Harmonic Analysis (undergraduate), Spring 2024-25.

Notes. They are renewed regularly.

Content: The first part of the course (two or two and a half weeks) is a quick review of the Lebesgue integral for functions defined on subsets of R. The second and main part of the course is the presentation of Fourier series for functions which are integrable or square-integrable on the interval [0,1) and of the Fourier transform for functions which are integrable or square-integrable on R. Fourier series and the Fourier transform are very important for many subjects of mathematics and the sciences and I hope that we shall find time to see a few applications.

Lectures: 3-5 Tuesday and Thursday, E204. Exercises session: 1-3 Friday, A212.

Office hours: 12-1 Tuesday and Thursday, in my office (Γ211).

Background: The courses Analysis I and II and, hopefully, a course in Real Analysis.

Bibliography: A good book is "Fourier series and integrals" of Dym and McKean. The parts of the book which are more relevant to our course are 1.1-1.5 from the first chapter and 2.1-2.6 from the second chapter.

February 23: The notes have been renewed.

March 9: The notes have been renewed.

I hope the following will help you when you read the notes. 

(i) Remember the basic properties of integrals as they are stated in sections 1.1 up to 1.4. You don't have to read the proofs. 

(ii) Read carefully section 1.5 which is about the actual calculations of integrals. 

(iii) Read sections 1.6 and 1.7 about the properties of the spaces L^1(A) and L^2(A). You don't have to read the proofs of Theorems 1.4 and 1.5. Also, we shall see later in class the part of section 1.7 which is after Proposition 1.40. Don't read it now.

(iv) In section 1.8 you don't have to read the proofs of Propositions 1.43, 1.44 and 1.45. Read all other proofs and also the alternative proof of Proposition 1.45 which uses the theorem of Fubini. 

(v) Read everything in section 1.9. 

(vi) Starting with chapter 2, read everything!

A set of exercises is coming soon!!

And here is the first set of exercises!!!

March 12: Today's lecture as well as last Thursday's lecture were lost (due to the very well known reasons). I am going to do two extra lectures: 11-1 Wednesday March 19 and 11-1 Wednesday March 26.

Moreover, I am going to add an extra session for exercises: 11-1 every Friday, starting next week.

March 19: The two extra lectures (on the two Wednesdays) will be done in A203. 

And the extra session for exercises: 1-3 every Friday in A212.

March 23: The notes have been renewed and there is a second set of exercises. You may try now the exercises 1, 2, 3, 4, 12. 

March 27: Tomorrow's exercise session will be done as announced.

April 4: The notes have been renewed. There are two small subsections, 2.1.6 and 2.2.2, under the title "a few extra things": you don't have to read them. 

April 19: The notes have been renewed. I have also rewritten the first two sets of exercises: I made some minor changes but, more important, I have included hints for most of the exercises. I hope that this helps.

A third set of exercises will follow soon: during these holy days I don't want to interrupt your dedication to praying :)

April 20: Praying is over, so here is the third set of exercises to think about over the Easter table!

May 9: Look at the new notes. (I have not done Propositions 3.14 and 3.15 yet.)

During the Tuesday lecture I was prosecuted by some of my students (of a specific gender). The accusation was: I have said a few times, during lecture time, that boys are smarter than girls. I think that I persuaded the accusers (and myself) that every time I say that boys are smarter than girls I am joking. Fortunately and for the time being I am out of jail. I look forward to see my student evaluations after the end of the semester.

May 10: Here is the fourth set of exercises.

May 25: The fifth and last set of exercises.

There will be an exercise session next week: 3-5 Tuesday, A212. (Perhaps one more.)

May 29: The last exercise session: tomorrow 1-3, A212.

Functional Analysis (graduate), Spring 2024-25.

Notes.

Content: Normed spaces and Banach spaces. Inner product spaces and Hilbert spaces. Bounded linear functionals and the dual space. Theorem of Hahn-Banach. Second dual space. Principle of uniform boundedness. Weak convergence and weak* convergence. Weak topologies σ(X,X'), σ(X',X), σ(X',X''). Theorem of Alaoglou. Bounded linear operators. Principle of uniform boundedness. Open mapping theorem. Closed graph theorem. Spectrum. Compact operators. Spectral theorem for compact selfadjoint (and normal) operators on Hilbert spaces.

Lectures: 9-11 Tuesday and Thursday, Β212.

Office hours: 12-1 Tuesday and Thursday, in my office (Γ211).

Background: I suggest to review your undergraduate linear algebra and metric spaces and also your graduate real analysis (measure spaces, L^p spaces etc). 

Bibliography: "Functional Analysis" of Yosida. "Methods of Modern Mathematical Physics" of Reed and Simon. "Functional Analysis" of Lax. "A course in Functional Analysis" of Conway. "Linear Analysis" of Bollobás. "Functional Analysis" of Riesz and Nagy. "Introduction to Functional Analysis" of Taylor and Lay. And many more.

March 8: The first set of exercises. Please give me the solutions of exercises 2, 3, 5, 8, 11 by next Thursday in class. 

March 24: There will be two extra lectures: 11-1 today and next Monday.

April 20: The second set of exercises. Please turn in the solutions of exercises 1, 7, 8, 10, 13, 15 by Tuesday April 29 in class.  

May 11: The third set of exercises. Please turn in the solutions of exercises 2, 3(i), 6, 7(i), 10, 12, 16, 17 by Tuesday May 20 in class.

May 31: The fourth set of exercises. Do not turn in any of them. Just look at 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 17, 19, 22, 23, 24, 27, 28. In a few days I shall upload a fifth (and last) set of exercises.

The final exam: 11-2 Friday July 4.

June 1: The fifth set of exercises. Look at 1, 2, 3, 4, 5, 6, 7.

Real Analysis (undergraduate), Autumn 2024-25.

Περιεχόμενο: Μέτρο Lebesgue και ολοκλήρωμα Lebesgue στο R.

Διαλέξεις: 1-3 Τρίτη και Πέμπτη, Α214. Δίωρο ασκήσεων: 1-3 Τετάρτη, Α203.

Ώρες γραφείου: 12-1 Τρίτη και Πέμπτη, στο γραφείο μου (Γ211).

Πρόοδος (προαιρετική): 5-7 Πέμπτη 14-11-2024.

Υπόβαθρο: Αν και δεν είναι απολύτως απαραίτητη, θα βοηθούσε πολύ η εξοικείωση με τα μαθήματα Ανάλυση Ι και ΙΙ.

27 Σεπτεμβρίου: Εδώ είναι η ύλη που καλύψαμε μέχρι τώρα (δύο πρώτα μαθήματα). Δοκιμάστε τις ασκήσεις 1.1.1, 1.1.2 (δύσκολη), 1.2.1 (ουσιαστικά έγινε και στο μάθημα), 1.2.2, 1.2.3 (δύσκολη), 1.2.4 (δύσκολη).

30 Σεπτεμβρίου: Θα γίνεται (από μεθαύριο) ένα δίωρο ασκήσεων κάθε Τετάρτη, 1-3 στην Α203.

4 Οκτωβρίου: Εδώ είναι η μέχρι τώρα ύλη, εκτός από το Θεώρημα 1.2 και ό,τι ακολουθεί. Δοκιμάστε τις ασκήσεις 1.3.1, 1.3.6, 1.3.7 (λιγάκι δύσκολη), 1.3.8 εκτός του (iii), 1.3.11.

11 Οκτωβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 1.3.2, 1.3.3, 1.3.8(iii), 1.3.10, 1.3.12, 1.3.13, 1.3.14 (ειδικά αυτήν), 1.3.15, 1.4.1 (ειδικά αυτήν). Οι ασκήσεις 1.3.21, 1.3.22, 1.3.23, 1.3.24 (ειδικά αυτή) ασχολούνται με σύνολα τύπου Cantor και έχουν ενδιαφέρον. Τέλος οι ασκήσεις 1.3.4, 1.3.5, 1.3.9, 1.3.16, 1.3.17, 1.3.18, 1.3.19, 1.3.20 είναι λίγο πιο εξεζητημένες και δεν χρειάζεται να ασχοληθείτε με αυτές άμεσα.

18 Οκτωβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 2.1.1, 2.1.2, 2.1.3, 2.1.6, 2.1.7 (ειδικά αυτήν), 2.1.13.

25 Οκτωβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 2.1.4, 2.1.5, 2.1.9, 2.1.10. Όποιος έχει όρεξη ας δει και την 2.1.14.

31 Οκτωβρίου: Άλλαξε η ημερομηνία (όχι η ώρα) εξέτασης της προόδου και έγινε 5-7 Πέμπτη 14-11-2024.

1 Νοεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 2.3.1, 2.3.2, 2.3.3, 2.3.4, 2.3.6.

8 Νοεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 2.4.1, 2.4.2, 2.4.3 (αυτή είναι λίγο πιο δύσκολη).

16 Νοεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 3.1.1, 3.2.1. Επίσης, εδώ είναι τα θέματα της προόδου.

23 Νοεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Δείτε τις ασκήσεις 3.3.1, 3.3.2.

30 Νοεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Έχουν γίνει αρκετές αλλαγές στην παρουσίαση των προηγουμένων, αλλά δεν αλλάζει η ουσία. Π.χ. το θεώρημα μονότονης σύγκλισης έχει μεταφερθεί σε μια νέα ενότητα, την οποία θα ξεκινήσουμε την ερχόμενη εβδομάδα, με τίτλο "οριακά θεωρήματα". Στην ενότητα 3.4 διαβάστε τους ορισμούς και τις διατυπώσεις των προτάσεων, χωρίς να δώσετε έμφαση στις αποδείξεις τους. Να κατανοήσετε καλά την ενότητα 3.5. Λύστε τις ασκήσεις 3.3.3, 3.3.4, 3.3.5, 3.5.1.

2 Δεκεμβρίου: Εδώ είναι οι βαθμοί της προόδου.

9 Δεκεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Διαβάστε καλά τα οριακά θεωρήματα.

20 Δεκεμβρίου: Εδώ είναι η μέχρι τώρα ύλη. Διαβάστε καλά τα οριακά θεωρήματα και την ενότητα για τη σχέση ανάμεσα στα ολοκληρώματα Riemann και Lebesgue. Διαβάστε προσεκτικά τα παραδείγματα. Κοιτάξτε και τις ασκήσεις 3.6.1 -  3.6.9 και 3.7.1 - 3.7.9. Λίγο πιο μετά θα αναρτήσω και την ύλη πάνω στον χώρο L^1(A) των ολοκληρώσιμων συναρτήσεων σε μετρήσιμο σύνολο A η οποία, όμως, θα είναι εκτός του τελικού διαγωνίσματος.

21 Δεκεμβρίου: Εδώ είναι η πλήρης ύλη. Επαναλαμβάνω ότι ο χώρος L^1(A) δεν περιλαμβάνεται στην εξεταστέα ύλη. Θα συμβούλευα πάντως όποιον ενδιαφέρεται σοβαρά για το μάθημα (και ειδικά όποιον ενδιαφέρεται να πάρει το μάθημα της Αρμονικής Ανάλυσης το επόμενο εξάμηνο) να διαβάσει και αυτό το μέρος της ύλης του μαθήματος.

11 Ιανουαρίου: Την ερχόμενη εβδομάδα θα είμαι στο γραφείο μου για απορίες: Δευτέρα 11:30-13:00, Τρίτη 11:00-13:00, Τετάρτη 11:00-13:00, Πέμπτη 10:00-11:30.

Analysis II (undergraduate), Spring 2023-24.

Βιβλίο, Κεφάλαια: 6,7,9,10,11.

Περιεχόμενο: ολοκλήρωση, ακολουθίες και σειρές συναρτήσεων, μετρικοί χώροι.

Διαλέξεις: 3-5 Τρίτη και Πέμπτη, Α201. Ασκήσεις: 9-11 Πέμπτη , Α214.

Ώρες γραφείου: 1-3 Τρίτη και Πέμπτη, Γ211.

Πρόοδος (προαιρετική): 5-7 Τρίτη 9-4-2024.

Φυλλάδιο 1, Φυλλάδιο 2, Φυλλάδιο 3, Φυλλάδιο 4, Φυλλάδιο 5, Φυλλάδιο 6, Φυλλάδιο 7, Φυλλάδιο 8.

25 Μαΐου: Το μάθημα έχει τελειώσει. Για το διαγώνισμα της ερχόμενης Δευτέρας διαβάστε πάλι προσεκτικά τα φυλλάδια αλλά κοιτάξτε και την σχετική θεωρία. Και μην αποστηθίζετε: ξεχάστε επιτέλους τις πρακτικές του λυκείου. Καλή σας επιτυχία!

20 Ιουνίου: Τα θέματα του τελικού διαγωνίσματος. Όποιος θέλει να δει το γραπτό του ας επικοινωνήσει μαζί μου από 1 Ιουλίου διότι αυτόν τον καιρό λείπω από το Ηράκλειο λόγω της δουλειάς μου.

8 Ιουλίου: Όποιος θέλει να δει το γραπτό του ας έρθει στο γραφείο μου 12-2 αύριο και μεθαύριο.

9 Σεπτεμβρίου: Τα θέματα περιόδου Σεπτεμβρίου.

29 Σεπτεμβρίου: Όποιος θέλει να δει το γραπτό του ας έρθει στο γραφείο μου 11-1 την Τρίτη και την Πέμπτη της ερχόμενης εβδομάδας.

Advanced Harmonic Analysis (advanced graduate), Spring 2023-24.

Notes. They are renewed regularly.

Content: the Fourier transform in the spaces L^1, L^2, L^p (1<p<2), in the space of complex measures, and in the space of tempered distributions. Depending on the time left, I plan to touch some of the following extra subjects: Bochner's theorem for positive definite distributions, the Paley-Wiener theorem for distributions of compact support, the uncertainty principle, Sobolev spaces, singular integrals (e.g. the Hilbert transform and the relation to Complex Analysis), the space of functions of bounded mean oscillation, etc.

Some very good books are: "Introduction to Fourier Analysis on Euclidean Spaces" by Stein and Weiss, "Lectures on Harmonic Analysis" by Wolff, "Functional Analysis" by Yosida, "Methods of Modern Mathematical Physics" by Reed and Simon, "Fourier Analysis" by Duoandikoetxea, "Harmonic Analysis" by Helson. There are many many more!! 

April 16: the notes have been renewed.

April 18: the notes have been renewed.

April 21: here is the first set of exercises. Most of them contain important results of harmonic analysis and most of them are covered as part of the theory in many very good classical books. You will gain a lot if you try them (as exercises) following the steps I propose instead of passively reading them in a book.

April 26: the notes have been renewed. They do not contain anything about the support of a tempered distribution, but I am going to include it soon.

May 2: the notes have been renewed.

May 3: the notes have been renewed.

May 14: the notes have been renewed.

May 21: the notes have been renewed.

May 25: the notes have been renewed. In the last chapter, there is an extra proof of Proposition 9.1 and two extra Propositions, 9.2 and 9.3. The course is over, but I shall keep renewing my notes from time to time. I want to include more things about Bochner's theorem, about the Paley-Wiener theorem and about distributions of compact support. I also want to include the main facts about the Hilbert transform and about more general singular integral transforms.  

Autumn 2022-23: Infinitesimal Calculus III (undergraduate).

Spring 2021-22: Differential Geometry (undergraduate).

Autumn 2021-22: Complex Analysis (undergraduate), Real Analysis (graduate).

Spring 2020-21: Functional Analysis (graduate).

Autumn 2020-21: Complex Analysis (undergraduate), Real Analysis (graduate).

Spring 2019-20: Infinitesimal Calculus II (undergraduate), Functional Analysis (graduate).

Autumn 2019-20: Infinitesimal Calculus I (undergraduate), Complex Analysis (graduate).

Spring 2018-19: Calculus of Several Variables (undergraduate).

Autumn 2018-19: Infinitesimal Calculus I (undergraduate), Harmonic Analysis (advanced graduate).

Spring 2017-18: Probability (undergraduate), Complex Analysis (graduate).

Autumn 2017-18: Infinitesimal Calculus III (undergraduate), Spaces of Analytic Functions (advanced graduate).

Spring 2016-17: Infinitesimal Calculus II (undergraduate), Functional Analysis (graduate).

Autumn 2016-17: Complex Analysis (undergraduate).

Spring 2015-16: Differential Equations (undergraduate), Partial Differential Equations (undergraduate).

Autumn 2015-16, at the Department of Mathematics of the University of Athens: Analytic (and continuous analytic) capacity (seminar).

Spring 2014-15: Infinitesimal Calculus II (undegraduate).

Autumn 2014-15: Infinitesimal Calculus III (undegraduate), Complex Analysis (graduate).

Spring 2013-14: Number Theory (undegraduate), Analysis II (undergraduate).

Autumn 2013-14: Analysis I (undegraduate).

Spring 2012-13: Complex Analysis (undegraduate), Harmonic Analysis (advanced graduate).

Autumn 2012-13: Harmonic Analysis (undegraduate).

Spring 2011-12: Calculus of Several Variables (undegraduate), Fourier Series (advanced graduate reading course).

Autumn 2011-12: Complex Analysis (graduate), Distributions and Fourier Transform (advanced graduate).

Spring 2010-11: Foundations of Mathematics (undergraduate), Infinitesimal Calculus III (undergraduate).

Autumn 2010-11: Infinitesimal Calculus I (undergraduate), Seminar of Infinitesimal Calculus I (undergraduate), Complex Analysis - Harmonic Analysis: H^p spaces (advanced graduate).

Spring 2009-10: Analysis II (undergraduate), Functional Analysis (graduate).

Autumn 2009-10: Complex Analysis (undergraduate).

Spring 2008-09: Real Analysis (undergraduate).

Autumn 2008-09: Infinitesimal Calculus I (undergraduate).

Spring 2007-08: Analysis I (undergraduate).

Autumn 2007-08: Infinitesimal Calculus I (undergraduate, Computer Science Department).

Spring 2006-07: Complex Analysis (graduate), Harmonic Analysis (advanced graduate).

Autumn 2006-07: Analysis II (undergraduate), Euclidean Geometry (undergraduate).

Spring 2005-06: Infinitesimal Calculus I (undergraduate).

Autumn 2005-06: Analysis I (undergraduate), Complex Analysis (graduate).

Autumn 2004-05: Measure Theory (graduate).

Spring 2003-04: Functional Analysis (graduate).

Autumn 2003-04: Analysis I (undergraduate).

Spring 2002-03: Classical Analysis (undergraduate), Set Theory (undergraduate).

Autumn 2002-03: Analysis I (undergraduate).

Spring 2001-02: Infinitesimal Calculus I (undergraduate), Analysis I (undergraduate).

Autumn 2001-02: Complex Analysis (graduate).

Spring 2000-01: Analysis II (undergraduate), Theory of Fields (undergraduate).

Autumn 2000-01: Analysis I (undergraduate), Set Theory (undergraduate).

Spring 1999-00: Infinitesimal Calculus II (undergraduate), Real Analysis (undergraduate).

Autumn 1999-00: Complex Analysis (undergraduate), Number Theory (undergraduate).

Spring 1998-99: Linear Algebra II (undergraduate), Inequalities (advanced undergraduate).

Spring 1997-98: Infinitesimal Calculus III (undergraduate), Set Theory (undergraduate).

Autumn 1997-98: Infinitesimal Calculus I (undergraduate).

Spring 1996-97: Analysis I (undergraduate).

Autumn 1996-97: Partial Differential Equations (undergraduate).

Spring 1995-96: Infinitesimal Calculus II (undergraduate).

Autumn 1995-96: Measure Theory (graduate).

Spring 1994-95: Fourier Series (undergraduate).

Autumn 1994-95: Analysis II (undergraduate).

Spring 1993-94: Functional Analysis (undergraduate).

Autumn 1993-94: Analysis I (undergraduate).

Spring 1992-93: Complex Analysis (advanced graduate).

Autumn 1992-93: Optimization Theory (undergraduate).

Academic years 1990-91 and 1991-92, at the Department of Mathematics of Washington University, St. Louis: Several undergraduate courses (Calculus, Differential Equations, Statistics, etc) and two advanced graduate courses (Classical Potential Theory, Spaces of Holomorphic Functions and Harmonic Measure).

Academic years 1987-88, 1988-89 and 1989-90, at the Department of Mathematics of UW, Madison: Several undergraduate courses (Trigonometry, Calculus, Advanced Calculus, Complex Analysis, Differential Equations, Linear Algebra, Number Theory, Stability of ODE, etc). As far as I remember, at least one advanced graduate course (Fourier Series and H^p spaces).

Spring quarter 1986-87, at the Department of Mathematics of UCLA: Analysis (131A, undergraduate), Linear Algebra (115A, undergraduate).