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Fourier and Wavelet Analysis, 6 ECTS credits.
/Fourier- och waveletanalys/
For:
MMAT
Y
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OBS! |
The course is only offered every second year. It will be offered during 2016.
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Prel. scheduled
hours: 36
Rec. self-study hours: 124
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Area of Education:
Main field of studies: Mathematics, Applied Mathematics
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Advancement level
(G1, G2, A): A
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Aim:
To give the student a deeper understanding of Fourier analysis and to give a theoretical background to wavelets and applications in signal processing. The student should reach good comprehension in the following fields.
- Basic Hilbert space theory
- Fourier series and the Fourier transform
- The theory of distributions
- Multiresolution analysis (MRA)
- Some commonly used wavelet systems
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Prerequisites: (valid for students admitted to programmes within which the course is offered)
Linear Algebra, Caculus in one and severel variables, Fourier Analysis or Transform Theory
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
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Organisation:
Lectures.
The course runs over the entire spring semester.
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Course contents:
Introduction to the Lebesgue integral. Hilbert spaces: Inner products, orthogonal projection, convergence, completeness, orthonormal systems, orthonormal bases. Fourier series: Convergence theorems, Parseval�?Ts identity. The Fourier transform: Basic properties, inversion, Plancherel�?Ts identity, the Schwartz class. Distributions: Operations on distributions, tempered distributions, the Fourier transform, convolutions, periodic distributions, the Poisson summation formula, the sampling theorem. Wavelets: The Haar system, MRA (multiresolutional analysis), the Shannon wavelet, Meyer's wavelets, and wavelets with compact support, e.g., Daubechies' wavelets. Applications to differential equations and filter theory.
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Course literature:
C. Gasquet, P. Witomski: Fourier Analysis and Application. Filtering, Numerical Computation, Wavelets, Springer-Verlag, 1998.
Hand out material.
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Examination: |
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Written assignments
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6 ECTS
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Course language is Swedish.
Department offering the course: MAI.
Director of Studies: Göran Forsling
Examiner: Mats Aigner
Link to the course homepage at the department
Course Syllabus in Swedish
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