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ma8105:2019v:lectures [2019-02-13]
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ma8105:2019v:lectures [2019-02-15] (nåværende versjon)
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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]]. ​ \\ \\ |
2019-02-13, Espen Robstad Jakobsen