M7120 Spectral Analysis I

Faculty of Science
Autumn 2007 - for the purpose of the accreditation
Extent and Intensity
2/0/0. 2 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Martin Kolář, Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Martin Kolář, Ph.D.
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. The exposition starts with the more transparent case of univariate functions being generalized to the multivariate case at the final stage. Numerous application examples are given throughout. The periodic case: Fourier series and the sequence of Fourier coefficients as a spectral representation of periodic functions. Fourier series are considered to be a special case of an orthogonal expansion and its equivalent forms are given. Various convergence concepts are introduced and relation to integral operators of periodic convolution and correlation is established. The view of Parseval identity as a power spectral density. Nonperiodic case: integral (continuous) Fourier transform (IFT) and its inverse being explained as a nonperiodic analogy to the spectrum of Fourier coefficients and the backward Fourier series expansion. Fundamental properties of IFT, relation to nonperiodic integral convolution and correlation operators.
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
Assessment methods (in Czech)
Výuka: přednáška, Zkouška: ústní s písemnou přípravou
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
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/~vesely/educ_cz.html#fa1
The course is also listed under the following terms 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, Autumn 2024.