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General |
MATH 634 , Fall 2021: 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. 31 | Weierstrass and Stone-Weierstrass Approximation Theorems. ONB for L^2[0,1] | get homework 1 | |
| 2 | Sep. 2 | Fourier ONB. Geometric Properties of Optimality. | S. | |
| 3 | Sep. 7 | L^2 convergence of Fourier Series. Parseval theorem. ONB, Riesz bases, Schauder Bases. | ||
| 4 | Sep. 9 | Pointwise convergence: Riemann Localization Principle (1) | |
S. |
| 5 | Sept.14 | Pointwise convergence: Riemann-Lebesgue Lemma. Riemann Localization Principle (2). Cases: L^1 functions differentiable at one point; Lipschitz class. | homework 1 due | |
| 6 | Sept.16 | Continuous Functions: Fejer Theorem. Abel Summations | ||
| 7 | Sept.21 | Discontinuous Functions: Dirichlet Theorem. Gibbs Phenomenon (1) | ||
| 8 | Sept.23 | Dirichlet Theorem, Gibbs Phenomenon (2). Wiener algebra: Wiener Lemma (1) | get homework 2 | |
| 9 | Sept.28 | Wiener algebra: Wiener Lemma (2). | ||
| 10 | Sept. 30 | Tauberian Theorems. Gelfand's Maximal Ideals Theory for Wiener algebra | pick up homework 3; homework 2 due | |
| 11 | Oct. 5 | Riesz-Thorin Interpolation Theorem (1). | ||
| 12 | Oct. 7 | Riesz-Thorin Interpolation (2): Hadamard Three-Line Theorem. Applications: Convolution (Young's) Inequality; Hausdorff-Young inequality (for Fourier series) | homework 3 due; pick up homework 4 | |
| 13 | Oct.12 | Fourier transform on R: L^1 theory. | ||
| 14 | Oct.14 | Fourier tranform on R: L^2 theory. Plancherel Theorem. | homework 4 due | |
| 15 | Oct.19 | L^2 theory: Unitary property of the Fourier transform. Inversion in L^2 sense. | ||
| 16 | Oct.21 | Hausdorff-Young inequality(2). Duality pairing. Product-Convolution rule. | homework 5 due | |
| 17 | Oct.26 | Schwartz class. Pointwise inversion. Poisson Summation Formula. Sine and Cosine Transforms. | ||
| 18 | Oct.28 | Mid-Term Exam | Mid-Term Exam | In CSIC 4122 |
| 19 | Nov. 2 | Fourier multipliers. Fractional Derivative. PDOs. | ||
| 20 | Nov. 4 | Bandlimited functions and Shannon's formula in B^2_Omega | ||
| 21 | Nov. 9 | Oversampling and approximation rates of signals by cardinal series | ||
| 22 | Nov. 11 | Paley-Wiener Theory (1) | homework 6 due | |
| 23 | Nov.16 | Paley-Wiener Theory (2) | ||
| 24 | Nov.18 | The Schwartz class in R^d. Tempered distributions | ||
| 25 | Nov.23 | Derivatives of Tempered Distributions | ||
| - | Nov.25 | THANKSGIVING BREAK | NO CLASS | |
| 26 | Nov.30 | Fourier Transform, Convolutions and other operations with tempered distributions | ||
| 27 | Dec. 2 | Operations with Distributions | ||
| 28 | Dec. 7 | log(x), 1/x, 1/x^2,... as distributions | homework 7 due | |
| 29 | Dec. 9 | Uncertainty Principles | homework due | Last Class |
| - | Dec.14 | Review Session | 9:00am-12:00noon | CSIC 4122 |
| - | Dec.16 | FINAL EXAM | 8:00am-10:00am | In CSIC 4122 |
