| TATA57 |
Transform Theory, 4 ECTS credits.
/Transformteori/
For:
FyN
Ii
MED
Yi
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Prel. scheduled
hours: 46
Rec. self-study hours: 61
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Area of Education: Science
Main field of studies: Mathematics, Applied Mathematics
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Advancement level
(G1, G2, A): G1
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Aim:
The course aims to give students a deeper knowledge of Fourier analysis and Transform Theory , which have many applications in both technology and mathematics. After successfully completing the course the student is expected to
- be acquainted with necessary conditions for the existence of the transforms,
- know and be able to derive simple properties of the transforms (e.g. behaviour at infinity, scaling and translation rules, rules for differentiation and integration as well as rules for multiplication by the time variable).
- be able to derive the transforms of the elementary funktions,
- know the invversion theorems, uniqueness theorems, the convolution formulas and the Parseval and Plancherel theorems,
- be able to use the transforms to solve problems such as differential equations, Difference equations and convolution equations
- be acquainted with and be able to use results about uniform convergence (continuity, differentiability and integrability of limit functions, Weierstrassâ?T Majorant Theorem).
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Prerequisites: (valid for students admitted to programmes within which the course is offered)
Calculus, linear algebra
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:
Teaching is done through lectures and problem classes.
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Course contents:
In this course we study some important linear transformations which allow us to translate linear problems (differential, integral and difference equations) into more tractable algebraic problems, whose solutions can then be translated back to solutions of the original problem.
We study: Fourier series, which translate periodic functions into function series. These series are used to analyze periodic behaviour. The problem of convergence of the function series is important and we look att uniform and pointwise convergence as well as convergence in the mean for Fourier series. Besselâ?Ts and Parsevalâ?Ts Theorems are key results. Fourier transforms: these transforms are used to analyze non-periodic behaviour. The inversion formula for Fourier transforms is of central importance, and other tools at our disposal include the rules of calculation, the convolution formula and Plancherelâ?Ts Theorem. The Laplace transform: this transforms functions of a real variable into functions defined in the complex plane and it is used amongst other things for solving initial value problems. The tools at our disposal include rules of calculation, the convolution formula as well as initial and final value theorems. The Z-transform: transforms functions of the natural numbers into power series, and it is used to solve difference equations. The tools at our disposal include rules of calculation and the convolution formula.
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Course literature:
Pinkus, A., Zafrany, S.: Fourier Series and Integral Transforms.
Additional material produced at the Department of Mathematics.
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Examination: |
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Written examination
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4 ECTS
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Course language is Swedish/English.
Department offering the course: MAI.
Director of Studies: Jesper Thorén
Examiner: Peter Basarab-Horwath
Link to the course homepage at the department
Course Syllabus in Swedish
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