Chapter 02
Convergence of Fourier Series
Dirichlet and Fejer kernels, pointwise and uniform convergence, the Gibbs phenomenon, and Cesaro summability.
Built · Not yet releasedApplied analysis · Graduate course · Staged public deployment
Fourier Analysis
Every signal is a sum of waves; every wave, a frequency.
Fourier analysis decomposes functions into pure frequencies and studies what that decomposition reveals. Building on measure theory and connecting to distribution theory, it develops Fourier series and their convergence, the Fourier transform on the line, convolution and approximate identities, the Schwartz space and inversion, the Plancherel theory on \(L^2\), the extension to tempered distributions, and applications to the heat and wave equations and to sampling. The recurring theme is a dictionary: smoothness on one side is decay on the other, and differentiation becomes multiplication.
8Chapters
GraduateLevel
EN · FRBilingual
Chapter 1 is free now.
All eight chapters are built. Chapters 2 through 8 will be released progressively as public deployment continues.
What the course builds
Three capabilities the course builds
Move between two worlds
Translate freely between a function and its spectrum, and read smoothness, decay, and support off the opposite side.
Master the transform
Own convolution, approximate identities, inversion, and Plancherel as the working toolkit of harmonic analysis.
Solve real equations
Turn the heat and wave equations into algebra, and understand the sampling theorem and Poisson summation.
Before you start
Prerequisites
Fourier analysis is analytic at heart. Measure theory supplies the \(L^1\) and \(L^2\) spaces on which the transform lives and the dominated convergence that justifies passing limits under the integral. The tempered-distribution chapters connect directly to the distribution theory course.
The syllabus
Eight chapters, built in sequence
All eight chapters are built. Chapter 1 is released free now. Chapters 2 through 8 remain visible here as the course roadmap and will be released progressively. Each chapter includes visual investigations, complete definitions and proofs, graded exercises, progressive hints, and detailed corrections.
Chapter 01
Fourier Series
Trigonometric systems, Fourier coefficients, orthogonality on the circle, and the \(L^2\) theory of periodic functions.
Free · Open chapterChapter 03
The Fourier Transform
The transform on the line, its algebraic and analytic properties, and the interplay between decay and smoothness.
Built · Not yet releasedChapter 04
Convolution and Approximate Identities
Convolution as smoothing, approximate identities, and the density of smooth compactly supported functions.
Built · Not yet releasedChapter 05
The Schwartz Space and Inversion
Rapidly decreasing functions, the Fourier inversion theorem, and the transform as an isomorphism of the Schwartz space.
Built · Not yet releasedChapter 06
Plancherel and \(L^2\) Theory
The Plancherel theorem, conservation of energy, and the Fourier transform as a unitary operator on \(L^2\).
Built · Not yet releasedChapter 07
Tempered Distributions
Extending the transform to tempered distributions, the delta and its transform, and differentiation as multiplication.
Built · Not yet releasedChapter 08
Applications: Heat, Wave, and Sampling
Solving the heat and wave equations by transform, the Poisson summation formula, and the Shannon sampling theorem.
Built · Not yet released