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Sounds like trying to tell someone what music sounds like, when there's a piano right in front of you. How 'bout a formula or two? Perhaps the Fourier transform? User:Kevin_baas -2003.05.06
I didn't understand the part about the basis functions. The article says that any normal function can be decomposed into an infinite series of sines and / or cosines. Does it mean that each sine term is a basis function for the original function? In other words (like we have n-dimensional vectors), is the function in an infinite-dimensional space?
Forgive my lack of mathematical rigour; I'm just in high school!
Gokul
The infinite-dimensional space is the set of all "quadratically integrable" functions, i.e., those satisfying
The functions sin(nx), cos(nx) for an "orthogonal basis", but not a Hamel basis. That it is not a Hamel basis means that not every quadratically integrable function is a linear combination of finitely many basis functions. (Some people say that phrase is a redundancy--that "linear combination" by definition means just finitely many. If so, that's why redundancy is sometimes useful!) Michael Hardy 20:31, 15 Nov 2003 (UTC)
Regarding that last edit by Wile, I don't see how it improved the article. It seems to me have made it less readable. What do other people think? -- Kevin Baas 08:58, 14 Jan 2004 (UTC)
In reference to other edits since the one I made -- this link [1] should yield the appropriate diff (at least until the page is changed again). The more recent edits seem problematic. The first paragraph says the FT "can be thought of" or "may be thought of" in a couple of ways: that is not appropriate for an encyclopedia article. First we have to say, definitely, what it is. Then we can talk about how that is interpreted or applied. Also, under the heading "Generalizations" some stuff has been cut out, including the definition of A and B, now orphaned in the article. I agree that section can be clarified but information has been lost. I'm inclined to revert some of these changes; I'll do so in a few days unless I hear otherwise. Wile E. Heresiarch 06:21, 15 Jan 2004 (UTC)
It is to be regretted that discrete and other non-continuous Fourier transforms have been purged from this page. Michael Hardy 01:22, 17 Jan 2004 (UTC)
This part here, "The real parts of the resulting complex-valued function F represent the amplitudes of their respective frequencies (s), while the imaginary parts represent the phase shifts." I don't believe is correct.. Isn't it the case that the Absolution value of the complex value is the magnitude, i.e. Mag = sqrt(Re^2+Im^2) while the phase is given by arctan(im/re)?
I think that, originally, Fourier transform was meant to discuss the general idea of Fourier transforms, and continuous Fourier transform was meant to discuss that particular case. However, as it stands, Fourier transform is now mostly about the continuous Fourier transform, and there is a tremendous amount of redundancy. I would suggest merging a lot of the present material into continuous Fourier transform. —Steven G. Johnson 00:35, May 19, 2004 (UTC)
The Wikipedia article on the Plancherel theorem and the one in MathWorld seem to be completely different. The Wikipedia one states a theorem about the Fourier transform being in L2. The MathWorld one states a specific identity that is a generalization of Parseval's theorem. The reference books I have on my desk don't even mention Plancherel, but I wonder if there aren't two completely different things, a Plancherel theorem about L2-ness and unitarity of the Fourier transform, and a Plancherel identity referenced on MathWorld. Could someone please look into this? (And who came first, Plancherel or Parseval?) And please give references; neither the Wikipedia nor the MathWorld articles cite anything for the Plancherel theorem. —Steven G. Johnson 21:42, May 19, 2004 (UTC)
The following books call it Plancherel:
It seems to be Math vs Physics and engineering....
CSTAR 01:18, 20 May 2004 (UTC)
Update: I did some checking, and it seems that
Parseval's theorem pre-dates Plancherel (and, in fact, predates Fourier transforms), but is also less general; it was a result about series that was used to prove "unitarity" of the Fourier series. I'm guessing that this historical precedence is why the name continues to be attached to all unitarity properties of Fourier transforms, at least in physics and engineering.
—Steven G. Johnson 02:26, Jun 6, 2004 (UTC)
I've just edited the properties of the Fourier Transform as I found there was something wrong with the convolution theorem : to be coherent with the previous definition of the trasform, the factor must be added. I've also written something about the important derivation property..very useful in many fields of pure and applied science. -- Giuscarl 20:22, 4 Jun 2004 (UTC)
Fourier transform is a decomposition into functions having a very specific form. Mentioning waves is misleading and should be changed..also sinusoidal should be replaced by something like complex exponential.
CSTAR 01:29, 11 Jun 2004 (UTC)
How does this sound?
or some combination thereof.
CSTAR 16:26, 11 Jun 2004 (UTC)
OK OK, it looks fine now. Thanks. CSTAR 23:12, 11 Jun 2004 (UTC)
I don't know where else to ask. What's the difference between and . Are they used correctly in this article? - Omegatron 01:56, Sep 19, 2004 (UTC)
Yeah. If you're picky you will note that t has no real purpose in the above formula. It really should be within the scope of a binding operator such as .; this however by rules of lambda-calclus reduces to the term f. CSTAR 02:47, 19 Sep 2004 (UTC)
I'm used to engineering notation, where , so you can use f for frequency and avoid confusing Fs. Then of course there's . :-) - Omegatron 02:40, Sep 19, 2004 (UTC)
In fact, I vote that we change f(t)->F(ω) into some other letter (I know you won't use X, but maybe g?), to avoid confusing newcomers to the Fourier transform. - Omegatron 02:42, Sep 19, 2004 (UTC)
One thing the "engineering notation" does not allow for is the idea that functions have values. E.g., if f(x) = x3 for all values of x, then f(2) is the value of that function at 2, and is equal to 8. If you say f(ω) is the function to be transformed, and g(t) is the transformed function, then g(2) should be the value of the transformed function at the point t = 2. But if you use the "engineers' notation" and write , then you cannot plug 2 into the left side. But watch this: is the result of plugging 2 into the transformed function.
One thing to be said for the difficulties introduced by the engineers' notation that are avoided by the cleaner, simpler, but more abstract "mathematicians' notation", is that perhaps sometimes one ought not to be evaluating these functions pointwise anyway! But that's a slightly bigger can of worms than what I want to open at this moment ... Michael Hardy 22:38, 19 Sep 2004 (UTC)
The comment(s) below were originally left at Talk:Fourier transform/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Comment(s) | Press [show] to view → |
---|---|
Should add history and expand on applications (inside maths as well as elsewhere).
Probably too much space is taken by the tables of explicit transforms — as such tables can be useful reference, they should perhaps be preserved in a linked page. There is atill clearly an issue (despite valiant efforts to sort it out, as evidenced by the discussion on this talk page!) with the overall structure of the Fourier-related articles. I'm afraid that most others than contributing editors are easily lost in the various articles and their interrelations. As an exmaple, where to find the (still quite elementary) Fourier transform of a function on Rn? As a solution, this page should provide more overview and links to details elsewhere. A compromise between too much and too little generality, a possibility would be to treat the Rn and (multi-)periodic cases on this page (with an informal comment on the relevant and quite straightforward Pontryagin duals), and link to Harmonic analysis (to be expanded) for a treatment of the abstract case. The role of tempered distributions could be alluded to here and treated properly in Harmonic analysis. Stca74 22:26, 14 May 2007 (UTC) It's not clear whether this is an "Applications of Maths" article or a pure Maths article. In Optics it's true that you can transform between frequency(time) and position domains. etc... A link should be inserted directing the user to "Applications of Fourier Transforms" or the to the actual theory behind all this (not hard - it's simpler than Laplace transforms!). JeremyBoden ( talk) 20:18, 16 August 2008 (UTC) |
Last edited at 20:18, 16 August 2008 (UTC). Substituted at 21:15, 4 May 2016 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | → | Archive 5 |
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | → | Archive 5 |
Sounds like trying to tell someone what music sounds like, when there's a piano right in front of you. How 'bout a formula or two? Perhaps the Fourier transform? User:Kevin_baas -2003.05.06
I didn't understand the part about the basis functions. The article says that any normal function can be decomposed into an infinite series of sines and / or cosines. Does it mean that each sine term is a basis function for the original function? In other words (like we have n-dimensional vectors), is the function in an infinite-dimensional space?
Forgive my lack of mathematical rigour; I'm just in high school!
Gokul
The infinite-dimensional space is the set of all "quadratically integrable" functions, i.e., those satisfying
The functions sin(nx), cos(nx) for an "orthogonal basis", but not a Hamel basis. That it is not a Hamel basis means that not every quadratically integrable function is a linear combination of finitely many basis functions. (Some people say that phrase is a redundancy--that "linear combination" by definition means just finitely many. If so, that's why redundancy is sometimes useful!) Michael Hardy 20:31, 15 Nov 2003 (UTC)
Regarding that last edit by Wile, I don't see how it improved the article. It seems to me have made it less readable. What do other people think? -- Kevin Baas 08:58, 14 Jan 2004 (UTC)
In reference to other edits since the one I made -- this link [1] should yield the appropriate diff (at least until the page is changed again). The more recent edits seem problematic. The first paragraph says the FT "can be thought of" or "may be thought of" in a couple of ways: that is not appropriate for an encyclopedia article. First we have to say, definitely, what it is. Then we can talk about how that is interpreted or applied. Also, under the heading "Generalizations" some stuff has been cut out, including the definition of A and B, now orphaned in the article. I agree that section can be clarified but information has been lost. I'm inclined to revert some of these changes; I'll do so in a few days unless I hear otherwise. Wile E. Heresiarch 06:21, 15 Jan 2004 (UTC)
It is to be regretted that discrete and other non-continuous Fourier transforms have been purged from this page. Michael Hardy 01:22, 17 Jan 2004 (UTC)
This part here, "The real parts of the resulting complex-valued function F represent the amplitudes of their respective frequencies (s), while the imaginary parts represent the phase shifts." I don't believe is correct.. Isn't it the case that the Absolution value of the complex value is the magnitude, i.e. Mag = sqrt(Re^2+Im^2) while the phase is given by arctan(im/re)?
I think that, originally, Fourier transform was meant to discuss the general idea of Fourier transforms, and continuous Fourier transform was meant to discuss that particular case. However, as it stands, Fourier transform is now mostly about the continuous Fourier transform, and there is a tremendous amount of redundancy. I would suggest merging a lot of the present material into continuous Fourier transform. —Steven G. Johnson 00:35, May 19, 2004 (UTC)
The Wikipedia article on the Plancherel theorem and the one in MathWorld seem to be completely different. The Wikipedia one states a theorem about the Fourier transform being in L2. The MathWorld one states a specific identity that is a generalization of Parseval's theorem. The reference books I have on my desk don't even mention Plancherel, but I wonder if there aren't two completely different things, a Plancherel theorem about L2-ness and unitarity of the Fourier transform, and a Plancherel identity referenced on MathWorld. Could someone please look into this? (And who came first, Plancherel or Parseval?) And please give references; neither the Wikipedia nor the MathWorld articles cite anything for the Plancherel theorem. —Steven G. Johnson 21:42, May 19, 2004 (UTC)
The following books call it Plancherel:
It seems to be Math vs Physics and engineering....
CSTAR 01:18, 20 May 2004 (UTC)
Update: I did some checking, and it seems that
Parseval's theorem pre-dates Plancherel (and, in fact, predates Fourier transforms), but is also less general; it was a result about series that was used to prove "unitarity" of the Fourier series. I'm guessing that this historical precedence is why the name continues to be attached to all unitarity properties of Fourier transforms, at least in physics and engineering.
—Steven G. Johnson 02:26, Jun 6, 2004 (UTC)
I've just edited the properties of the Fourier Transform as I found there was something wrong with the convolution theorem : to be coherent with the previous definition of the trasform, the factor must be added. I've also written something about the important derivation property..very useful in many fields of pure and applied science. -- Giuscarl 20:22, 4 Jun 2004 (UTC)
Fourier transform is a decomposition into functions having a very specific form. Mentioning waves is misleading and should be changed..also sinusoidal should be replaced by something like complex exponential.
CSTAR 01:29, 11 Jun 2004 (UTC)
How does this sound?
or some combination thereof.
CSTAR 16:26, 11 Jun 2004 (UTC)
OK OK, it looks fine now. Thanks. CSTAR 23:12, 11 Jun 2004 (UTC)
I don't know where else to ask. What's the difference between and . Are they used correctly in this article? - Omegatron 01:56, Sep 19, 2004 (UTC)
Yeah. If you're picky you will note that t has no real purpose in the above formula. It really should be within the scope of a binding operator such as .; this however by rules of lambda-calclus reduces to the term f. CSTAR 02:47, 19 Sep 2004 (UTC)
I'm used to engineering notation, where , so you can use f for frequency and avoid confusing Fs. Then of course there's . :-) - Omegatron 02:40, Sep 19, 2004 (UTC)
In fact, I vote that we change f(t)->F(ω) into some other letter (I know you won't use X, but maybe g?), to avoid confusing newcomers to the Fourier transform. - Omegatron 02:42, Sep 19, 2004 (UTC)
One thing the "engineering notation" does not allow for is the idea that functions have values. E.g., if f(x) = x3 for all values of x, then f(2) is the value of that function at 2, and is equal to 8. If you say f(ω) is the function to be transformed, and g(t) is the transformed function, then g(2) should be the value of the transformed function at the point t = 2. But if you use the "engineers' notation" and write , then you cannot plug 2 into the left side. But watch this: is the result of plugging 2 into the transformed function.
One thing to be said for the difficulties introduced by the engineers' notation that are avoided by the cleaner, simpler, but more abstract "mathematicians' notation", is that perhaps sometimes one ought not to be evaluating these functions pointwise anyway! But that's a slightly bigger can of worms than what I want to open at this moment ... Michael Hardy 22:38, 19 Sep 2004 (UTC)
The comment(s) below were originally left at Talk:Fourier transform/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Comment(s) | Press [show] to view → |
---|---|
Should add history and expand on applications (inside maths as well as elsewhere).
Probably too much space is taken by the tables of explicit transforms — as such tables can be useful reference, they should perhaps be preserved in a linked page. There is atill clearly an issue (despite valiant efforts to sort it out, as evidenced by the discussion on this talk page!) with the overall structure of the Fourier-related articles. I'm afraid that most others than contributing editors are easily lost in the various articles and their interrelations. As an exmaple, where to find the (still quite elementary) Fourier transform of a function on Rn? As a solution, this page should provide more overview and links to details elsewhere. A compromise between too much and too little generality, a possibility would be to treat the Rn and (multi-)periodic cases on this page (with an informal comment on the relevant and quite straightforward Pontryagin duals), and link to Harmonic analysis (to be expanded) for a treatment of the abstract case. The role of tempered distributions could be alluded to here and treated properly in Harmonic analysis. Stca74 22:26, 14 May 2007 (UTC) It's not clear whether this is an "Applications of Maths" article or a pure Maths article. In Optics it's true that you can transform between frequency(time) and position domains. etc... A link should be inserted directing the user to "Applications of Fourier Transforms" or the to the actual theory behind all this (not hard - it's simpler than Laplace transforms!). JeremyBoden ( talk) 20:18, 16 August 2008 (UTC) |
Last edited at 20:18, 16 August 2008 (UTC). Substituted at 21:15, 4 May 2016 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | → | Archive 5 |