Non-polynomial, non-sinusoidal expansion of an arbitrary function
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Given a family of functions, what functions can be derived as a linear combination of these functions? Or rather, how to determine the "completeness" of a group of functions as to their ability to linearly combine into an arbitrary function?
For example, is it possible to express an arbitrary simple curve around the origin as the sum of ellipses $r(theta; b,e)=frac{b}{sqrt{1-(ecos{theta})^2}}$?
calculus linear-algebra
asked Dec 17 '18 at 7:51
araxarax
1,46011119
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add a comment |
$begingroup$
Given a family of functions, what functions can be derived as a linear combination of these functions? Or rather, how to determine the "completeness" of a group of functions as to their ability to linearly combine into an arbitrary function?
For example, is it possible to express an arbitrary simple curve around the origin as the sum of ellipses $r(theta; b,e)=frac{b}{sqrt{1-(ecos{theta})^2}}$?
calculus linear-algebra
asked Dec 17 '18 at 7:51
araxarax
1,46011119
$endgroup$
$begingroup$
You can start by showing their $L_2$ completeness..
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– mm-crj
Dec 17 '18 at 7:57
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@mm-crj Could you elaborate or point me to some reading materials? I'm not really familiar with the mathematical "completeness". Thanks!
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– arax
Dec 23 '18 at 1:04
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en.m.wikipedia.org/wiki/Complete_metric_space This is what I was talking about. Also this en.m.wikipedia.org/wiki/Function_space
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– mm-crj
Dec 23 '18 at 7:12
add a comment |
$begingroup$
Given a family of functions, what functions can be derived as a linear combination of these functions? Or rather, how to determine the "completeness" of a group of functions as to their ability to linearly combine into an arbitrary function?
For example, is it possible to express an arbitrary simple curve around the origin as the sum of ellipses $r(theta; b,e)=frac{b}{sqrt{1-(ecos{theta})^2}}$?
calculus linear-algebra
asked Dec 17 '18 at 7:51
araxarax
1,46011119
$endgroup$
Given a family of functions, what functions can be derived as a linear combination of these functions? Or rather, how to determine the "completeness" of a group of functions as to their ability to linearly combine into an arbitrary function?
For example, is it possible to express an arbitrary simple curve around the origin as the sum of ellipses $r(theta; b,e)=frac{b}{sqrt{1-(ecos{theta})^2}}$?
calculus linear-algebra
calculus linear-algebra
asked Dec 17 '18 at 7:51
araxarax
1,46011119
asked Dec 17 '18 at 7:51
araxarax
1,46011119
asked Dec 17 '18 at 7:51
araxarax
1,46011119
asked Dec 17 '18 at 7:51
araxarax
1,46011119
asked Dec 17 '18 at 7:51
araxarax
1,46011119
1,46011119
$begingroup$
You can start by showing their $L_2$ completeness..
$endgroup$
– mm-crj
Dec 17 '18 at 7:57
$begingroup$
@mm-crj Could you elaborate or point me to some reading materials? I'm not really familiar with the mathematical "completeness". Thanks!
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
en.m.wikipedia.org/wiki/Complete_metric_space This is what I was talking about. Also this en.m.wikipedia.org/wiki/Function_space
$endgroup$
– mm-crj
Dec 23 '18 at 7:12
add a comment |
$begingroup$
You can start by showing their $L_2$ completeness..
$endgroup$
– mm-crj
Dec 17 '18 at 7:57
$begingroup$
@mm-crj Could you elaborate or point me to some reading materials? I'm not really familiar with the mathematical "completeness". Thanks!
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
en.m.wikipedia.org/wiki/Complete_metric_space This is what I was talking about. Also this en.m.wikipedia.org/wiki/Function_space
$endgroup$
– mm-crj
Dec 23 '18 at 7:12
$begingroup$
You can start by showing their $L_2$ completeness..
$endgroup$
– mm-crj
Dec 17 '18 at 7:57
$begingroup$
You can start by showing their $L_2$ completeness..
$endgroup$
– mm-crj
Dec 17 '18 at 7:57
$begingroup$
@mm-crj Could you elaborate or point me to some reading materials? I'm not really familiar with the mathematical "completeness". Thanks!
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
@mm-crj Could you elaborate or point me to some reading materials? I'm not really familiar with the mathematical "completeness". Thanks!
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
en.m.wikipedia.org/wiki/Complete_metric_space This is what I was talking about. Also this en.m.wikipedia.org/wiki/Function_space
$endgroup$
– mm-crj
Dec 23 '18 at 7:12
$begingroup$
en.m.wikipedia.org/wiki/Complete_metric_space This is what I was talking about. Also this en.m.wikipedia.org/wiki/Function_space
$endgroup$
– mm-crj
Dec 23 '18 at 7:12
add a comment |
1 Answer
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Some 20 years ago, when I was a math student, I too saw the good properties of the families of polynomial functions $f_n(x){=x^{n}}$ (for expansion into Taylor series) the and the sinusoidal functions (for expansion into Fourier series), so I asked myself the same question. As my career didn't proceed deeper into mathematics, I didn't find an elaborated answer to it.
Actually there might be plenty of families of functions that yield another (larger) family of functions as their linear combinations, or even larger considering limits of their linear combinations. But the issue is in the human readibility as well as computer calculability of these families of functions.
Also there should be a consistent method for calculating the coefficients of the expansion. So if you are trying to express curves around the origin as sum of ellipses, you'd better make an educated guess how to calculate the coefficients of your expansion, and confirm it by writing software.
These are the reasons why the two well known families are interesting to us, human mathematicians, and we actually aren't interested into many other possible families.
answered Dec 17 '18 at 8:49
MuckoMucko
11
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Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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$begingroup$
Some 20 years ago, when I was a math student, I too saw the good properties of the families of polynomial functions $f_n(x){=x^{n}}$ (for expansion into Taylor series) the and the sinusoidal functions (for expansion into Fourier series), so I asked myself the same question. As my career didn't proceed deeper into mathematics, I didn't find an elaborated answer to it.
Actually there might be plenty of families of functions that yield another (larger) family of functions as their linear combinations, or even larger considering limits of their linear combinations. But the issue is in the human readibility as well as computer calculability of these families of functions.
Also there should be a consistent method for calculating the coefficients of the expansion. So if you are trying to express curves around the origin as sum of ellipses, you'd better make an educated guess how to calculate the coefficients of your expansion, and confirm it by writing software.
These are the reasons why the two well known families are interesting to us, human mathematicians, and we actually aren't interested into many other possible families.
answered Dec 17 '18 at 8:49
MuckoMucko
11
$endgroup$
$begingroup$
Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
add a comment |
$begingroup$
Some 20 years ago, when I was a math student, I too saw the good properties of the families of polynomial functions $f_n(x){=x^{n}}$ (for expansion into Taylor series) the and the sinusoidal functions (for expansion into Fourier series), so I asked myself the same question. As my career didn't proceed deeper into mathematics, I didn't find an elaborated answer to it.
Actually there might be plenty of families of functions that yield another (larger) family of functions as their linear combinations, or even larger considering limits of their linear combinations. But the issue is in the human readibility as well as computer calculability of these families of functions.
Also there should be a consistent method for calculating the coefficients of the expansion. So if you are trying to express curves around the origin as sum of ellipses, you'd better make an educated guess how to calculate the coefficients of your expansion, and confirm it by writing software.
These are the reasons why the two well known families are interesting to us, human mathematicians, and we actually aren't interested into many other possible families.
answered Dec 17 '18 at 8:49
MuckoMucko
11
$endgroup$
$begingroup$
Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
add a comment |
$begingroup$
Some 20 years ago, when I was a math student, I too saw the good properties of the families of polynomial functions $f_n(x){=x^{n}}$ (for expansion into Taylor series) the and the sinusoidal functions (for expansion into Fourier series), so I asked myself the same question. As my career didn't proceed deeper into mathematics, I didn't find an elaborated answer to it.
Actually there might be plenty of families of functions that yield another (larger) family of functions as their linear combinations, or even larger considering limits of their linear combinations. But the issue is in the human readibility as well as computer calculability of these families of functions.
Also there should be a consistent method for calculating the coefficients of the expansion. So if you are trying to express curves around the origin as sum of ellipses, you'd better make an educated guess how to calculate the coefficients of your expansion, and confirm it by writing software.
These are the reasons why the two well known families are interesting to us, human mathematicians, and we actually aren't interested into many other possible families.
answered Dec 17 '18 at 8:49
MuckoMucko
11
$endgroup$
Some 20 years ago, when I was a math student, I too saw the good properties of the families of polynomial functions $f_n(x){=x^{n}}$ (for expansion into Taylor series) the and the sinusoidal functions (for expansion into Fourier series), so I asked myself the same question. As my career didn't proceed deeper into mathematics, I didn't find an elaborated answer to it.
Actually there might be plenty of families of functions that yield another (larger) family of functions as their linear combinations, or even larger considering limits of their linear combinations. But the issue is in the human readibility as well as computer calculability of these families of functions.
Also there should be a consistent method for calculating the coefficients of the expansion. So if you are trying to express curves around the origin as sum of ellipses, you'd better make an educated guess how to calculate the coefficients of your expansion, and confirm it by writing software.
These are the reasons why the two well known families are interesting to us, human mathematicians, and we actually aren't interested into many other possible families.
answered Dec 17 '18 at 8:49
MuckoMucko
11
answered Dec 17 '18 at 8:49
MuckoMucko
11
answered Dec 17 '18 at 8:49
MuckoMucko
11
answered Dec 17 '18 at 8:49
MuckoMucko
11
11
$begingroup$
Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
add a comment |
$begingroup$
Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
Is there a way to show that the elliptical expansion can't be done systematically?
$endgroup$
– arax
Dec 23 '18 at 1:04
add a comment |
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$begingroup$
You can start by showing their $L_2$ completeness..
$endgroup$
– mm-crj
Dec 17 '18 at 7:57
$begingroup$
@mm-crj Could you elaborate or point me to some reading materials? I'm not really familiar with the mathematical "completeness". Thanks!
$endgroup$
– arax
Dec 23 '18 at 1:04
$begingroup$
en.m.wikipedia.org/wiki/Complete_metric_space This is what I was talking about. Also this en.m.wikipedia.org/wiki/Function_space
$endgroup$
– mm-crj
Dec 23 '18 at 7:12