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Wiki source code of Fourier analysis, fall 2015

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1 * (((
2 = Fourier analysis, fall 2015 =
3 )))
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7 {{panel}}
8 **Teacher:** [[Kari Astala>>doc:mathstatHenkilokunta.Astala, Kari]]
9
10 **Scope:** 10 cr
11
12 **Type:** Advanced studies
13
14 **Teaching:** Lectures on weeks 36-42 and 44-50, in room C123, on Tuesdays 10-12 and on Wednesdays 12-14.
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16 First lecture on Tuesday, September 1.
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20 Exercise classes are part of the course; these are on Mondays 12-14, in room C122. Exercise// assignments// are delivered on Wednesdays of the previous week.
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22 First exercise class on Monday, September 7.
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25
26 **Topics: **The course is an introduction to modern Fourier Analysis. Fourier analysis is a basic tool both in pure and applied mathematics, from PDE's and harmonic analysis to stochastics, signal processing, physics,...
27
28 The first part of the course, roughly weeks 36-42, covers discrete Fourier analysis (% style="line-height: 1.4285;" %)(Fourier series and also Fast Fourier transform), while the second period, weeks 44-50, studies the continuous Fourier analysis, i.e. Fourier transforms of functions and (Schwartz) distributions in //R//(%%)//^^n^^//(% style="line-height: 1.4285;" %). Throughout the course, and time allowing, many of the applications of Fourier analysis are discussed.
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30 (% style="line-height: 1.4285;" %)
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32
33 **Prerequisites: **Basic knowledge on measure theory and Lebesgue integration (e.g. the course "Mitta ja integraali") is required. Knowledge of basics of //L^^p^^-//spaces// //will be advantageous (e.g. as discussed in the course Reaalianalyysi I; see below). In addition some facts from functional analysis will be used - if necessary, these can be briefly discussed during the course.
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35
36
37 **Contents:**
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39 * Representing a periodic function as a Fourier series
40 * Basic properties of Fourier series
41 * Dirichlet kernel; Fejer kernel and good convolution kernels
42 * Pointwise convergence, Convergence in //L^^p^^//, //C[0,1]//, etc. -norm
43 * Continuous functions with diverging Fourier series
44 * (% style="line-height: 1.4285;" %)Discrete Fourier transform, Fast Fourier transform
45 * (% style="line-height: 1.4285;" %)Continuous Fourier transform in //L^^1^^(R^^n^^)//
46 * (% style="line-height: 1.4285;" %)Fourier transform in //L^^2^^(R^^n^^)//; Plancherel's theorem
47 * (% style="line-height: 1.4285;" %)Interpolation and Fourier transform in// L^^p^^(R^^n^^)//
48 * (% style="line-height: 1.4285;" %)Tempered distributions
49 * (% style="line-height: 1.4285;" %)Applications of Fourier Analysis
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51
52
53 == **Course material:** ==
54
55 The course will basically follow the lecture notes: 'Kari Astala, Fourier Analyysi' (course in 2012; notes are in Finnish)
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57 Minor modifications will be made, and new Lecture notes (in Finnish) will be provided on this page as the course proceeds.
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59 **Here are **the complete [[Lecture notes>>attach:2015.Luennot.FourierAnalyysi.pdf]] with Appendix.
60
61 **NOTE:**
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63 * **Section VII.1. **was left for a self study.
64 * **Proof** of Riesz-Thorin Theorem 11.2 (pages 100-106) was left for a self study.
65 (% style="text-align: right;" %)(Riesz-Thorin theorem is proved also e.g. in "Rudin: Real and Complex Analysis")
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67 Except for the distribution theory, the material covered in this course can be found in the books:
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69 * Stein-Shakarchi: Fourier Analysis, An Introduction.
70 * Rudin: Real and Complex Analysis.
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72 Other books related to the course:
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74 * Grafakos: Classical and Modern Fourier Analysis.
75 * Rudin: Functional Analysis.
76 * Duoandikoetxea: Fourier Analysis.
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80 Additional information on real analysis, in particular on basics of //L^^p^^-//spaces, can be found from lecture notes by Ilkka Holopainen attached here: [[Reaalianalyysi.I (2011).pdf>>attach:Reaalianalyysi.I.2011.pdf]]
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82 The information on //L^^p^^-//spaces needed can also be found e.g. in Rudin's book 'Real and Complex analysis' mentioned above; a brief sketch of //L^^p^^-//properties needed is also given in the Appendix of the course notes above.
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85 {{/panel}}
86
87 == News ==
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89
90
91 == (% style="line-height: 1.5;" %)Exams(%%) ==
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93 (% style="line-height: 1.5;" %)The course will have two exams, one at the end of first period, week 43, and another in the end of second period, week 51.
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95 (% style="line-height: 1.5;" %)The second course exam is on Thursday December 17, at 10.00-12.30, in the room C123.
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97 **Topic of Exam/Koealue **for **second exam**: Section VII.2 and Chapters IX - XIII.2 (i.e. Last topic: Fundamental solution)
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99 (% style="line-height: 1.5;" %)
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102 (% style="line-height: 1.5;" %)The first exam is on Thursday Oct. 22., at 10.00-12.30, in the room C 123.
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104 (% style="line-height: 1.5;" %)**Topic of Exam/Koealue**: Chapters I - VI from Lecture notes (i.e. Beginning - L^^2^^ theory of Fourier series)
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106 (% style="line-height: 1.5;" %)
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109 (% style="line-height: 1.5;" %)**Note**: (%%)There is also the possibility to pass the course by a **general examination** (Loppukoe/erilliskoe), where topic of exam is the whole course.
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111 If you wish to do so, please contact Kari Astala; an agreement in advance is necessary. We plan to arrange a general examination on //Fourier Analysis// on Dec 21 at 10-14, if there are students who wish to take this.
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115 (% style="line-height: 1.5;" %)[[(% style="line-height: 1.5;" %)** Here** >>attach:Kurssikokeet.2011-2013.pdf]](%%)you find the corresponding course exams from 2011-2013
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117 (% style="line-height: 1.5;" %)**[[Here >>attach:Kertaus.pdf]]** you find //voluntary review problems//; Aleksis Koski will be on call in his office A415 on Monday 19th October at 12-14 to give help/answers/advice on these problems.
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119 (% style="line-height: 21.0px;" %)
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122 (% style="line-height: 21.0px;" %)The results of the course and its exams can be found on the department web-page "Kurssitulokset/Exam results"
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124 == [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57353||shape="rect"]] ==
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127 (% style="color: rgb(96,96,96);" %)Did you forget to register? (%%)[[What to do?>>doc:mathstatOpiskelu.Etusivu.Kysymys4.WebHome]]
128
129 == Exercises ==
130
131 === Assignments ===
132
133 * [[Exercises 1>>attach:Harjoitus1.2015pdf.pdf]]
134 * [[Exercises 2>>attach:Harjoitus2.2015.pdf]]
135 * [[Exercises 3>>attach:Harjoitus3.2015.pdf]]
136 * [[Exercises 4>>attach:Harjoitus4.2015.pdf]]
137 * [[Exercises 5>>attach:Harjoitus5.2015.pdf]]
138 * [[Exercises 6>>attach:Harjoitus6.2015.pdf]]
139 * [[Exercises 7 >>attach:Harjoitus7.2015.pdf]]
140 * [[Exercises 8>>attach:Harjoitus8.2015.pdf]]
141 * [[Exercises 9>>attach:Harjoitus9.2015.pdf]]
142 * [[Exercises 10>>attach:Harjoitus10.2015.pdf]]
143 * [[Exercises 11>>attach:Harjoitus11.2015.pdf]]
144 * [[Exercises 12>>attach:Harjoitus12.2015.pdf]] (voluntary/includes some review problems)
145
146 === Exercise classes ===
147
148 Note: You get extra points from solved assignments as follows:
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150 25% = 1p, 35% = 2p, 45% = 3p, 55% = 4p, 65% = 5p and 75% = 6p.(% style="line-height: 1.4285;" %)
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152 (% style="line-height: 1.4285;" %)
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155 (% style="line-height: 1.4285;" %)First exercise class on Monday, September 7.
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157 |=(((
158 Group
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160 Day
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162 Time
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164 Room
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166 Instructor
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168 |(((
169 1.
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171 Monday
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173 12-14
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175 C122
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177 Aleksis Koski
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179
180 == Course feedback ==
181
182 Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].