The Fourier Series and Fourier Transform Demystified
Added 2022-06-30 12:51:49 +0000 UTCHi everyone! Here is the latest video. Sorry it took me so long. As much as I enjoyed making it, it was quite a difficult subject and took a lot of time to understand. I hope you like it! I will try to upload the next one sooner :)
Comments
I would love to share but my reference material was my husband haha
Up and Atom
2022-07-07 09:33:26 +0000 UTCHey Cy thanks for the comment, and yes you're right! Those are great videos for anyone wanting to know more :)
Up and Atom
2022-07-07 09:33:00 +0000 UTCThank you Donald!
Up and Atom
2022-07-07 09:32:12 +0000 UTCPythagoreans were gathered around a fire, murmuring. A man stood up to the fire, and with a flurry of embers, pulled out a stick and began to draw on the ground and in the air with mesmerizing long and wide movements. He mumbled how in the path of his life he would measure (in his mind) who might be larger or smaller than him, faster or slower, funny or mean. If two people were nearly equal, their traits could be slid across parallel lines, like beads, and come out on the far side of an equal sign, unchanged, completely the same. If a person or event looms large in importance, then to indicate this largeness and expansiveness he uses a greater-than symbol and places the large objects on the open-triangle side of a greater-than symbol < . What if something remains neither greater than, nor equal nor less than something else? A woman steps forth with her staff and draws her own line sometimes coming close to the man's line, yet for the most part making her own dance around the fire. And then it happens, she crosses the man's path at a right angle. Quickly the geometers pull out straight edges and compasses to mark the intersection. A lengthy discussion ensues on measuring the "dot-product" of the male and female curves. They decide that the dot-product of both curves may be difficult to determine, yet in the immediate area of their right-angled intersection, neither curve adds to the other; the male and female paths head off in a perpendicular manner into independent dimensions of their own. No two points on their distinct paths have a simple greater than or lesser than relation to each other. A turtle shell with three taut strings subtly vibrates the air. Sometimes it gets plucked, sometimes it just hangs in the wind and sings like an Aeolian harp. Each string behaves as a creature defining its own dimensions in terms of its volume and the number of times and manner in which its stretched figure may be stroked. The three distinct strings, each with a full plane of expression gives one 3x2 = 6 variables, six dimensions, each perpendicular to each other. A Pythagorean approaches the fire and the right-angled intersection of paths. With a dramatic motion. he leaps up then seemingly plunges into the ground and then bursts up right through the intersection point and into the air like a fountain. The rest likewise join the dance each plunging down and emerging from a point of intersection. Their paths intertwine as every bit of the earth gets touched by the dancing feet. The participants represent a complete orthonormal basis of functions that fills the air, the entire space around the fire. Figures and movements can be summed and superimposed in an infinite series to come asymptotically close to any function, matrix or system of equations. Each dance becomes a projection, a mere slice and figment of a larger whole, a Hilbert space. The dances become convoluted and correlated as the participants branch off and fan out from each other driven by a cosmological constant Lambda /\ . The ubiquitous intersection points behave as saddle points spreading everything out into an infinite hyperbolic space. Embers twinkle and vanish while point-like stars hover above as silent and faithful witnesses.
Scott Ready
2022-07-03 15:51:48 +0000 UTCThats a very cool introduction. I must say the phrase "without loss of generality" took me back in memory to a few maths lectures :-). That was an interesting point about infinite dimensinal bases of function space. I was fiddling round a while ago trying to think of primes as infinite dimensional vector bases but didnt get very far. I would be interested to know reference material you found on the infinite dimensional bases bit if you happy to share.
James Matheson
2022-07-01 11:49:08 +0000 UTCHi Jade, that's a very accessible and intuitive introduction! As a suggestion, you may want to add pointers in the video description to deeper dives into the FT: 3b1b's progression from https://youtu.be/spUNpyF58BY to https://youtu.be/MBnnXbOM5S4 to https://youtu.be/r6sGWTCMz2k, the latter accompanied by an animation-only video https://youtu.be/-qgreAUpPwM that nevertheless shows that an arbitrary functions can be approximated. Then, one of the Mathologer's master classes at https://youtu.be/qS4H6PEcCCA tops it all :)
Cy 'kkm' K'Nelson
2022-06-30 22:12:58 +0000 UTCPlease take all the time you need. I believe your quality is worth it over any arbitrary quantity.
DONALD McLeod
2022-06-30 12:55:34 +0000 UTC
