MATH-205: Analysis IV - Lebesgue measure, Fourier analysis

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Lectures in this course (41)

Heat Equation: Separation of Variables

Covers the application of separation of variables method to solve the heat equation.

Analysis IV: Le Spaces

Introduces Le spaces, measurable functions, Holder inequality, and LP space properties.

Completeness of LP

Explores the completeness of LP, discussing implications of mathematical conditions and the convergence of series.

Laplace Equation: Decomposition and Solutions

Covers the Laplace equation, decomposition of linear problems, and solutions through separation of variables.

Approximation by Smooth Functions

Discusses approximation by smooth functions and the convergence of function sequences in normed vector spaces.

Analysis IV: Laplace Equation Solutions

Explores Laplace equation solutions through separation of variables and general solution writing for complex problems.

Analysis IV: Exam Instructions

Covers exam instructions, wave equation solutions for vibrating ropes, and determining the final solution based on initial conditions.

Analysis IV: Convolution and Partial Derivatives

Covers the properties of functions in Rn and the computation of partial derivatives by induction.

Lebesgue Measure and Fourier Analysis

Explores Lebesgue measure, Fourier analysis, PDE applications, and optimal transport in PDEs.

Analysis IV: Convolution and Hilbert Structure

Explores convolution, uniform continuity, Hilbert structure, and Lebesgue measure in analysis.

Lebesgue Integration: Cantor Set

Explores the construction of the Lebesgue function on the Cantor set and its unique properties.

Partial Differential Equations: Separation of Variables

Explores the method of separation of variables for solving partial differential equations with boundary conditions and discusses convergence properties of functions in different spaces.

Approximation of Functions: Principles and Theorems

Explores the principles and theorems of function approximation, focusing on Littlewood 3 principles and Egorov's theorem.

Analysis IV: Convergence and Approximation in L² Space

Explores convergence and approximation in L² space, emphasizing the limitations of continuous functions and the importance of closed sets.

Fourier Analysis and PDEs

Explores Fourier analysis, PDEs, historical context, heat equation, Laplace equation, and periodic boundary conditions.

Laplace Equation in Polar Coordinates

Explores the Laplace equation in polar coordinates and its solution process through separation of variables.

Trigonometric Polynomials: Periodicity and Fourier Coefficients

Explores trigonometric polynomials, periodicity, and Fourier coefficients in mathematical analysis.

Trigonometric Polynomials: Fourier Inversion and Plancherel Formulas

Explores trigonometric polynomials, emphasizing Fourier inversion and Plancherel formulas.

Convergence of Fourier Series

Explores the convergence of Fourier series in L² space with trigonometric polynomials and approximation theorems.

Fourier Coefficients: Trigonometric Series

Explores Fourier coefficients in trigonometric series, emphasizing even and odd functions.

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