Yozgat Bozok University Lifelong Learning Programme

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type of Course Unit	Year of Study	Semester	Number of ECTS Credits
MAT508	ADVANCED CALCULUS II	Elective	1	1	6

Level of Course Unit

Second Cycle

Objectives of the Course

To introduce to student the Fourier series and its applications.

Name of Lecturer(s)

Doç. Dr. Abdullah Sönmezoğlu

Learning Outcomes

1	To define of Fourier series.
2	To learn Bessel's inequality and Parseval's equality.
3	To learn Summability theory.

Mode of Delivery

Formal Education

Prerequisites and co-requisities

None

Recommended Optional Programme Components

None

Course Contents

Abel transformation, the mean value theorem. Convex curves and convex arrays, monotone decreasing polynomial series. Summability method Summability method of arithmetic mean. Abel summability method, numerical inequalities. Hölder's inequality, Minkowski's inequality, for the series and integrals "O" and "o" concepts. The upper limit of the cluster sequences, according to the measure of convergence. Switch to the limit under the Lebesgue integral, Lebesgue points, the Riemann integral of Stiltjes. Helly's theorems, Fubini's theorem, Trigonometric series. Conjugate series, trigonometric series of writing complex. Fourier representation, the Fourier series of complex software. Trigonometric series expansions of periodic functions, Fourier expansions based on orthogonal systems. Trigonometric systems in L-space completeness, uniform convergence of Fourier series. Bessel's inequality, the convergence of Fourier series of L2-space, closed systems. Connection between the closed and complete, Riesz-Fischer theorem, the coefficients with the help of the integral evaluation module. Nörlund summability method, Hölder mean, Euler, Taylor and Borel transformations, Hausdorff mean, Tauberian theorems

Weekly Detailed Course Contents

Week	Theoretical	Practice	Laboratory
1	Abel transformation, the mean value theorem.
2	Convex curves and convex arrays, monotone decreasing polynomial series.
3	Summability method Summability method of arithmetic mean.
4	Abel summability method, numerical inequalities.
5	Hölder's inequality, Minkowski's inequality, for the series and integrals "O" and "o" concepts.
6	The upper limit of the cluster sequences, according to the measure of convergence.
7	Switch to the limit under the Lebesgue integral, Lebesgue points, the Riemann integral of Stiltjes.
8	Mid-Term Exam
9	Helly's theorems, Fubini's theorem, Trigonometric series.
10	Conjugate series, trigonometric series of writing complex.
11	Fourier representation, the Fourier series of complex software.
12	Trigonometric series expansions of periodic functions, Fourier expansions based on orthogonal systems.
13	Trigonometric systems in L-space completeness, uniform convergence of Fourier series.
14	Bessel's inequality, the convergence of Fourier series of L2-space, closed systems.
15	Connection between the closed and complete, Riesz-Fischer theorem, the coefficients with the help of the integral evaluation module.
16	Final Exam

Recommended or Required Reading

A. Zigmund, Trigonometric Series 1-2, Cambridge Univ. Press, 1988. N. K. Bary, Treatise on Trigonometric Series. Pergamon Press, 1964. R. E. Edwards, Fourier series: A modern introduction Vol. 1&2, Springer, 1979,1982. J. P. Kahane, Series de Fourier Absolument Convergentes, Springer, 1970. E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, 1971.

Planned Learning Activities and Teaching Methods

Assessment Methods and Criteria

Term (or Year) Learning Activities	Quantity	Weight
Midterm Examination	1	100
SUM		100
End Of Term (or Year) Learning Activities	Quantity	Weight
Final Examination	1	100
SUM		100
Term (or Year) Learning Activities		40
End Of Term (or Year) Learning Activities		60
SUM		100

Language of Instruction

Work Placement(s)

None

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Midterm Examination	1	2	2
Final Examination	1	2	2
Self Study	14	5	70
Individual Study for Mid term Examination	7	8	56
Individual Study for Final Examination	7	9	63
TOTAL WORKLOAD (hours)			193

Contribution of Learning Outcomes to Programme Outcomes

LO1

LO2

LO3

* Contribution Level : 1 Very low 2 Low 3 Medium 4 High 5 Very High