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Lecturer(s)
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Boháč Pavel, doc. RNDr. Ph.D.
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Novák Pavel, prof. Ing. Ph.D.
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Course content
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Chapter 1. Measure and Lebesgue integral 2.1 Fundaments of measure theory 2.2 Measurable functions and integral 2.3 Integrals depending on parameters 2.4 Lebesgue integral in R and functions with bounded variation Chapter 2. Spaces of integrable functions 2.1 Basic properties 2.2 Completeness, separability 2.3 Mappings in these spaces, continous embeddings Chapter 3. Fourier series 3.1 Orthogonal a orthonormal systems of functions 3.2 Pointwise and uniform konvergence of Fourier series
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Learning activities and teaching methods
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Interactive lecture, Lecture supplemented with a discussion, Lecture with practical applications, Task-based study method
- Contact hours
- 65 hours per semester
- Preparation for comprehensive test (10-40)
- 40 hours per semester
- Preparation for an examination (30-60)
- 55 hours per semester
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| prerequisite |
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| Knowledge |
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| define and explain basic notions of mathematical analysis in one and/or several dimensions |
| explain the definition and basic properties of Newton integral |
| explain the definition and basic properties of Riemann integral |
| explain basics of Fourier series. |
| Skills |
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| calculate indefinite and/or definite integrals (of certain types) in one dimension using integrations-by-parts and/or substitution methods |
| calculate multiple integrals using Fubini theorem within Riemann theory |
| derive and prove the convergence of Fourier series for piecewise smooth functions |
| Competences |
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| N/A |
| learning outcomes |
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| Knowledge |
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| define and explain basic notions of abstract measure theory |
| define and explain basic notions of theory of the abstract Lebesgue integration |
| define and explain basic notions of theory of the Lebesgue spaces |
| define and explain basic notions of theory of Lebesgue integration in R |
| define and explain basic notions of theory of Fourier series |
| Skills |
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| work with abstract structures of measure theory |
| use of limit theorems in calculating integrals |
| use of the Fubini and Tonelli theorems in calculating multiple integrals |
| analyze integrals depending on parameters |
| Competences |
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| N/A |
| teaching methods |
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| Knowledge |
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| Lecture supplemented with a discussion |
| Interactive lecture |
| Task-based study method |
| Skills |
|---|
| Practicum |
| Competences |
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| Task-based study method |
| assessment methods |
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| Knowledge |
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| A) Basics of abstract measure theory. B) Basics of theory of the abstract Lebesgue integration. C) Basic theory of the Lebesgue spaces. D) Lebesgue integration in R. E) Basic theory of Fourier series. |
| Combined exam |
| Skills demonstration during practicum |
| Skills |
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| A) Work with abstract structures of measure theory. B) Use of limit theorems in calculating integrals. Smazat C) Use of the Fubini and Tonelli theorems in calculating multiple integrals. D) Analysis of integrals depending on parameters. |
| Combined exam |
| Competences |
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| Combined exam |
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Recommended literature
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Folland, G. B. Real analysis : modern techniques and their applications. Second edition. 2007. ISBN 978-0-471-31716-6.
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Jarník, Vojtěch. Diferenciální počet II. Praha : Academia, 1976.
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Jarník, Vojtěch. Integrální počet. II. Praha : Academia, 1976.
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Kolmogorov, A. N.; Fomin, S.V. Základy teorie funkcí a funkcionální analýzy. Vyd. 1. Praha : SNTL, 1975.
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Nagy, Jozef; Nováková, Eva; Vacek, Milan. Lebesgueova míra a integrál. 1. vyd. Praha : SNTL, 1985.
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Rudin, Walter. Analýza v reálném a komplexním oboru. Vyd. 2., přeprac. Praha : Academia, 2003. ISBN 80-200-1125-0.
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