Transform Theory, 4 credits
Transformteori, 4 hp
TATA57
Main field of study
Mathematics Applied MathematicsCourse level
First cycleCourse type
Programme courseExaminer
Peter Basarab-HorwathDirector of studies or equivalent
Jesper ThorénEducation components
Preliminary scheduled hours: 46 hRecommended self-study hours: 61 h
Available for exchange students
YesMain field of study
Mathematics, Applied MathematicsCourse level
First cycleAdvancement level
G1XCourse offered for
- Industrial Engineering and Management - International, M Sc in Engineering
- Biomedical Engineering, M Sc in Engineering
- Applied Physics and Electrical Engineering - International, M Sc in Engineering
- Physics, Bachelor´s Programme
Entry requirements
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.
Prerequisites
Calculus, linear algebraIntended learning outcomes
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, Weierstrass’ Majorant Theorem).
Course content
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. Bessel’s and Parseval’s 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 Plancherel’s 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.
Teaching and working methods
Teaching is done through lectures and problem classes.
Examination
TEN1 | Written examination | 4 credits | U, 3, 4, 5 |
Grades
Four-grade scale, LiU, U, 3, 4, 5Department
Matematiska institutionenDirector of Studies or equivalent
Jesper ThorénExaminer
Peter Basarab-HorwathEducation components
Preliminary scheduled hours: 46 hRecommended self-study hours: 61 h
Course literature
Pinkus, A., Zafrany, S.: Fourier Series and Integral Transforms. Kompletterande material (exempelsamling) utgivet av MAI.Code | Name | Scope | Grading scale |
---|---|---|---|
TEN1 | Written examination | 4 credits | U, 3, 4, 5 |
Regulations (apply to LiU in its entirety)
The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.
LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva.
Note: The course matrix might contain more information in Swedish.
I | U | A | Modules | Comment | ||
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1. DISCIPLINARY KNOWLEDGE AND REASONING | ||||||
1.1 Knowledge of underlying mathematics and science (G1X level) |
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1.2 Fundamental engineering knowledge (G1X level) |
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1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level) |
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1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level) |
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1.5 Insight into current research and development work |
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2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES | ||||||
2.1 Analytical reasoning and problem solving |
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2.2 Experimentation, investigation, and knowledge discovery |
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2.3 System thinking |
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2.4 Attitudes, thought, and learning |
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2.5 Ethics, equity, and other responsibilities |
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3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION | ||||||
3.1 Teamwork |
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3.2 Communications |
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3.3 Communication in foreign languages |
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4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT | ||||||
4.1 External, societal, and environmental context |
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4.2 Enterprise and business context |
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4.3 Conceiving, system engineering and management |
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4.4 Designing |
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4.5 Implementing |
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4.6 Operating |
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5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS | ||||||
5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development |
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5.2 Economic conditions for knowledge development |
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5.3 Identification of needs, structuring and planning of research or development projects |
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5.4 Execution of research or development projects |
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5.5 Presentation and evaluation of research or development projects |
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