--- title: Convolution description: Convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. Learners will understand its applications in filtering, signal processing, and differential equations. category: mathematics subcategory: harmonic-analysis difficulty: beginner, intermediate, advanced url: /subject/convolution --- # Convolution Convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. Learners will understand its applications in filtering, signal processing, and differential equations. ## Available Resources 2 Books • 2 Courses • 3 Websites ## Websites ### 1. 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. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/convolution **Tags:** category-theory, harmonic-analysis, day-convolution, group-algebra, abstract-algebra ### 2. Convolution - Mathematics of the DFT (CCRMA) **Author:** Julius O. Smith III 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ccrma.stanford.edu/~jos/mdft/Convolution.html **Tags:** convolution, dft, digital-signal-processing, audio-dsp, fft ### 3. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Books ### 1. Classical Fourier Analysis, 3rd Edition **Author:** Loukas Grafakos 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1493911937?tag=edmonddante07-20 **Tags:** harmonic-analysis, fourier-analysis, convolution, singular-integrals, graduate-textbook ### 2. Fourier Analysis: An Introduction (Princeton Lectures in Analysis I) **Author:** Elias M. Stein, Rami Shakarchi 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/069111384X?tag=edmonddante07-20 **Tags:** fourier-analysis, convolution, real-analysis, harmonic-analysis, textbook ## Courses ### 1. Convolution and Green's Formula (MIT 18.03SC) **Author:** Arthur Mattuck, Haynes Miller, Jeremy Orloff et al. 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/pages/unit-iii-fourier-series-and-laplace-transform/convolution/ **Tags:** convolution, differential-equations, laplace-transform, impulse-response, greens-formula ### 2. EE261: The Fourier Transform and its Applications **Author:** Brad G. Osgood Brad Osgood's thirty-lecture Stanford course, released free with video, a complete course reader, nine problem sets and solutions. Develops convolution alongside the Fourier transform, the convolution theorem, filtering, sampling and distributions, so you can compute and justify convolutions. This is the masterclass for convolution as it is actually used. Osgood spends multiple lectures on convolution specifically (Lecture 10 pairs it with the central limit theorem), then keeps returning to it through filtering, sampling and the delta function, so the operation is never left as an isolated definition. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://see.stanford.edu/Course/EE261 **Tags:** fourier-transform, convolution, signal-processing, sampling, distributions ## Youtubes ### 1. But what is a convolution? **Author:** Grant Sanderson 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. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=KuXjwB4LzSA **Tags:** convolution, visual-intuition, fft, probability, image-processing --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/convolution