SVD for Compact Operators // The Singular Value Decomposition @JoelRosenfeld

Uploaded December 2021 | Updated September 2026, 1 day ago

Here we break down the singular value decomposition for operators over infinite dimensional spaces. In the case of compact operators, this is a direct extension of the finite dimensional setting, where we replace the sum of outer products of vectors with rank one tensors between Hilbert Spaces. The singular value decomposition is the workhorse for data science, and it turns out that the extension for infinite dimensional data points is really straightforward. Come have a watch and see for yourself!

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//Books and References
Lang's Real and Functional Analysis: amzn.to/3ooTzJa
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//Chaptering
0:00 Start
1:01 Domain Basis
2:38 Range Basis
3:50 The SVD for Compact Operators

SVD for Compact Operators // The Singular Value Decomposition

$How DMD restricts your Dynamics Selecting a kernel has a huge impact on the types of dynamics you can analyze. Here we go over what happens when you use the exponential dot product kernel and the Gaussian RBF (and many other spaces) for Dynamic Mode Decompositions. In the end, this comes down to products and quotients of power series yield... power series (provided the denominator is nonzero). We go over a couple of basic concepts in kernel spaces, and establish a couple of theorems. Note that here we took the perspective of having a densely defined Liouivlle operator (i.e. Koopman generator). In the bounded case, the theorem is much easier to establish. Also, this style of argument applies to pretty much any function space you select, and yes, you must select a function space, since otherwise convergence of sequences of functions is meaningless. NOTE: I did leave out one detail in the discussion. Here we are leveraging the universality of our kernels to ensure that there is at least one function that doesnt vanish at a selected point (for multiplication operators), and the universality of their gradients to ensure that for each x and v, there is at least one function such that grad g(x)*v doesnt vanish. This ensures that the respective kernel at the selected point is not the zero function, and that h_{x,v} is not the zero function either. Universality isnt the only way to achieve this, and many spaces will satisfy the conditions. Music: Come 2gether by Ooyy Remove the Complexities by Peter Sandberg Supine by Peter Sandberg Guardians + Tek by Craig Hardgrove$

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$Laplace Transform of Derivatives of Distributions We talk about Distributional Derivatives and impact on the Laplace transform. Interestingly, this is a bit different than what happens for Laplace transforms of functions. //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 Vegas - Onda Norte Late Nights - Daxtan Twist - Cushy We Come Alive - Waycap Opposite - Jones Meadows Video Call from Las Angeles - Trevor Kowalski From the Cliffside - Tape Machines Cant Wait for You - Deanz 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! 0:00 Introduction 0:48 About Laurent Schwartz 1:26 Derivatives of Regular Distributions 2:56 Autobiography of Laurent Schwartz 3:45 Derivatives of Distributions 5:11 Laplace Transforms of Derivatives 6:58 Too Simple yet Too Complicated 7:30 The Transfer Function$

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