International Intensive Courses

dedicated to Dirac operators, Hypercomplex and Harmonic Analysis

Please pay attention to the following rules for the online courses:


  1. It is highly appreciated if you use your real and complete name when entering the system. 

  1. Switch your microphone off when the lecture begins. You can switch it back when asking questions or replying. 

(!) un-authorized recording and distribution is illegal and it will be reported to the competent authorities.

Time zone converter

Most common time zones (recall, some countries changed already to Standard Time, or Winter Time):

  1. CET zone (Central European Time)

  2. PST zone (Pacific Standard Time)

  3. EST zone (Eastern Standard Time)

  4. CST (China Standard Time)

The online series of International Intensive Courses dedicated to Dirac operators, Hypercomplex and Harmonic Analysis are jointly organized by Chapman University, Politecnico di Milano, and University of Aveiro. These sequence of events will take place online and are directed to postgraduate students, young researchers, and all who wishes to expand their knowledge on current topics in these and related areas.

Courses will occur with a trimestral frequency. Interested participants are kindly request to register via the on-line form:

2nd Course - December 06 to December 09, 2021. November 23 to December 04, 2020.

(online platform ZOOM)

Lecturer: Professor David F. Walnut, George Mason University, Fairfax, VA 22030, USA

Title: Fundamentals of Time-Frequency Analysis and Wavelet Theory

Abstract: In this course we provide an introduction to some of the motivations and central results in the field of time-frequency analysis. References will be provided for each topic covered so that the student can consult them for further information and details. The course will be delivered in online format in the form of two back-to-back one hour lectures each day, with a short break in between lectures.

Tentative schedule: (see below for a time converter)

Monday December 06,  10AM - 12noon or 10:00-12:00 (EST)

Tuesday December 07,  10AM - 12noon or 10:00-12:00 (EST)

Wednesday December 08,  10AM - 12noon or 10:00-12:00 (EST)

Thursday December 09,  10AM - 12noon or 10:00-12:00 (EST)

Lecture 7. Multiresolution Analysis. In this lecture, we present the construction of orthonormal bases of wavelets by means of multiresolution analysis. We define this concept and show how it is used to construct the wavelet bases described in Lecture 6. We also mention in this context the notion of wavelet packets and the Daubechies scale of smooth, compactly-supported wavelet bases. 

Lecture 8. Co-orbit Spaces. In this final lecture, we present the theory of co-orbit spaces due to Feichtinger and Gröchenig which realizes the construction of Gabor and wavelet bases as manifestations of notions related to integrable, irreducible group representations. From this unifying theory come results about function spaces with natural Gabor and wavelet atomic decompositions and operators on these spaces.  

Lecture 1. Some Time-Frequency Transformations. In this lecture, we motivate and introduce some of the historically important time-frequency distributions, including the Wigner distribution, the Short-time Fourier transform, and the Radar Ambiguity function. We also discuss related uncertainty principles.

Lecture 2. Gabor Analysis. We begin this lecture with the notion of information area first put forward by D. Gabor in the 1940s. This is a discrete time-frequency representation that is motivated by uncertainty principles. Properties of the resulting Gabor atoms are discussed and some of their advantages and shortcomings are highlighted.

Lecture 3. Frame Theory. The notion of a frame in a Hilbert space is the most natural context in which to define representations of functions in terms of Gabor atoms. Abstract frame theory in finite dimensions and in infinite-dimensional Hilbert spaces is presented. Some historical remarks and application of frames are given.

Lecture 4. Structure Theorems for Gabor Frames. Frames for the Hilbert space             that consist of Gabor atoms dis- play a rich and deep mathematical structure. In this lecture, we discuss existence of Gabor frames, representations of the frame operator, density theorems for Gabor frames, and the Wexler-Raz and Ron-Shen duality principles.

Lecture 5. Wavelet Transforms – Time and Scale. In this lecture we introduce the Continuous Wavelet Transform which constitutes a time-scale distribution similar to the time-frequency distributions described earlier. Motivation for this transform comes from the Calderon Reproducing Formula and Littlewood-Paley Theory. Atomic decompositions of certain function spaces related to time and scale structure serve as a historical backdrop for further developments of
the theory. 

Lecture 6. Wavelet Orthonormal Bases. In this lecture we discuss construction of orthonormal wavelet bases including the Haar wavelet basis, the Shannon wavelet basis, and the Meyer wavelet basis. These bases correspond to a prescribed partition of time-frequency space into rectangles of unit area, as envisioned by Gabor. We also present the construction of Local Cosine bases of Coifman and Weiss, and the orthonormal Wilson bases of Daubechies, Jaffard, and Journé. 

1st Course - November 23 to December 04, 2020. 

Lecturer: Professor Daniel Alpay, Chapman University, Orange, CA, USA

Title: A Course on Positive Definite Functions and Reproducing Kernel Spaces



This event is supported by the

International Society for Analysis, its Applications, and Computation