MMA0016 | Special Topics in Mathematical Analysis |
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 |