Fourier Series: Part 1 

Fourier series as inner products and projections onto orthogonal basis. I'm in awe! Can't thank you enough Prof. Steve Brunton!

It makes a lot of sense to think about the coefficient A_k and B_k as projections on the orthogonal basis formed by cos and sine. Never thought about it this way ;) it always good to learn and relearn the same concept from different perspectives.

I learned Fourier Transforms 5 years ago and today is the first day I fully understand it from the vector space point of view. Thank you so much.

If there is such a thing as pedagogical wizardry you're witnessing it. This channel is superbly good.

After all the calculus, physics, signals and systems courses I have taken, the only thing I understood is how to do them and not what they actually mean. I was worried about not getting answers from the books, online, or my professors because I want to teach in the future. This is a game changer and was like inventing the light bulb to me. I can't thank you enough for not only explaining it in detail, but doing it in a simple and understandable way.

Thinking of sine and cosine as basis, function space is span ( sine , cosine). This really make sense when I was trying to gain a solid understanding of Fourier ‘ s method. Thank you very much for your instruction video.

The best Fourier Analysis videos taught by a floating torso on RU-vid. Thank you Steve. You've made this topic much easier to understand.

Suddenly these formulas appear to be very simple and I'll probably never forget them. Thank you very much.

I just got the book and I'm starting Chapter 2 with a major head scratch. Then I watch this video and it's like taking a blindfold off. Thank you, Steve!!

That's literally the best explanation I've ever heard about Fourier series. Thank you!

I have read dozens of books and viewed dozens of videos on this topic. This series is the only one that marries intuition with process and procedures and explain why. Congrats. I highly recommend to everyone.

I've just found out about this awesome channel. You sir are wonderful. Please don't stop the good work 😉

Hi Steve! I've been a sound designer working and building with Fourier EQ's and Compressors for years and years BUT never has someone explained it to me like this. I can't thank you enough. I feel like I get my job and my art now.... and it feels great.

I'm not trying to impress anyone, but after a lot of thinking over the years (I'm a game programmer) I came to this idea on my own but I was never sure if I was correct. This is the first time I've seen a presentation that clearly confirms what I had conjectured. I'm learning a lot of details now. I will continue on with this series as I am quite curious to know more and confident that I can understand your presentation format.

What a great interpretation of Fourier series with vector spaces, I am proud to say I do follow quite a few channels and your's work is best among them.

This is the most simple, elegant and beautiful explanation (IF YOU KNOW LINEAR ALGEBRA) I've ever seen. Thank you Prof. Brunton.

The analogue for projecting f onto orthogonal vector basis to explain projecting f onto orthogonal functions was really helpful and made everything click for me. Thanks!

Mr. Brunton. Thank you for clear, concise, organized presentation of DFT. Appreciative of how much time and effort such a presentation / explanation takes to create and deliver. Appreciative of the format you use and precision in getting explanation correct. Explanation of terms and where terms originate has always been helpful in your presentations. Going through the whole DFT, FFT series again to refresh my thinking on the topics. Thanks again. (Erik Gottlieb)

What a great and clear explanation. I graduated this past spring with my B.S. and am currently working now. I wanted to just brush up on this topic just for fun and I didn’t realize how unclear my professors were during university. This is definitely helping me regain the understanding of the fundamentals of FFT.

The intuition you provide regarding the inner products is incredible. So helpful in understanding this difficult topic. Wish you were my Prof. when I learned Fourier Series in class!

I'll join into the choir and just want to say that this is an amazing explanation! Thank you so much!

You made me fall in love with mathematics again. Great work, Thank You so much.

This guy is actually the best on RU-vid!

Man you make my mind blow ... please make a video on reference frame theory ....

Magníficas aulas do Prof. Brunton e seu excelente livro merece ser lido. Magnificent videos of Prof. Brunton and his excellent book deserves detailed reading.

Professor Steve Brunton, you are a GEM! I can't thank you enough.

I have never seen anybody so excited to teach something. Amazing

YES we get it Prof. Brunton! Fourier Series is writing f(x) as an orthogonal basis of sines and cosines of increasing frequencies. I will use that phrase instead of "The mitochondria is the powerhouse of the cell".

Thanks for explaining this, looking forward to the rest of the playlist!

After watch Steve's online videos. I don't know any reason to pay the tuition for Stanford. Actually your videos is the only reason I upgrade my RU-vid account to premium account. Really appreciate for what you are doing. So far you and Gilbert Strang are the source I got my math foundation in English. I got my education in China, Which didn't explain everything very well and all the math vocabulary is in Chinese, which make it extremely hard to read papers. Thank you again.

Thank you so much for sharing your knowledge in a more clear way and I loved the explanation.

you always have things to learn from you.thanks

Steven L. Brunton i love you, thank you for everything you saved some lives and you helped people being hired

Awesome intuition explained about Fourier transform! Very good video

Very helpful. I wish my other professors would have explained it this way... Thank you Steve.

Really thank you for these videos !

Thanks Professor, for such lucid presentations!

It's marvelous! I guess I finally undersand the core meaning of fourier series. It just an abstraction of coordinate system of orthogonal functions

Thank you for your thoughts and time

That's great explanation.. I think i will never forget that...

Thanks, papi. Completely forgot most of the EE stuff I learned from my CompE degree, this is a nice refresh without reopening my Analog signals & systems (Kudeki & Munson) book. Lol

Never heard FS taught like this! 😊 Great!

Absolutely mindblowing, thank you Steve.

I really like maths and machine learning and after long time trying, i finally understand fourier series with this video .Thank you so much!!. Greetings from Peru

Just found your channel, I must say you are Amazing MAN :)

This video was very helpful! Thank you so much!

Great finally understood fourier series with linear algebra I'm lost at ||1/cos(kx) ||^2 = 1/pi or the sin

Very well explained!!

Wow, I am working With FMCW radar and FT is the heart of its working theory and this video helps me a lot why Chirps is increasing linearly over the time with respect to band width woow thanks a lot !

Great channel and great explications. i love the way to explain the concepts of him.

Recently found this channel. Really liking your teaching. Can you make a video about lanczos algorithm?

Thank you for the amazing video! I have a question about normalizing inner product of with ||cos(kx)||^2, if resembling to vector space inner product, I would expect ||cos(kx)|| instead of ||cos(kx)||^2; could you give some hint about the source of power 2? it is possible to see that we are looking for a transformation that if a cos(kx) enters then the results should be 1 for the cofficient, then it makes sense. but to make an analogy with vector inner products, if it is not the unite vector, it is divided by L2 norm. Thanks in advance.

Even I understand this now! :-) Very fine explanation. Thanks.

Great video Steve!

World class teaching!

Great content

Nice presentation!

Amazing explanation, thank you !!

Hello, this is a great video. I want to know which references you use for this section, can you recommend me some books about Fourier series and Fourier transform? Thank you so much 😀

Amazing video!

at 8:15, should it be the norm's square or should it be the 2-norm(Euclidean norm)?

Nice explanation!

I could not understand the concept of orthogonal in a non vecter funtion all the last term. It is crazy explanation!

Great Video

Brilliant explanation

Fantastic series

Remind me I came up with an equation to solve when image is greater than one in unsupervised SVD for recreating an image with guassian noise. Though I will say, from memory, the mean and standard deviation of some images do not equal one, and therefore was confusing. So I placed guassian noise on your dog picture and recreate the image from that. Hence the whole variable being greater than one.

Excellent , thank you

you are so cool,I love your course

Wearing a black top behind a black screen, he is like a ghost with a head and two hands He is brilliant by the way, great explanation of Fourier series

sir ji ek number!

I understood for the first time, wonderful

Thanks ! Excellent!!

I love math so much, I might be changing it a lot with an idea I have... A new coordinate system based on koopman linear complex plain transformation, where our number line disappears completely into polar but gives a different view to enable the elimination of apparent randomness in several fields. I think I came up with something that if correct may be the most significant mathematical discovery in the century..... If I'm right, i could win a fields medal and more as well as crack all elliptical curve cartography and create a system that should be able to provide solutions to at least four of the millennium problems... Not bad for a guy who went to alternative school and hasn't finished traditional postsecondary. Thanks for the knowledge man!

love this "series" :)

This guy is a genius. The way he explains this complex concept by comparing it to linear algebra and projections is incredible and unique. Thank you very much!

Thank you.

Master piece ❤❤❤❤

This video is great! I love it! But one week earlier would be great, because I wrote an exam about it and I didn't went well.

very very good lecture

So why is A naught scaled by 1/2 ?

I might have missed it but when you scaled the A_0 term by 2 you mentioned we'd see why later, was this in reference to the constant outside of the integral for the Fourier coefficients? Just wanted to make sure I asked this to clarify why you decided to scale the A_0 naught term. Thanks for the awesome recap on Fourier Series!

Prof Steve,I clearly understood how Ak and Bk come but I wonder why Ak and Bk are again multiplying with sine and cosine to get f(x)?

Great great video

Thanks for the video but for unit vector,, shouldn't it be norm of x or y rather than norm of x or y squared?

Excellent video, thanks Steve ! Makes me wonder if the fourier series can be constructed with arbitrary functions as basis (instead of sine/cosine)?

Dear Profesor, thank you very much for your work. I would like to ask you, why is orthogonality of basis vector so important? In ME565 lectures you mentioned, that cosines are orthogonal to other cosines, sines to other sines but that they are not orthogonal to each toher (with coresponding frequencies). That is possible to be shown with inner product. But what is the final effect on expressing general functions as a sum of these basis fnctions? Why is orthogonality so important, since it is possible to express general vector as a sum of non orthogonal basis vectors? Once again, thank you very much. I do enjoy your lectures so much!

A little correction: the unit vector consists of the vector divided by its norm, not its norm squared.

You are a very good teacher, you know that!?

God I wish I didn't go to art school sometimes, this is so much more interesting than what I did in school :(

Eureka! I am getting it!

I am still trying to understand if you are writing backwards or the glass is converting it into backwards so can we see it normally

Funtastic!

Mr. Brunton What material and program did you use while shooting this video?

how do you write on a glass board where text appears visible to you as well as in the video?

Nice video

How come this channel was never recommended to me before, given I am subscribed to a number of popular math and science channels like 3b1b, blackpenredpen, numberphile, computerphile etc etc

I love you man hahahaha You always open my eyes to soooo much

the fourier weighs the contributing trigonometric functions by the value of f(x) at that particular value of frequency... sound like a weighted mean.. which makes sense.. as it the amount/quantity of the frequency's contribution to the overall response... could you do a series on the acutal derivation of the series, or if that has been done.. could someone point me in that direction. Always worthwhile for me, to get the development history..like to know how it came into existence.

What about some Videos on Fourier Optics?