I'm honestly very impressed by the concise nature of your videos and I'm glad that you make these videos accessible on TH-cam to a large audience. With a bit of will power and a desire to learn anybody with a sufficient mathematical background can have a better grasp on the extremely powerful tools of harmonic analysis. Great work. I honestly wish my instructors at university had been as understandable.
@@Eigensteve Seriously. I too wish my instructors had put in half as much effort in being concise and coherent as you have. I've been watching your videos in order for about a week now.
I've finished my university degree 31 years ago. I wish I had this quality of explanation available during my education! Amazing! Thank you!!! His book is excellent as well! Consider it a good investment in your education.
This is one of the best series of lectures! Question professor: 1) what's the meaning of delta omega? Pi/infinity = 0 here. especially at 7:46 based on what it came with dw integral range from negative infinity to infinity? from omega definition I don't see this integral domain. 2) Shouldn't it be delta K when you converge your first summation equation to integral? Hope you can help illustrate! Thank you very much in advance!
In the Complex Fourier Series video ψₖ was defined as eᶦᵏˣ, I understand that π/L was introduced in for frequency but why did the exponent become negative. It would become more clear to me if someone could explain the general formula for the inner product with period L rather than 2π ( I don't believe he produced this formula in the Complex Fourier Series video).
this is extremely important also in quantum mechanics! the difference between “free particle” (such as photons) with continuous spectrum and quantized particle with discrete spectrum within potential well is connected to Fourier Transform and Fourier Series.. super important
Wow, I've heard a lot of explanations and derivations for FT but this by far takes the cake, and I'm not someone who's into pointless compliments but this is really worth it, thank you and please keep up the great work👍
I'm amazed that you pronounced ξ as it's supposed to be pronounced. I'm Greek and have studied and lived in both the UK and Australia for long enough to have heard the Greek alphabet being massacred in all sorts of ways hahahah. Props to ya and your amazing lectures :)
The summation that you turned into an integral was k=-inf to inf ....but you wrote the integral wrt d(dummy variable) and not wrt dk. Could you clarify?
I am very confused in your derivation as to why you replace all of the k∆w in the summation with w in the integral. If anyone could help me understand this I would very much appreciate it.
He at the beginning took w_k = k(pi)/L, then he wrote it as kΔw. Where Δw= PI/L. In the limit of Δw -> 0, inside the integral, he easily replaced w_k from the first equation. You can also write it as w only in the continuous limit.
I must agree with some other comments... Gibbs phenomen does not disappear when you go to infinite number of terms but if I remember correctly tends to a constant value.
Steve, wish you and your family happy holiday! If you have time, want to bother you one question, but no rush. 7:19 - 8:37 from summation to integral. I am struggling to understand 1) why delta_omega = Pi/L, 2) why the summation is over K (the frequency), but integral is over omega? I thought the idea is to transfer to continuous frequency basis K, shouldn't it be something like delta_omega = delta_K * Pi / L ? How did you come up with omega which kind of wrap up K inside omega. Thank you so much!
This video is very insightful, however, I don't understand how the Fourier transform can represent continuous Fourier coefficients if the term Delta Omega / 2π is omitted or used by the inverse transform? Why is this possible?
Hi Steve, I've worked with the Fourier transform for years, but I just realized that I don't understand something. Please, consider your triangular function f(x). Suppose I multiply your f(x) by a factor B. Suppose I want a transform that in independent of B. In other words, I want a transform that is independent of a scaling factor. Does that make sense? I have a physics research situation where I believe it should make sense. How would you define a transform that is independent of a scaling factor? Thanks.
Hello, thank you for the video. Can anyone tell where can i find a more specific explanation of the step using the Riemann sum? I do not understand it. Thanks.
Same question here. The part of taking the Riemann integral makes sense on the expression as a whole, but I'm confused as to what justifies that substitution (hiding the fact that the delta omega has shrunk to zero). Probably has to do with the fact that the k is infinetly summed (and then transformed to Riemann integral).
The Riemann integral is simply defined by the limit of a Riemann Sum as the delta variable approaches zero. The instructor just recognized the form and replaced it with the integral notation. Not sure if that answers your question.
@@NoNTr1v1aL Your question is different from what Dr. Brunton did: like Kyle said, he's just recognizing a Riemann sum that already exists, he's not integrating the Fourier series. It's just a definition: as L -> inf, the series turns into an integral. Your question is different and gets into deep waters. The short answer is: if f is "nice enough" (e.g. C^inf smooth.) The long answer: this is the entire subject of Fourier and harmonic analysis. In general, integrating and differentiating series term by term depends on the analytic properties of the f being expanded, the types of convergence (uniform, pointwise, etc.) and it only gets more subtle with Fourier. The Gibbs phenomenon is already an example of this: a jump discontinuity in f screws up uniform convergence of the fourier series, and if you remember, Dr. Brunton mentioned that the "top hat" function was related to the derivative of the "triangle hat" function (which was continuous, but not continuously differentiable), so already you can see some of these subtleties creeping in. Check out Stein and Sharkachi's books, Tao's books, etc.
Sorry to hear that, but don’t despair! A little more background in linear algebra and vectors would likely help. You could check out my courses to see where this fits in: faculty.washington.edu/sbrunton/me564/ and faculty.washington.edu/sbrunton/me565/
I miss from most lectures like this a more precise definition of what you mean in this context by taking the limit as the period tends to infinity. If you let 'N' be the absolute value of the upper and lower bound of sum index 'n' then what you get to see is both N and the period tend to infinity. It is very hard to think about how this all converges because of the 2 variables. If I wanted, I could always choose a larger period and a larger 'N' so that the frequency domain passed on to the Fourier Transform Function will not increase. You need to imagine the tendency of the period and 'N' so that you always get a larger AND denser frequency domain so that your sum can be seen as a Riemann sum and it indeed converges to the limit defined by the final inverse integral from minus infinity to infinity. If you take in this very sense "take limit as period tends to infinity" then yes you must obtain the Fourier Transform but it is not straightforward.
If you need brain surgery, let's hope your guy is as good as Dr. Brunton. It could get a little hairy, but he will be able to pull you through the complex stuff. Hmmm, I think I may have constructed an unintended pun here.......
I'm honestly very impressed by the concise nature of your videos and I'm glad that you make these videos accessible on TH-cam to a large audience. With a bit of will power and a desire to learn anybody with a sufficient mathematical background can have a better grasp on the extremely powerful tools of harmonic analysis. Great work. I honestly wish my instructors at university had been as understandable.
Thank you!!
@@Eigensteve Seriously. I too wish my instructors had put in half as much effort in being concise and coherent as you have. I've been watching your videos in order for about a week now.
I never thought of actually deriving the Fourier transform this way. This is amazing.
Glad you liked it!!
I've finished my university degree 31 years ago. I wish I had this quality of explanation available during my education! Amazing! Thank you!!! His book is excellent as well! Consider it a good investment in your education.
I don't usually comment on videos but I just want to let you know how amazing this series has been. Thank you!
i regret coming across this channel so late. it's just perfect. i am learning a lot! thanksss!
Happy to hear it :)
Finally the chance to commentate under your video: Thanks for the awesome content!
You're much better than my lecturer.. I dont get how you do better in 15 minutes than he does in 2 hours. Crazy.
This is one of the best series of lectures! Question professor:
1) what's the meaning of delta omega? Pi/infinity = 0 here. especially at 7:46 based on what it came with dw integral range from negative infinity to infinity? from omega definition I don't see this integral domain.
2) Shouldn't it be delta K when you converge your first summation equation to integral?
Hope you can help illustrate! Thank you very much in advance!
It's very nice to see Steve makes very technical videos that do really well. Gives me some hope for my vids :)
This playlist is really amazing.
Thanks!
This an amazing series. Thank you so much ❤️❤️
Glad you enjoy it!
In the Complex Fourier Series video ψₖ was defined as eᶦᵏˣ, I understand that π/L was introduced in for frequency but why did the exponent become negative. It would become more clear to me if someone could explain the general formula for the inner product with period L rather than 2π ( I don't believe he produced this formula in the Complex Fourier Series video).
this is extremely important also in quantum mechanics! the difference between “free particle” (such as photons) with continuous spectrum and quantized particle with discrete spectrum within potential well is connected to Fourier Transform and Fourier Series.. super important
I have watched your videos, they are really good explained,I haven't seen such a good explanation of this topic so far.
Wow, I've heard a lot of explanations and derivations for FT but this by far takes the cake, and I'm not someone who's into pointless compliments but this is really worth it, thank you and please keep up the great work👍
I really liked your lectures! Very clear and easy to understand. Thanks!
I'm amazed that you pronounced ξ as it's supposed to be pronounced. I'm Greek and have studied and lived in both the UK and Australia for long enough to have heard the Greek alphabet being massacred in all sorts of ways hahahah. Props to ya and your amazing lectures :)
The summation that you turned into an integral was k=-inf to inf ....but you wrote the integral wrt d(dummy variable) and not wrt dk. Could you clarify?
Thank you for this great explanation! Greetings from germany
Hallo! Which city?
at timestep 3:03 shuouldn't Ck formula have e^i in positive sign?
I am very confused in your derivation as to why you replace all of the k∆w in the summation with w in the integral. If anyone could help me understand this I would very much appreciate it.
He at the beginning took w_k = k(pi)/L, then he wrote it as kΔw. Where Δw= PI/L. In the limit of Δw -> 0, inside the integral, he easily replaced w_k from the first equation. You can also write it as w only in the continuous limit.
"Welcome back" gets me everytime :D
I must agree with some other comments... Gibbs phenomen does not disappear when you go to infinite number of terms but if I remember correctly tends to a constant value.
Thank you, this was a great video!
Also, just realized you've been writing backwards... That's impressive.
Amazing, thank you for this lecture!
You're welcome!
really good video! streight to the point, quick and easy to understand!
Steve, wish you and your family happy holiday! If you have time, want to bother you one question, but no rush. 7:19 - 8:37 from summation to integral. I am struggling to understand 1) why delta_omega = Pi/L, 2) why the summation is over K (the frequency), but integral is over omega? I thought the idea is to transfer to continuous frequency basis K, shouldn't it be something like delta_omega = delta_K * Pi / L ? How did you come up with omega which kind of wrap up K inside omega. Thank you so much!
What is the technology behind this transparent mirrored board? What is his using exactly? Thx, excellent video btw
Thank you for the video but i have this question is pi/0 considered infinity or unidentified?
Ok, it helps a little bit, which is a lot, but every time you say ok I have so very many questions. So many.
Do magnitude and phase for particular frequency in continuous frequency spectrum represent exact magnitude and phase of sinusoid with this frequency?
Is a Fourier transform of f(x) = x worth evaluating? It shouldn't give you any Frequency, should it?
This is extremely helpful, thank you!
Shouldn't the second equation on the board be c_k = 1/2L , rather than 1/2pi as we're on the domain [-L,L] rather than [-pi,pi] ?
yes it had to be so!
Indeed!
This video is very insightful, however, I don't understand how the Fourier transform can represent continuous Fourier coefficients if the term Delta Omega / 2π is omitted or used by the inverse transform? Why is this possible?
Hi Steve, I've worked with the Fourier transform for years, but I just realized that I don't understand something. Please, consider your triangular function f(x). Suppose I multiply your f(x) by a factor B. Suppose I want a transform that in independent of B. In other words, I want a transform that is independent of a scaling factor. Does that make sense? I have a physics research situation where I believe it should make sense. How would you define a transform that is independent of a scaling factor? Thanks.
Hello, thank you for the video. Can anyone tell where can i find a more specific explanation of the step using the Riemann sum? I do not understand it. Thanks.
Riemann sum is just when you sum something that's infinitesimal that it becomes an integral
Sir, what is the difference between π/L and κ in e^ikπx/L?
Aren't they both supposed to represent angular frequency?
Really amazing explanation wow.
I really enjoyed learning from you....thanks sir
From a physical standpoint since omega and time are conjugates, perhaps it would have been better to use “t” rather than “x”.
Very precise and nice explanation ...
How does he write k times delta omega as omega?
Same question here. The part of taking the Riemann integral makes sense on the expression as a whole, but I'm confused as to what justifies that substitution (hiding the fact that the delta omega has shrunk to zero). Probably has to do with the fact that the k is infinetly summed (and then transformed to Riemann integral).
When can you interchange summation and integral in Fourier series?
The Riemann integral is simply defined by the limit of a Riemann Sum as the delta variable approaches zero. The instructor just recognized the form and replaced it with the integral notation. Not sure if that answers your question.
@@kylegreen5600 I don't understand. When can you integrate a Fourier series term by term?
@@NoNTr1v1aL Could you respond with a rough time in the video where you're not following the steps and I'll try to help.
@@NoNTr1v1aL Your question is different from what Dr. Brunton did: like Kyle said, he's just recognizing a Riemann sum that already exists, he's not integrating the Fourier series. It's just a definition: as L -> inf, the series turns into an integral.
Your question is different and gets into deep waters. The short answer is: if f is "nice enough" (e.g. C^inf smooth.) The long answer: this is the entire subject of Fourier and harmonic analysis. In general, integrating and differentiating series term by term depends on the analytic properties of the f being expanded, the types of convergence (uniform, pointwise, etc.) and it only gets more subtle with Fourier. The Gibbs phenomenon is already an example of this: a jump discontinuity in f screws up uniform convergence of the fourier series, and if you remember, Dr. Brunton mentioned that the "top hat" function was related to the derivative of the "triangle hat" function (which was continuous, but not continuously differentiable), so already you can see some of these subtleties creeping in. Check out Stein and Sharkachi's books, Tao's books, etc.
when the integrable function is uniformly continuous, then we can do like this interchanging.
Omg ! We have the same name ! I am so happy
Awesome explanation
:( I did not get it. Any background I need to understand this topic? this guy seems to explain clearly but not clear for me
Sorry to hear that, but don’t despair! A little more background in linear algebra and vectors would likely help. You could check out my courses to see where this fits in: faculty.washington.edu/sbrunton/me564/ and faculty.washington.edu/sbrunton/me565/
@@Eigensteve Thanks man!
Thanks from India
I miss from most lectures like this a more precise definition of what you mean in this context by taking the limit as the period tends to infinity. If you let 'N' be the absolute value of the upper and lower bound of sum index 'n' then what you get to see is both N and the period tend to infinity. It is very hard to think about how this all converges because of the 2 variables. If I wanted, I could always choose a larger period and a larger 'N' so that the frequency domain passed on to the Fourier Transform Function will not increase. You need to imagine the tendency of the period and 'N' so that you always get a larger AND denser frequency domain so that your sum can be seen as a Riemann sum and it indeed converges to the limit defined by the final inverse integral from minus infinity to infinity. If you take in this very sense "take limit as period tends to infinity" then yes you must obtain the Fourier Transform but it is not straightforward.
Great video. Thanks
... is that backwards in your perspective
Amazing video, thank you!
beautiful ! beautiful!
BRILLIANT😍
thanks for ur explaination
Very good
thank you!!
thank you
Thanks so much
9:10
now I got it ha... thank you !
I want to focus on the video, but all I'm thinking about is how he is writing on the board backwards with perfection...
bro wtf its just mirrored camera, of course he's writing in the right direction
@@alegian7934 jokes
@@Thespookygoat oof I wooshed rather hard there... sry this is math after all
Deviation
my brain is melting
The first five seconds of this video made me realize that CBD actually makes u high
This is legit.
Contdsir
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arifin manchestera attığı golü arıyodum buraya nasıl geldim amk
23k people view this but only 500 likes. This is why we have a pandemic.
If you need brain surgery, let's hope your guy is as good as Dr. Brunton. It could get a little hairy, but he will be able to pull you through the complex stuff. Hmmm, I think I may have constructed an unintended pun here.......
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This is some first-rate BS.
??