---
title: Distributions
description: Distributions, or generalized functions, extend the concept of derivative to all continuous functions. Learners will understand how to formulate and solve weak solutions for partial differential equations that lack classical solutions.
category: mathematics
subcategory: functional-analysis
difficulty: beginner, intermediate, advanced
url: /subject/distributions
---

# Distributions

Distributions, or generalized functions, extend the concept of derivative to all continuous functions. Learners will understand how to formulate and solve weak solutions for partial differential equations that lack classical solutions.

## Available Resources

1 Books • 2 Courses • 5 Websites

## Courses

### 1. Harmonic Analysis

Learn harmonic analysis with Prof. Melrose! This course covers distributions and key concepts.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PL6sQgeH9dxy4BAbUlPy5R1KkL6uqzuRgi

**Tags:** mathematics, harmonic-analysis, distributions, functional-analysis

### 2. Differential Analysis (MIT 18.155)

**Author:** Richard Melrose

First semester of graduate differential analysis: fundamental solutions for elliptic, hyperbolic and parabolic operators, the method of characteristics, Lebesgue integration, distributions, homogeneous distributions, the Fourier transform and asymptotic methods. Provides 18 lecture note files and problem sets with solutions, preparing readers for modern PDE theory.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/

**Tags:** distribution-theory, partial-differential-equations, fourier-transform, lebesgue-integration, fundamental-solutions

## Websites

### 1. Fourier Analysis Notes

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.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://terrytao.wordpress.com/tag/fourier-analysis/

**Tags:** fourier-analysis, harmonic-analysis, fourier-transform, lecture-notes, real-analysis

### 2. Paul Garrett Notes on Distributions

Graduate-level lecture notes on distribution theory from Paul Garrett's functional analysis course at the University of Minnesota, covering generalized functions, test functions, Fourier transforms, and tempered distributions with worked examples and proofs.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.math.umn.edu/~garrett/m/fun/distributions.pdf

**Tags:** distribution-theory, functional-analysis, generalized-functions, lecture-notes

### 3. nLab: Distribution

A collaborative wiki reference article on distributions (generalized functions) in functional analysis, covering their definition via test functions, derivatives, Fourier transforms, and connections to partial differential equations and category-theoretic perspectives.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/distribution

**Tags:** distribution-theory, functional-analysis, generalized-functions, reference

### 4. stattrek.com

StatTrek is an online statistics resource that provides clear tutorials and explanations of probability and statistics. It also features free calculators for distributions, hypothesis tests, confidence intervals, and other statistical methods.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://stattrek.com

**Tags:** websites, science-research, scientific-method

### 5. 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. The Analysis of Linear Partial Differential Operators I

**Author:** Lars Hörmander

A foundational graduate textbook on distribution theory and Fourier analysis, building the mathematical framework for linear partial differential operators, including Sobolev spaces, wave front sets, and the theory of pseudodifferential operators.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/3540006621?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, functional-analysis

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