M7120 Spectral Analysis I

Faculty of Science
Autumn 2024
Extent and Intensity
2/1/0. 5 credit(s). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
doc. Mgr. Peter Šepitka, Ph.D. (lecturer)
Guaranteed by
doc. Mgr. Peter Šepitka, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 16:00–17:50 M6,01011
  • Timetable of Seminar Groups:
M7120/01: Tue 18:00–18:50 M6,01011, P. Šepitka
Prerequisites
Complex numbers, differential and integral calculus, Lebesgue integral, metric spaces, linear functional analysis.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The course is an introduction to the Fourier spectral analysis of both periodic and non-periodic functions.
Learning outcomes
After completing the course the students will understand basic princiles of the Fourier analysis and will be able to apply them in particular problems, for example in the theory of differential equations. Students will understand the connections between the operators of the Fourier transform and its inverse, and will understand convolutions and their utility.
Syllabus
  • 1. Fourier series - equivalent forms of the Fourier series, Dirichlet kernel and pointwise convergence, Fejér kernel and convergence in mean, convergence in norm, L1 and L2 spaces, convolution and correlation, Parseval identities.
  • 2. Fourier transform - existence and inversion, the Fourier theorem, the Plancherel theorem, convolution and correlation, Parseval identities, examples.
  • 3. Generalization of the Fourier series and Fourier transformation - higher dimension, distributions.
Literature
    recommended literature
  • HOWELL, Kenneth B. Principles of Fourier Analysis. Boca Raton-London-New York-Washington: Chapman & Hall, 2001, 776 pp. Studies in Advanced Mathematics. ISBN 0-8493-8275-0. info
  • BRACEWELL, Ronald N. The Fourier transform and its applications. 3rd ed. Boston: McGraw Hill, 2000, xx, 616. ISBN 0073039381. URL info
  • BRACEWELL, Ronald N. Fourier transform and its applications. 2nd ed. New York: McGraw-Hill, 1986, xx, 474. ISBN 0070070156. info
    not specified
  • BRIGHAM, E. Oran. Fast Fourier transform. Englewood Cliffs: Prentice Hall, 1974, 252 s. ISBN 0-13-307496-X. info
  • KUFNER, Alois and Jan KADLEC. Fourierovy řady (Fourier series). Praha: Academia, 1969. info
  • LASSER, Rupert. Introduction to Fourier series. New York: Marcel Dekker, 1996, vii, 285. ISBN 0824796101. info
  • HARDY, G. H. and Werner ROGOSINSKI. Fourierovy řady. Translated by Alois Kufner. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1971, 155 s. URL info
  • BOYCE, William E. and Richard C. DIPRIMA. Elementary differential equations and boundary value problems. 6th ed. New York: John Wiley & Sons, 1996, xvi, 749. ISBN 0471089559. info
Teaching methods
Lectures and exercises.
Assessment methods
Two-hour written final exam (it is needed to reach at least 50 % of points) with oral evaluation of the exam with each student. • The conditions (especially regarding the form of the exam) will be specified according to the epidemiological situation and valid restrictions.
Language of instruction
Czech
Follow-Up Courses
Further Comments
Study Materials
The course is taught once in two years.
Listed among pre-requisites of other courses
Teacher's information
http://www.math.muni.cz/~mkolar
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2010 - only for the accreditation, Autumn 2000, Autumn 2001, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, Autumn 2022.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2024/M7120