Fourier Transform and its Mathematical Foundation @JoelRosenfeld

Uploaded June 2025 | Updated September 2026, 21 hours ago

We talk about the Fields medal winning concept of the Schwarz space for the Fourier Transform.

//Watch Next
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764

//Books

Hunter and Nachtergaele - Applied Analysis
amzn.to/3FX9qdI (Amazon)
https://www.math.ucdavis.edu/~hunter/book/pdfbook.html (Free PDF)

Walter Rudin - Principles of Mathematical Analysis
amzn.to/3G3RuxX (Amazon)

J. R. Higgins - Sampling Theory in Fourier and Signal Analysis
amzn.to/3Zz5p5N (Amazon)

Zigmund - Trigonometric Series
amzn.to/4l70Jwa (Amazon)

Wendland - Scattered Data Approximation
amzn.to/3ZAqYTD (Amazon)

DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!

Fourier Transform and its Mathematical Foundation

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$Convolutions and Anti Derivatives of Distributions The definition of the convolution of two distributions arises from that of regular distributions and in turn, the convolution of functions. We leverage this definition to give a definition of an antiderivative of a distribution, and we determine the related Laplace transform property of both of these operations. //Additional Reading Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations (JSTOR) https://www.jstor.org/stable/2027808?seq=1#metadata_info_tab_contents (SIAM) https://epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread //Exercises - Show that differentiation is a continuous operation on L. - Show that 1/n phi(t/n) e^{-st} goes to zero in L (i.e. with respect to each seminorm) as n goes to infinity for every s in C. //Watch Next The Schwartz Space (Full Lecture) https://youtu.be/StzEO8B1pFY Undergraduate Intro to the Laplace Transform https://youtu.be/mJSqt7nchRQ Introduction to Control Theory https://youtu.be/0v4WFmOm764 Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE https://youtu.be/_I790SqNDjs //Music Provided by Epidemic Sound Late Nights - Daxten bam bam - Yomoti Tiny Things - Giants Nest Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referral/644nao //Books Advanced Mathematical Analysis - Beals https://amzn.to/3fn2jdP From Vector Spaces to Function Spaces - Yutaka Yamamoto https://amzn.to/3sU2CVj Feedback Control Theory - Doyle, Francis, Tannenbaum https://amzn.to/3HFvuov Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy https://amzn.to/3EOJU3K Nonlinear Systems - Khalil https://amzn.to/32Zyk8E Modern Control Systems - Dorf and Bishop https://amzn.to/331JQAm Fundamentals of Differential Equations - Nagle, Saff, and Snider https://amzn.to/3qeGD9C //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://amzn.to/3D3upUu DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!$