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General |
MATH 634 , Fall 2023: Notes and Summary
| Lecture | Date | What was covered | Notes | Textbook Section (S=Stein; Si I.=Simon vol.1; Si III. = Simon vol.3) |
| 1 | Aug. 29 | Weierstrass and Stone-Weierstrass Approximation Theorems. ONB for L^2[0,1] | ||
| 2 | Aug. 31 | Fourier ONB. Geometric Properties of Optimality. | S. | |
| 3 | Sep. 5 | L^2 convergence of Fourier Series. Parseval theorem. ONB, Riesz bases, Schauder Bases. | ||
| 4 | Sep. 7 | Pointwise convergence: Riemann Localization Principle (1) | S. | |
| 5 | Sept.12 | Pointwise convergence: Riemann-Lebesgue Lemma. Riemann Localization Principle (2). Cases: L^1 functions differentiable at one point; Lipschitz class. Holder class. Dini criterion. Dirichlet theorem. | ||
| 6 | Sept.14 | Continuous Functions: Fejer Theorem. Cesaro Summation | ||
| 7 | Sept.19 | Wiener algebra (1): Construction | ||
| 8 | Sept.21 | Gelfand's Representation theory: Properties of Maximal Ideals. Gelfand-Mazur Theorem. | ||
| 9 | Sept.26 | Multiplicative Linear functionals. Weak* compactness, and the character representation. Wiener Lemma as a consequence of the Gelfand Representation Theory. | ||
| 10 | Sept. 28 | Almost Periodic Functions. Bohr's compactification. | ||
| 11 | Oct. 3 | Riesz-Thorin Interpolation Theorem and applications: Young's and Hausdorff - Young. | ||
| 12 | Oct. 5 | More on AlmostPeriodic Functions | ||
| 13 | Oct.10 | Fourier transform on R: L^1 theory. | ||
| 14 | Oct.12 | Mid-Term Exam | Mid-Term Exam | Regular classroom EGR 1102 |
| 15 | Oct.17 | Fourier tranform on R: L^2 theory. Plancherel Theorem. | ||
| 16 | Oct.19 | L^2 theory: Unitary property of the Fourier transform. Inversion in L^2 sense. | ||
| 17 | Oct.24 | Hausdorff-Young inequality(2). Duality pairing. Product-Convolution rule. Schwartz class. | ||
| 18 | Oct.26 | FFT Conference. | Attend talks: schedule | MATH 3206 |
| 19 | Oct. 31 | Pointwise inversion. Poisson Summation Formula. Sine and Cosine Transforms. |
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| 20 | Nov. 2 | Bandlimited functions and Shannon's formula in B^2_Omega | ||
| 21 | Nov. 7 | Paley-Wiener Theory | ||
| 22 | Nov. 9 | The Schwartz class in R^d. Tempered distributions | ||
| 23 | Nov.14 | Fourier Transform, Convolutions and other operations with tempered distributions | ||
| 24 | Nov.16 | Operations with Distributions | log(x), 1/x, 1/x^2,... as distributions | |
| 25 | Nov.21 | NO CLASS | ||
| - | Nov.23 | THANKSGIVING BREAK | NO CLASS | |
| 26 | Nov. 28 | Square Integrable Representations: Schur's lemma and general theory | ||
| 27 | Nov. 30 | Square Integrable Representations: Weyl-Heisenberg group | ||
| 28 | Dec. 5 | Square Integrable Representatons: ax+b group | ||
| 29 | Dec. 7 | Uncertainty Principles | Last Class | |
| - | Dec. 13 | FINAL EXAM | 9:00am-11:00am | In EGR 1102 |
