Course notes in Fourier Analysis
Short description
The Fourier transform was originally invented by Fourier to study the heat partial differential equation. Since then it evolved into one of the main tools used to study general functions and (hidden) patterns in these functions (usually periodic functions like sines and cosines). In this course we will see why and how to decompose functions into their periodic components, the properties and results arising from such transforms, and some of its applications with emphasis on differential equations.
Syllabus
Geometry in infinite dimensional spaces: Inner products, norms, distance, orthonormal systems
Fourier series: The Fourier basis of sines and cosines, Fourier coefficients, The Fourier series convergence (in norm, pointwise, uniformly), applications in differential and integral equations
Fourier transforms: Definition, basic arithmetic properties of the transform (scaling, rotation, translation, conjugate), the Riemann-Lebesgue lemma, the inverse Fourier transform, Plancherel theorem, convolution, applications to signal processing and partial differential equations.
Laplace transform: Definition and properties, Heaviside function, convolution, application to differential and integral equations.
Prerequisites
- Basic Linear Algebra: Linear system of equations, matrices, eigenvalues and eigenvectors.
- Inner Products: Inner products and norms, orthonormal bases, orthogonal projection.
- Calculus: Limits, continuity, derivatives and integrals on finite and infinite segments.