Instructor: Dr. Péter Simon and Dr. Kálmán Cziszter
Text: handouts
Prerequisite: Calculus
Course description: After quickly reviewing basic notions and concepts
in analysis, we turn to the study of sequences of functions
and prove rudiments of Fourier series. Topological concepts are also
discussed in detail.
Topics:
1. Basic calculus
- metrics and sequences
- convergent and Cauchy sequences, complete metric spaces
- Basic topology (open and closed sets, compact sets, connectedness)
- Continuous functions
- absolute and conditional convergence
- Tests for convergence
- Power series
- Fundamental theorem of calculus
- Differentiation in R^{n} (Implicit and Inverse function theorems)
- Riemann-Stieltjes integrals (integration by parts, fundamental theorem of integral calculus)
- Integration in R^{n }(Jordan measure)
- Series of functions (uniform, pointwise and L_{2} convergence)
- Arzela-Ascoli theorem, compact sets in C(X)
- Fourier series (Fourier coefficients, Bessel's inequality, Parseval formula, applications)
- Theorem of Fejér and Riesz-Fisher
- Weierstrass approximation theorem