---
tag: YCK709QB06
kind: concept
name: Integral transform
msc: [44A05]
specializes: [H8P36XFV8K]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_transform
same_as:
  wikidata: Q877802
  wikipedia: Integral transform
---

## Summary

An integral transform turns a function $f$ into a new function $Tf$ by integrating $f$ against a fixed function of two variables, the [[kernel|9M0NNNFJGM]].
In symbols, $(Tf)(u) = \int f(t) K(t, u)\, dt$.
The [[Laplace transform|0V49JEZW06]], the [[Fourier transform|DPVM34ADZV]] and the [[Mellin transform|N5PVGRJE1T]] are the main examples.
A good transform turns a hard operation, such as differentiation or convolution, into an easy one, such as multiplication.
It can be undone by an [[inverse transform|GC96MQC0PP]].

## Explanations

### L1 human

Think of the kernel as a family of test patterns, one for each value of $u$.
The number $(Tf)(u)$ measures how much $f$ looks like the pattern with label $u$.
For the Laplace transform the patterns are the decaying exponentials $e^{-st}$.
For example, the constant function $1$ on $[0, \infty)$ has Laplace transform $\int_0^\infty e^{-st}\, dt = 1/s$ for $s > 0$.
A problem about $f$ often becomes simpler when it is rewritten as a problem about $Tf$.