Lectures in course MATH-205: Analysis IV - Lebesgue measure, Fourier analysis

Differentiation under Integral Sign

Explores differentiation under the integral sign, comparing it with the Riemann integral and discussing key assumptions and theorems.

Analysis: Measure and Integration

Introduces the course on measure and integration, focusing on developing a new theory to overcome the limitations of the Riemann integral.

Lebesgue Measure: Definition and Properties

Explores the Lebesgue outer measure, its properties, and applications in measure theory.

Lebesgue Measure: Properties and Existence

Covers the properties of the Lebesgue measure and its existence.

Analysis IV: Measurable Sets and Properties

Covers the concept of outer measure and properties of measurable sets.

Fourier Series: Periodic Functions

Explores Fourier series for periodic functions, emphasizing coefficient calculation and function representation using sine and cosine terms.

Measurable Sets: Countable Additivity

Explores the countable additivity of measurable sets and the properties of sigma algebra, highlighting the significance of understanding measurable functions in analysis.

Analysis IV: Measurable Sets and Functions

Introduces measurable sets, functions, and the Cantor set properties, including ternary development of numbers.

Uniform Convergence of Fourier Series

Covers the concept of uniform convergence of Fourier series and Dirichlet's theorem application.

Fourier Transform: Analysis IV

Covers the Fourier transform in Analysis IV, discussing Fourier series and solutions for various problems.

Lebesgue Integration: Simple Functions

Covers the Lebesgue integration of simple functions and the approximation of nonnegative functions from below using piecewise constant functions.

Fourier Transform: Properties and Applications

Covers the properties and applications of the Fourier transform and its relation to Heisenberg's uncertainty principle.

Lebesgue Integral: Properties and Convergence

Covers the Lebesgue integral, properties, and convergence of functions.

Analysis IV: Convergence Theorems and Integrable Functions

Covers convergence theorems and integrable functions, including the Lebesgue integral and Borel-Cantelli sets.

Fourier Inversion Formula

Covers the Fourier inversion formula, exploring its mathematical concepts and applications, emphasizing the importance of understanding the sign.

Fourier Inversion Formula and Plancherel Identity

Covers the Fourier inversion formula and the Plancherel identity, emphasizing their importance in mathematical analysis.

Fourier Transform and Partial Differential Equations

Explores the application of Fourier transform to PDEs and boundary conditions.

Lebesgue Integral: Comparison with Riemann

Explores the comparison between Lebesgue and Riemann integrals, demonstrating their equivalence when the Riemann integral exists.

Optimal Transport: Analysis IV

Explores optimal transport, integration theorems, proof strategies, and function applications in multiple dimensions.

Heat Kernel and Gaussian Analysis

Explores the Heat Kernel and Gaussian analysis in solving the heat equation.