Fourier Analysis Notes
What's new (Terence Tao's blog)
Archive of Terence Tao's blog posts and graduate lecture notes on Fourier analysis, covering the Fourier transform, Fourier series, Littlewood-Paley theory, and restriction problems. Readers gain rigorous, research-level working knowledge of harmonic analysis techniques and their applications to analysis and PDE.
More resources on Fourier Transforms
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Terry Tao's Blog: Harmonic Analysis Notes
Terence Tao's blog posts tagged harmonic analysis, including graduate lecture notes and research discussions on Fourier transforms, singular integrals, restriction theory, and related estimates. Readers with real analysis background gain the working tools and intuitions of modern harmonic analysis from a leading practitioner.
Fourier Analysis
Lecture series by Steve Brunton (University of Washington): Fourier series, the Fourier transform, the discrete and fast Fourier transforms, spectral derivatives, and PDE solutions. Viewers learn to apply Fourier methods to heat and wave equations and signal data in Python and MATLAB.
But what is the Fourier Transform? A visual introduction.
3Blue1Brown's animated introduction to the Fourier transform, building it from winding a signal around a circle to find frequency components. After watching, you can picture why the transform separates a waveform into constituent frequencies.
Classical Fourier Analysis, 3rd Edition
Graduate reference establishing the Lp theory of convolution on Euclidean space: Young's inequality, approximate identities, convolution operators and multipliers, maximal functions, and singular integrals of convolution type. The standard citation when a sharp convolution estimate is needed.
Fourier Analysis: An Introduction (Princeton Lectures in Analysis I)
Volume one of the Princeton Lectures in Analysis. Treats convolution as the multiplication underlying Fourier theory, building good kernels and approximate identities to prove convergence theorems, then applying convolution to the heat and wave equations and to Poisson summation.