The relations shown in this video are completely mindblowing.... As a hobby mathemathician and professionalelectronic engineer this has been a bit lifechanging .... thanks a lot.
Beautiful video as always Also I did try and experiment to see what happens when the spectrum of frequencies is continuous, and tried a continuous interval from 0 to 1 of cosine waves (Yes this breaks the nice inner product space but I got curious) Had to use riemann sum to account for partial multiple frequencies, the result looked like sin(x)/x. Now I can see why all the engineers are excited about this function, to the point they gave it a name sinc(x)
Wow, that's a really useful experiment. What you've (re)discovered here is called the (continuous) Fourier transform. Basically, what you did is calculate a Fourier series for a function that isn't periodic at all. The official way to do this, is to let the period T go to infinity. As you do so, the different frequencies in the spectrum get closer and closer together, until they become a continuum. The result is a transform that takes a non-periodic continuous function to a non-periodic continuous spectrum.
Is there any other inner product for functions than the integral in the video, which also has all the nice properties? Or is that one the "best" one? Great series 😀
Great question! There are many different inner products that you could define. For starters, you can introduce a "weight function" under the integral. Different weights give different results, which basically makes the number of possibilities infinite. In engineering, these weights can be used e.g. to put more emphasis on a certain part of the interval. If you want your approximations to be better near the edge points, you just use a weight function which is larger there. I'm pretty sure that you can also come up with inner products that don't use an integral. e.g. If you sum up the function values at a discrete set of input points, you probably already get a good inner product, although I'm not sure.
I really thought that this was going in the direction of using the power series for sine and cosine, since their powers of x are completely independent. That is, sine only has odd powers of x and cosine has even powers of x. So, in the conventional sense of a dot product, all pairwise coefficients are guaranteed to have a zero. Multiplying coefficient pairs will always give a zero product. I know this is different from the integral definition of an inner product of functions, but it seems to support the idea that sin(nx) and cos(nx) are truly orthogonal. It makes me wonder what other power series representations of two functions have a similar property.
Thanks for the video. Am I understanding correctly that any function can be written as a linear combination of 1,x,x^2.... (or the terms in the Fourier series)? If so, is there a proof for this?
Thanks for the question. The answer is no: there are many exceptions. Plenty of functions don't have a Taylor series, and plenty more have a Taylor series that doesn't converge for all real inputs.
Music AND math ? Im sold.
😊 thank you!
The relations shown in this video are completely mindblowing.... As a hobby mathemathician and professionalelectronic engineer this has been a bit lifechanging .... thanks a lot.
Beautiful video as always!
Thank you!
Thank you!
This is an awesome series! I love the pace, and the perfect ratio between the density of information and the clarity of the steps and examples..
Thank you so much!
It's often difficult to make a good trade-off between going too fast and going too slow. I'm glad you like the balance we found.
Nice, QM fundamentals from scratch haha. Better than my university's mathematical techniques course
Thanks for the video
Beautiful video as always
Also I did try and experiment to see what happens when the spectrum of frequencies is continuous, and tried a continuous interval from 0 to 1 of cosine waves (Yes this breaks the nice inner product space but I got curious)
Had to use riemann sum to account for partial multiple frequencies, the result looked like sin(x)/x. Now I can see why all the engineers are excited about this function, to the point they gave it a name sinc(x)
Wow, that's a really useful experiment. What you've (re)discovered here is called the (continuous) Fourier transform. Basically, what you did is calculate a Fourier series for a function that isn't periodic at all. The official way to do this, is to let the period T go to infinity. As you do so, the different frequencies in the spectrum get closer and closer together, until they become a continuum. The result is a transform that takes a non-periodic continuous function to a non-periodic continuous spectrum.
These videos are so good! I refer them to all my math friends
That's great to hear, thank you so much.
This is so good I was always wondering how orthogonality relates to Fourier series thank you!
You're welcome. Always happy to clarify something.
Is there any other inner product for functions than the integral in the video, which also has all the nice properties? Or is that one the "best" one? Great series 😀
Great question!
There are many different inner products that you could define. For starters, you can introduce a "weight function" under the integral. Different weights give different results, which basically makes the number of possibilities infinite.
In engineering, these weights can be used e.g. to put more emphasis on a certain part of the interval. If you want your approximations to be better near the edge points, you just use a weight function which is larger there.
I'm pretty sure that you can also come up with inner products that don't use an integral. e.g. If you sum up the function values at a discrete set of input points, you probably already get a good inner product, although I'm not sure.
Zo moet Fourier het gezien hebben.
I really thought that this was going in the direction of using the power series for sine and cosine, since their powers of x are completely independent. That is, sine only has odd powers of x and cosine has even powers of x. So, in the conventional sense of a dot product, all pairwise coefficients are guaranteed to have a zero. Multiplying coefficient pairs will always give a zero product. I know this is different from the integral definition of an inner product of functions, but it seems to support the idea that sin(nx) and cos(nx) are truly orthogonal. It makes me wonder what other power series representations of two functions have a similar property.
Thanks for the video. Am I understanding correctly that any function can be written as a linear combination of 1,x,x^2.... (or the terms in the Fourier series)? If so, is there a proof for this?
Thanks for the question. The answer is no: there are many exceptions. Plenty of functions don't have a Taylor series, and plenty more have a Taylor series that doesn't converge for all real inputs.
I AHVENT' WATCHED THE VIDEO DID YOU COOK IT UP SHOULD I FINSIH IT
'All angles' is right angle