Fourier Analysis

Fourier Analysis (MATH790, SFSU)

• Lecture 01: Overview
• [Example 1] Find the Fourier Coefficients
• [Example 2] Find the Fourier Coefficients of \(f(x)=x\)
• [Example 3] Find the Fourier Coefficients of \(f(x)=\frac{(\pi-x)^2}{4}\)
• [Example 4] Find the Fourier Coefficients of \(f(x)=\frac{\pi}{\sin(\pi\alpha)}e^{i(\pi-\theta)\alpha}\)
• Lecture 02: Absolute Convergence of Fourier Series
• More on Convergence (Recap)
• Weierstrass M-Test
• Uniform convergence of derivatives
• [Example 5] Given the Fourier series \(\sum_{n=1}^{\infty}\frac{1}{2^n}\sin(n\theta)\), is \(f\) continuous? is it differentiable?
• [Example 6] Given a Fourier series, is \(f\) continuous?, is it differentiable?
• Lecture 03: Convolution
• Dirichlet's Test (Proof and Examples)
• Dirichlet's Kernel
• General Properties of Convolution
• Lecture 04: Good Kernels
• Lecture 05: Examples of Good Kernels
• Lecture 06: Inner Product Space
• Orthogonality (Pythagorean Theorem and Cauchy Schwarz)
• Lecture 07: Lebesgue Measure, Integration and \(L_1\) Review [TODO]
• Lecture 08: Convergence in Inner Product Spaces
• \(\ell^2(\mathbb{Z})\) is Complete
• Issues with \(C[0,2\pi]\)
• Fundamental Theorem of Hilbert Space
• Parseval’s for Continuous Periodic Functions
• Lecture 09: Banach Spaces
• If \(V\) is finite dimensional, all norms are equivalent.
• If \(V\) is finite dimensional if and only if the unit ball is compact.
• Riesz's Lemma
• Lecture 10: Linear Operators
  • Example: The Fourier Coefficient Operator
• Lecture 11: The Space of Bounded Linear Operators
  • Does the Fourier partial sum converge for every continuous function?
• Lecture 12: Uniform Boundedness Theorem
  • Construction of Divergent Fourier Series
• Lecture 13: The Open Mapping Theorem
  • Bounded Inverse Theorem
  • Application to Fourier Series