The course lies in Blok 1A. The teaching period is from August 31 to October 30, in total 9 weeks.
There will be lectures Tuesday 8.15-10 (Aud 8), Thursday 13.15-15 (Aud 5) by Prof. Christian Berg and Exercises Thursday 10.15-12 in A 110 by Ph.d. student Phan Thanh Nam, all starting in week 36.
The lectures will be in English unless all participants agree that they can be given in Danish. The exercise class will be in English.
Problems for Thursday 17.09.2009: Problem set for 17.09.09
Problems for Friday 09.10.2009 at 15.00: (Note that I gave you one more day!) Problem set for 09.10.09
Problems for Thursday 29.10.2009: Problem set for 29.10.09(Corrected Oct. 22 at 10.12 )
Exercises: The following exercises will be discussed: E 1.1-1.4 and E 2.1.
This is a general comment about the Exercise classes: The participants are supposed to have studied the problems before the class and hopefully to have solved them so that some of you can present the solutions to your fellow students. The TA is supposed to guide you to the solutions if you need help and eventually explain you the solutions. The TA will also answer questions in connection with the course material, but you are always wellcome to discuss with me after the lectures and to come to my office 04.1.08.
Comments: Misprints: on the pages 4-5 I sometimes write D(f\ast g) sometimes D(f,g). They are the same and I prefer the first symbol.
Page 7, last line before exercises: The formula referred to shall be (1.1.6).
Page 13 third line: It is better to sum from 1 to infinity and to replace log n by log (n+1). It is discussed in Example 1.9.4.
Page 17. The reference to Exercise E.2 shall be E 5.2
Page 23. The reference to Exercise 5 shall be E 6.2.
Exercises: E 1.5, E 4.1, E 5.1-5.3. The first two can be found in the following page Extra Exercises 1
Comments: Misprints: Formula (1.5.3). In the second sum one shall only sum to k=n.
Formula (1.5.6). "dy" shall be to the right of the absolute value.
In Exercise E 5.3 the space \mathcal L^1([a,b]) shall be replace by the smaller space \mathcal L^\infty([a,b]).
Exercises: E 5.4, E 6.1-6.4. Remember that 5.4 can be found on the sheet: Extra Exercises 1
Comments: On Tuesday Sept. 15 I finished Theorem 1.8.1. On Thursday Sept. 17 I finished Theorem 1.9.2.
Misprints: On page 35 example b): alpha shall be a. One cannot directly apply the function 1/log(1+x) to Theorem 1.8.3 since it is not defined for x=0. But this function is still decreasing and convex on the open interval ]0,\infty[, so we just have to define c_0\ge 0 such that c_0+c_2\ge 2c_1.
Page 33, line 4 of the proof shall be: we choose N such that c_n\le epsilon/2 for n\ge N. Then in line 6 from the bottom replace 2(c_N-c_{N+p}) by ((2p-1)/p)c_N.
Page 38, line 2: The fraction x/\pi shall be \pi/x.
The Lectures on Thursday Sept. 24 are cancelled.
Exercises: E 6.5, E 7.1, E 8.1, E 8.2 and E. 9.1. Remember that the exercises 6.5 and 7.1 can be found on the sheet: Extra Exercises 1
Comments: I finished Theorem 1.10.5.
Misprints: Page 40, third line from bottom: The first "have" shall be deleted.
Page 41 line 2: The absolute value around c_n in the series shall be removed.
Page 48, middle: the intersection of the sets F_p shall be the union of the sets.
Thursday October 1st I will continue with Section 1.12.
Exercises: 10.1, 10.2, 10.3, 11.1, 11.2.
Comments: Nam finished 1.11 and I talked about 1.12 on October 1st until end of page 53.
Misprints: Page 49, line 5 from bottom. The mapping p is defined on T^2.
Page 50: In (1.12.4) right-hand side: the first \nu shall be \mu. The second last sentence in the proof of Lemma 1.12.4 shall reed: A real-valued function f \in \mathcal L^1(\mu\ast\nu) can be written...
Page 53, line 6: The formula shall begin \alpha=Re f(0)=
Page 54 line 5 from bottom: This statement should be marked (iii'')
Page 58 line 3 from bottom: bounding shall be bounded
Page 60 The proof of Theorem 1.12.14 assumes a_n to be real. Afterwards it is easy to deduce the complex-valued case.
Page 61 Exercise E 12.1. In the last formula on the page we sum from - infinity
Page 62 E 12.3 2^\circ, second line: c(T) shall be C(T).
Thursday October 8: I plan starting on Chapter 2 which will be made available soon.
Exercises: 7.2, 12.1-12.4. The exercises 7.2 and 12.4 can be found on the sheet: Extra Exercises 2
Comments: The evaluation of the course by those who follow the course is taking place during this week: go to evaluation
I finished Section 1.12 and covered the two first sections of Chapter 2.
Thursday October 15: I plan to cover Sections 2.5,2.6. I was quicker: I covered most of section 2.7 also, except for Example 2.7.5.
Exercises: 12.5-12.7 and 1.1, 1.2. The exercises 12.5-12.7 can be found on the sheet: Extra Exercises 2 and the exercise 1.2 can be found on Extra Exercises for Chapter 2
Comments: Page 70, line 3 from bottom: The factor 2(R+2) shall be (2R+2)
Page 71 line 1 and line 4: epsilon/2 shall be epsilon/3.
Concerning Dini's test page 77: The word "pointwise" in the theorem shall be deleted. One should add a note about applications of Dini's theorem: If f is differentiable at x then s_u(x) converges to s=f(x). More generally: If f has limits from right and left at x and f is differentiable from left and right, then s_u(x) converges to the mean-value s=(f(x+0)+f(x-0))/2. All this is completely parallel to results about Fourier series.
Page 82 , last 3 lines: An absolute value is missing around the first 3 expressions. The last equality on the page shall be \le.
Page 85. In the last integral 2 lines above Theorem 2.7.2 one shall replace the integrand by sin(t)/t dt.
Page 85 line 3 from bottom: Replace "truncated" by "partial".
Thursday October 22: I will finish Section 2.8 and begin on 2.9. I will also say a little about extending the Fourier transformation to Tempered distributions-this is something which is not included in the notes.
Exercises: E 1.3, E 2.1, E 3.1-3.3. All these exercises except E 2.1 can be found on an updated version of Extra Exercises for Chapter 2
Comments: Misprints: Page 87 Exercise 7.1 (ii).One shall assume that f is continuous for all x in [a,b].
Page 89. Start Theorem 2.8.3 by saying: Let F be ...
Page 91, line 4 from the bottom: Remove the last "to"
Page 96, line 2: The space L_2 shall be L^2 etc.
Page 96 In (2.8.9) script L^2 shall be roman L^2
Thursday October 29: I will finish Section 2.9 and give you an outlook to almost periodic function and Fourier analysis on locally compact abelian groups.
Exercises: E 6.1, E 7.1, E 7.2, E 8.1, E 8.2
Comments: Misprints:
Page 102, line 4 from bottom: Insert: Let h\in C_b(\mathbb R) be given.
Page 105: In the definition of alpha_2, the absolute value should be moved from the denominator to the numerator. Just below, the word lambda should have been the greek letter.
Page 107 line 4 from bottom: Instead on referring to Exercise 9.1, we shall use Lemma 2.9.4.
berg@math.ku.dk/ August 19, 2009.