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ma8105:2019v:lectures [2019-02-13]
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ma8105:2019v:lectures [2019-02-15]
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| ^ 5 | **Distribution theory** \\ Definitions, properties, \\ operations, regular/singular \\ \\ operations(cont.), derivatives of regular distr., \\ the fundamental theorem, convolution \\ \\ | 3 | \\ \\ | **Remark:** We define convolution both as a function as in Holden and as a distribution (see e.g. Wikipedia). The two definitions are related, one is the "density function" of the other. | | ^ 5 | **Distribution theory** \\ Definitions, properties, \\ operations, regular/singular \\ \\ operations(cont.), derivatives of regular distr., \\ the fundamental theorem, convolution \\ \\ | 3 | \\ \\ | **Remark:** We define convolution both as a function as in Holden and as a distribution (see e.g. Wikipedia). The two definitions are related, one is the "density function" of the other. | |
| ^ 6 | Convolutions (cont.), convergence, \\ approximations \\ \\ Primitive in 1D, equations in \(D'\), \\ fundamental solutions | | | | | ^ 6 | Convolutions (cont.), convergence, \\ approximations \\ \\ Primitive in 1D, equations in \(D'\), \\ fundamental solutions | | | | |
| ^ 7 | **Lebesgue Spaces** \\ Strong and weak \(L^p\), properties, inequalities. \\ \\ Convolutions, approximation in \(L^p\). \\ \\ Approximation in \(L^p\), compactness in \(L^p\) | 4 | 4.1, Prop 4.4 and Thm 4.6, \\ 4.2, 4.7, 4.3 \\ \\ | **Proof of Kolmogorov:** We take the classical proof and not the one in the Holden notes, see e.g. Theorem A.5 in Holden-Risebro: Front Tracking for Hyperbolic Conservation Laws. \\ \\ The classical proof is an approximation argument that reduces the proof to an application of Arzela-Ascoli. \\ \\ | | ^ 7 | **Lebesgue Spaces** \\ Strong and weak \(L^p\), properties, inequalities. \\ \\ Convolutions, approximation in \(L^p\). \\ \\ Approximation in \(L^p\), compactness in \(L^p\) | 4 | 4.1, Prop 4.4 and Thm 4.6, \\ 4.2, 4.7, 4.3 \\ \\ | **Proof of Kolmogorov:** We take the classical proof and not the one in the Holden notes, see e.g. Theorem A.5 in Holden-Risebro: Front Tracking for Hyperbolic Conservation Laws. \\ \\ The classical proof is an approximation argument that reduces the proof to an application of Arzela-Ascoli. \\ \\ OBS: 3 lectures this week, only one next week. \\ \\ | |
| ^ 8 | Modes of convergence | | p 86-89 \\ \\ p 59-63 | | | ^ 8 | Modes of convergence | | p 86-89 \\ \\ p 59-63 | Only one lecture this week, three last week. \\ \\ | |
| ^ 9 | Modes of convergence (cont.), \\ convergence and compactness in \(L^p,\ p\in(1,\infty)\). \\ \\ Limiting cases: Convergence and compactness in\(L^1\), \\ Dunford-Pettis with proof, uniform integrability, de la Vallee-Poussin. | | p 60-65 \\ \\ p 65-70 \\ \\ | **My lecture notes** on Dunford-Pettis and equiintegrability [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote02.03.2017_DunfordPettis_and_Equiintegrability.pdf|PDF]] \\ \\ Note that the proof in the notes of Holden lack the conclusion. \\ \\ | | ^ 9 | Modes of convergence (cont.), \\ convergence and compactness in \(L^p,\ p\in(1,\infty)\). \\ \\ Limiting cases: Convergence and compactness in\(L^1\), \\ Dunford-Pettis with proof, uniform integrability, de la Vallee-Poussin. | | p 60-65 \\ \\ p 65-70 \\ \\ | **My lecture notes** on Dunford-Pettis and equiintegrability [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote02.03.2017_DunfordPettis_and_Equiintegrability.pdf|PDF]] \\ \\ Note that the proof in the notes of Holden lack the conclusion. \\ \\ | |
| ^ 10 | Convergence and compactness in\(L^1\) (cont.), \\ Convergence and compactness in \(L^\infty\) \\ Examples. \\ \\ Radon measures, the space \(\mathcal M\),\\ Weak * compactness in \(\mathcal M\) and the subspace \(L^1\) | | p 69-75 \\ \\ p 75-78 | This material is partly taken from Folland: //Real Analysis// chp 7, partly from Holden. \\ \\ **Note:** Riesz representation theorem, the space \(M\) needs be defined as the space of //finite// Radon measures. \\ \\ **My lecture notes** on Radon measure and compactness: [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote09.03.2017_RadonMeasuresAndCompactness.pdf|PDF]]. \\ \\ | | ^ 10 | Convergence and compactness in\(L^1\) (cont.), \\ Convergence and compactness in \(L^\infty\) \\ Examples. \\ \\ Radon measures, the space \(\mathcal M\),\\ Weak * compactness in \(\mathcal M\) and the subspace \(L^1\) | | p 69-75 \\ \\ p 75-78 | This material is partly taken from Folland: //Real Analysis// chp 7, partly from Holden. \\ \\ **Note:** Riesz representation theorem, the space \(M\) needs be defined as the space of //finite// Radon measures. \\ \\ **My lecture notes** on Radon measure and compactness: [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote09.03.2017_RadonMeasuresAndCompactness.pdf|PDF]]. \\ \\ | |