Fourier Analysis: An Introduction
by Elias M. Stein, Rami Shakarchi · Elias M. Stein, Rami Shakarchi
First volume of the Princeton Lectures in Analysis, developing Fourier series and integrals with applications to the heat and wave equations, finite Fourier analysis, and Dirichlet's theorem on primes. Readers learn convergence and summability results and how Fourier methods connect analysis to number theory.
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More resources on Fourier Series
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.
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.
Paul's Online Math Notes - Fourier Series
Paul Dawkins' differential equations notes on Fourier series, with worked examples computing sine, cosine, and full Fourier series on symmetric intervals. Learners will be able to find series coefficients, use orthogonality and even/odd symmetry, and prepare for separation of variables in partial differential equations.
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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.
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.
An Introduction to Harmonic Analysis
Concise graduate text on harmonic analysis covering Fourier series on the circle, convergence and summability, Fourier transforms on the line, Banach algebra methods, and locally compact abelian groups. Readers will be able to prove core convergence theorems and work with Fourier analysis in functional-analytic settings.