---
tag: 6Y1RY5767R
kind: concept
name: Degenerate kernel
msc: [45B05, 45P05]
specializes: [GSMBXT7FPK]
sources:
  - url: https://en.wikipedia.org/wiki/Fredholm_integral_equation
---

## Summary

A degenerate kernel is a [[kernel|GSMBXT7FPK]] that is a finite sum of products $a_i(x) b_i(s)$ of functions of one variable.
It is also called a separable kernel or a kernel of finite rank, since its integral operator is a [[finite-rank operator|Z8FABSYQTS]].
An integral equation of the second kind with a degenerate kernel reduces to a finite [[system of linear equations|0GT4WWGWH2]].
Approximating a general kernel by degenerate ones is a main route to Fredholm theory and to numerical methods.

## Explanations

### L1 human

A degenerate kernel splits into finitely many pieces, each a function of $x$ times a function of $s$.
Then the integral in the equation only produces finitely many unknown numbers, the integrals $\int b_i \varphi$.
Solving for these numbers is plain linear algebra.
For example, $K(x, s) = 1 + x s$ is degenerate with two terms, while $e^{x s}$ is not degenerate.