M7120 Spectral Analysis I

Faculty of Science
Autumn 2010 - only for the accreditation
Extent and Intensity
2/0/0. 2 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
Mgr. Jiří Zelinka, Dr. (lecturer)
Guaranteed by
prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
M4170 Measure and Integral && M6150 Linear Functional Analysis I
Calculus of complex numbers, Differential calculus and Lebesgue integral, 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 nonperiodic functions.
After completing the course, students will be able to use methods of Fourier analysis to solve various problems, eg when solving differential equations.
Syllabus
  • Fourier series (FS): 3 equivalent forms of FS (complex, trigonometric and amplitude-phase form), Dirichlet kernel and pointwise convergence, Fejér kernel and convergence in mean, convergence in spaces $L^1$ and $L^2$, statements on cyclic convolution and correlation, Parseval identities.
  • Fourier transform (FT): existence and inversion (theorems by Fourier and Plancherel), properties, statements on convolution and correlation, Parseval identities, examples.
  • Multivariate Fourier series and transforms.
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. Fourier transform and its applications. 2nd ed. New York: McGraw-Hill, 1986, xx, 474. ISBN 0070070156. info
  • 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
Teaching methods
Teaching is through lectures.
Assessment methods
Oral exam.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
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 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, Autumn 2024.