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	Sem.	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., fzoltan@math.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],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)

	
Bibliography	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.