Tomography Lecture - The Schwartz Space
Автор: ThatMathThing
Загружено: 2020-08-31
Просмотров: 2170
Here we introduce the Schwartz space. Up until now we have not discussed exactly what kind of functions we are considering when we take the Radon transform. One class of functions for us to do this with is the space of rapidly decreasing functions (aka the Schwartz space). The Schwartz space will give us the ability to give rigorous definitions of the Fourier transform and the Radon transform, and will set us up for signal processing.
Music:
Flames by Dan Henig
Flowers in Heaven by Craig Hardgrove
Empire Seasons by Dan Henig
Guardians + Tek by Craig Hardgrove
0:00 Start
11:13 Whiteboard Start - Discussion of Test Functions
20:10 Definition of Schwartz Space
25:23 Notations for Vector Calculus multi-indices
39:35 Norms and Seminorms
48:54 Seminorm Definition of the Schwartz Space and its Metric
56:55 Convergence in the Schwartz Space
1:03:50 Completeness of the Schwartz Space
1:24:47 Continuity of Differentiation over the Schwartz Space
1:30:00 Closing Remarks
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