question

asked Apr 1 at 5:41

AlbertCityAlbertCity

11

asked Apr 1 at 5:41

AlbertCityAlbertCity

11

asked Apr 1 at 5:41

AlbertCityAlbertCity

11