MA  4355,    Spring 2026

MWF 12-1,    Room: SEC 206



1.    Instructor:  Demetrio Labate
Office Hours: M 2:30-3:30, F: 1-2 or by appointment
Phone Number:  (713) 743-3492
E-mail address:  dlabate@math.uh.edu
Homepage:  http://www.math.uh.edu/~dlabate

TA: , e-mail:
TA Office hours:
2. Goals and Objectives:
This course is a self-contained introduction to Fourier analysis and wavelets, with a focus on applications from image and signal processing. The concepts presented in this course are underlying some fundamental questions that are highly relevant for data science. Specifically, this course is motivated by fundamental questions about the analysis and processing of signals/data such as: what is the best way to store and transmit a signal? how can we remove unwanted noise from data? how can we automatically identify features of interests in a signal? Fourier analysis and wavelets offer a very powerful and conceptual sound framework to deal with these problems. The ideas covered in this course are at the core of a variety of technologies used in applications including image and video compression, object detection, electronic surveillance, remote sensing and data transmission.
3. Textbook: A first course in wavelets with Fourier analysis, by A. Boggess and F. Narcowich, Wiley, 2nd edition 2009.
Note: Additional material provided by the instructor in this course are for the use of the students enrolled in the course only. Course materials may not be further disseminated without instructor permission. This includes sharing content to commercial course material suppliers such as Course Hero or Chegg. Students are prohibited from sharing materials derived from the instructor’s content including lecture notes, problems and exams
Useful background material:
  • Very brief linear algebra review [from http://alumni.media.mit.edu]
  • Another linear algebra review (with Matlab examples) [by G. Recktenwald Portland State University]

    • 4. Prerequisites:
    MATH 2331 and one of the following: MATH 3333, MATH 3334, MATH 3330, MATH 3363. Students who wish to enroll without having one of the above junior-level courses are encouraged to discuss it with the instructor. While a prior knowledge of Matlab/Pyhton is not required, be aware that Matlab/Python will be used for some homework. The use of the basic Matlab/Python functions is very simple and it will be easy to acquire this basic-level knowledge during the course.
    5. Lectures and Attendance:
    Recording of Class: Students may not record all or part of class, livestream all or part of class, or make/distribute screen captures, without advanced consent of the instructor. If you have or think you may have a disability such that you need to record class-related activities, please contact the Center for Students with Disabilities. If you have an accommodation to record class-related activities, those recordings may not be shared with any other student, whether in this course or not, or with any other person or on any other platform. Classes may be recorded by the instructor. Students may use instructor’s recordings for their own studying and notetaking. Instructor’s recordings are not authorized to be shared with anyone without the prior written approval of the instructor. Failure to comply with requirements regarding recordings will result in a disciplinary referral to the Dean of Students Office and may result in disciplinary action. Attendance is ecouraged as class attendance is strongly correlated with student success but will not be enforced. Note that attendance is required for the weekly quiz.
    6. Homework, tests, exams and grading:
    The only way to understand and master the material presented in class is by working out the homework problems on your own. You are strongly encouraged to work out the homework problems that are assigned regularly and carefully. Copying the homework or watching someone else doing the work for you will bring you minimal benefit. There will be a (almost) weekly homework assignments posted at the link below. At the end of the semester, your worst HW score will be dropped. The homework (evaluation described below) will count 30% towards the final grade.

    Homework submission and evaluation policy: Every week I will administer a short quiz (15 min) based on the homework and I might collect the homework. The quiz will be at the end of the lecture on the day the homework is due. If you are forced to missed class on the day of a quiz for a justified reason (according to UH Excused Absence Policy.), you are allowed to submit your homework by email as a single PDF file no later than the due date, at 1pm. No late submissions will be accepted. A late or missed HW will receive a 0 score. Submitted Quiz/homework should be delivered in a "professional" form which allows a grader to read your solutions without unnecessary effort or ambiguity. In particular, your solution should either be handwritten or typed in a neat and legible form; if you submit scanned pages, they should be perfectly legible, with pages ordered and clear indication of which problem is being solved. Quizzes/Homework which does not satisfy these guidelines might receive a penalty in the score. Requests for review or reconsideration of answers for a regrade must be submitted within one week of the return of the graded assignment.

    HOMEWORK PROBLEMS:
    (The list below will be updated during the semester. Solutions will be posted after due date)

    Tests. There will be three tests counting 40% towards the final grade set (tentatively) on WED FEB 25, WED MAR 25, MON APR 20. The worst of your 3 tests will be half-dropped; that is, the 3 tests counts 40% towards the final grade, where the best two tests will count 16% each, the worst one will count 8%.

    Here are the tests with solutions (to be updated during the semester):


    Final exam. The final exam counts 30% towards the final grade. This is scheduled on TUE MAY 12 at 2 pm. Make up tests will only be allowed for reasons justified by the official UH Excused Absence Policy.


    Here are some old Tests and their solutions: Tests+Solutions
    Grading: Each student will receive a score based 30% on the homework/quizzes, 40% on the tests and 30% on the final. The grade will be determined according to a set point scale:
           90%-100%: A, 80%-89%: B, 70%-79%: C, 60-69% D; F is less than 60% (+ and - will also be used).
    Policy on grades of I (Incomplete): The grade of "I" (Incomplete) is a conditional and temporary grade given when a student, for reasons beyond his or her control, has not completed a relatively small portion of all requirements. Sufficiently serious, documented situations include illness, death in the family, etc.


    7. Course outline (note: the schedule and topics below are subject to change in the event of extenuating circumstances, including, but not limited to class cancellations due to inclement weather):

    Inner product spaces [Ch.0, Sec.0.1-0.5]

    • Inner product spaces.
    • The spaces of square integrable functions and square summable series.
    • Schwarz and triangle inequalities.
    • Orthogonal projections and the least squares fit.

    Fourier series and transform [Ch.1, Sec 1.1-1.3; Ch.2, Sec. 2.1-2.4]

    • Computation of Fourier series.
    • Convergence of Fourier series.
    • The Fourier transform.
    • Convolutions.
    • Linear filters.
    • The sampling theorem: Analog to digital and digital to analog conversions.
    • From analog to digital filters.
    • The Discrete Fourier transform (DFT), FFT, its use for the approximate computation of integral Fourier transforms [Ch.3,Sec.3.1].

    Wavelets [Ch.4, Sec 4.1-4.3; Ch.5, Sec. 5.1-5.2]

    • The Haar wavelet.
    • Multiresolution analysis.
    • The scaling relation.
    • Properties of the scaling function.
    • Decomposition and reconstruction.
    • Wavelet design in the frequency domain.
    • The Daubechies wavelet.


    COMMUNICATION: I will communicate with you (class announcements and graded quizzes/tests) using your official UH email or by replying to your email address. If I or the TA do not respond to your email within two working days, please resend the email (it is possible that your email is not coming through). Please, be sure to use a subject line and to identify your name and course if do not use your offocoal UH email. It is your responsibility to ensure that I and your TA are aware of issues you may have with the course. If you have complaints about a grade on any evaluation, you need to communicate with me within 1 week since the graded quiz or test was sent to you. Failure to effectively initiate timely communication is not a valid basis for a grade grievance and cannot be used as such.
     

    Resources for Online Learning: The University of Houston is committed to student success, and provides information to optimize the online learning experience through our Power-On website. Please visit this website for a comprehensive set of resources, tools, and tips including: obtaining access to the internet, AccessUH, and Blackboard; requesting a laptop through the Laptop Loaner Program; using your smartphone as a webcam; and downloading Microsoft Office 365 at no cost. For questions or assistance contact UHOnline@uh.edu.

    UH Email: Email communications related to this course will be sent to your Exchange email account which each University of Houston student receives. The Exchange mail server can be accessed via Outlook, which provides a single location for organizing and managing day-to-day information, from email and calendars to contacts and task lists. Exchange email accounts can be accessed by logging into Office 365 with your Cougarnet credentials or through Acccess UH. They can also be configured on IOS and Android mobile devices. Additional assistance can be found at the

    Academic Integrity: University of Houston students are expected to adhere to the Academic Honesty Policy as described in the UH Undergraduate Catalog. “Academic dishonesty” means employing a method or technique or engaging in conduct in an academic endeavor that contravenes the standards of ethical integrity expected at the University of Houston or by a course instructor to fulfill any and all academic requirements.
    Academic dishonesty includes,     but is not limited to, the following: Plagiarism; Cheating and Unauthorized Group Work; Fabrication, Falsification, and Misrepresentation; Stealing and Abuse of Academic Materials; Complicity in Academic Dishonesty; Academic Misconduct.
    Please, refer to UH Academic Honesty website     and the UH Student Catalog for the definition of these terms and university’s policy on Academic Dishonesty. Anyone caught cheating will be reported to the department for further disciplinary actions, receive sanctions as explained on these documents, and will have an academic dishonesty record at the Provosts office. The sanctions for confirmed violations of this policy shall be commensurate with the nature of the offense and with the record of the student regarding any previous infractions. Sanctions may include, but are not limited to: a lowered grade, failure on the examination or assignment in question, failure in the course, probation, suspension, or expulsion from the University of Houston, or a combination of these. Students may not receive a W for courses in which they have been found in violation of the Academic Honesty Policy. If a W is received prior to a finding of policy violation, the student will become liable for the Academic Honesty penalty, including F grades.
    Cases of using websites/apps like GroupMe and Chegg to facilitate cheating     will be punished to the maximum degree possible. This includes grades of 0 on any assignments or exams and potentially a grade of F for the course.

    Students with disabilities:     Written requests issued by the Office of Disability Services will be honored.

    Additional resources:     Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach CAPS (www.uh.edu/caps) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus. http://www.uh.edu/caps/outreach/lets_talk.html