But what is a convolution?
by Grant Sanderson · 3Blue1Brown
Grant Sanderson builds discrete convolution visually from sliding-window averages, dice-sum probability, and image blur kernels, then shows why the FFT turns an O(n^2) convolution into O(n log n). Leaves you able to picture the flip-and-slide operation. Free, 23 minutes, and it installs the single mental model everything else on this topic assumes: convolution as flip-slide-multiply-sum, arrived at independently from probability, image filtering, and polynomial multiplication. The FFT segment also motivates why fast convolution matters before any formal machinery appears.
More resources on Convolution
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.
nLab: Convolution
The nLab's abstract treatment of convolution: the operation on functions over a group or groupoid, its integral form on the reals, group and groupoid algebras, distributions, and the categorified Day convolution on functor categories. Assumes graduate algebra.
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.
Convolution - Mathematics of the DFT (CCRMA)
Chapter from Stanford CCRMA's free DFT text covering cyclic convolution: commutativity, the convolution-multiplication duality, graphical flip-and-slide construction, polynomial multiplication, and audio uses including smoothing, ADSR envelopes and matched filtering. Bridges the continuous theory to what a computer actually runs.
Convolution and Green's Formula (MIT 18.03SC)
Derives the convolution integral from Green's formula and shows that a linear time-invariant system's response to arbitrary input is that input convolved with the unit impulse response. The session provides notes, lecture and recitation videos, a mathlet and practice problems with solutions.