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M7120 Spectral Analysis I

Faculty of Science
Autumn 2008

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)

Supervisor

prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics - Departments - Faculty of Science

Timetable

Thu 12:00–13:50 M2,01021

Prerequisites

M4170 Measure and Integral && M6150 Linear Functional Analysis I
Calculus of complex numbers, Differential calculus and Lebesgue integral, Linear functional analysis

Course Enrollment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

Fields of study the course is directly associated with

Mathematics (programme PřF, M-MA)
Mathematics (programme PřF, N-MA)

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 (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 Newbold. Fourier transform and its applications. 2nd ed. New York: McGraw-Hill, 1986. xx, 474 s. ISBN 0-07-007015-6. info
BRIGHAM, E. Oran. Fast Fourier transform. Englewood Cliffs: Prentice Hall, 1974. 252 s. ISBN 0-13-307496-. 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 s. ISBN 0-8247-9610-1. info
HARDY, G. H. and W. W. ROGOSINSKI. Fourierovy řady : Fourier series (Orig.). Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1971. 155 s. info

Assessment methods