Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate

SUBJECT

Code
Subject
MMA0016 Special Topics in Mathematical Analysis
Section
Semester
Hours: C+S+L
Category
Type
Mathematics - in Hungarian
4
2+1+0
speciality
optional
Mathematics-Computer Science - in Hungarian
4
2+1+0
speciality
optional
Teaching Staff in Charge
Assoc.Prof. FINTA Zoltan, Ph.D.,  fzoltanmath.ubbcluj.ro
Aims
Getting to know some knowledges of the theory of Fourier series.
Content
1) Orthogonal systems of functions (Gram-Schmidt orthogonalization process, Fourier series in the orthogonal system, Bessel@s inequality, Parseval@s equality)
2) Orthogonal systems of functions (trigonometric series, completeness of the trigonometric system, trigonometric Fourier series)
3) Orthogonal systems of functions (orthogonal polynomial systems, Haar@s system of orthogonal functions)
4) The convergence of a trigonometric Fourier series (properties, example of Fejer)
5) The convergence of a trigonometric Fourier series (Dirichlet formulas, the Riemann-Lebesgue lemma and the localization principle)
6) The convergence of a trigonometric Fourier series (Dini@s conditions, the uniform convergence of a trigonometric Fourier series)
7) The convergence of a trigonometric Fourier series (the theorem of Dirichlet-Jordan, consequences)
8) Summation of a trigonometric Fourier series (Fejer@s formulas, Fejer@s theorem, consequences)
9) Another summation methods (A-summation, (H,r)-summation, (C,r)-summation, the theorem of Abel, the theorem of Frobenius)
10) Another summation methods (Abel-Poisson method of summation, properties)
11) Complex Fourier series (the space L^{2}([-\pi,pi],C), definitions, properties)
12) The Fourier transform (the Fourier transformation, the inverse Fourier transformation, the Fourier transform, examples, properties)
13) The Fourier transform (convergence of the inverse Fourier transformation, the space
S(R;C), the inversion formula)
14) The Fourier transform (applications)



References
1) Szokefalvi-Nagy B.: Valos fuggvenyek es fuggvenysorok, Tankonyvkiado, Budapest, 1977.
2) Balazs M.-Kolumban J.: Matematikai Analizis, Dacia Konyvkiado, Kolozsvar, 1978.
3) Precupanu A.: Analiza matematica (Functii reale), Editura Didactica si Pedagogica,
Bucuresti, 1976.
4) Finta Z.: Matematikai Analizis II, Presa Universitara Clujeana, Kolozsvar, 2007.
5) Yosida K.: Functional Analysis, Springer, Berlin, 1965.
6) Zorich V.A.: Mathematical Analysis, I-II, Springer, Berlin, 2004.
Assessment
Exam.
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject