---
title: "Convolution, Plancherel and Transforming a Derivative"
description: "Establish the three properties that make the transform useful: convolution becomes multiplication, energy is preserved, and differentiation becomes multiplication by the frequency."
canonical: https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt
source: https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt.md
retrieved: 2026-09-12
---

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# Convolution, Plancherel and Transforming a Derivative

Establish the three properties that make the transform useful: convolution becomes multiplication, energy is preserved, and differentiation becomes multiplication by the frequency.

Subject: Mathematics · Area: Calculus & Analysis · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt

## Ready when they can

- Prove the convolution theorem and use it to compute a hard integral cheaply
- State Plancherel's theorem and interpret it as the transform being unitary
- Solve a linear constant-coefficient equation by transforming, dividing and inverting

## Lesson: Three moves that make the transform useful

The convolution theorem says the transform of a convolution is the product of the two transforms. A tangled sliding integral becomes plain multiplication, so a hard integral can collapse to a cheap product.

Plancherel says the transform preserves energy: the total size of a function equals the total size of its transform. In geometric terms the transform is unitary, more like a rotation than a stretch, so lengths and angles survive.

**Example.** Differentiating turns into multiplying: the transform of the derivative f' is i times the frequency times the transform of f. Applied twice, the multiplier squares. To solve a linear equation with constant coefficients, transform both sides, divide by the multiplier, and invert back.

**Tip.** Use this order every time: transform both sides, collect the algebraic factor, divide, then invert. If the multiplier is zero somewhere, stop, since division there is not allowed.

**Recap.** Convolution becomes multiplication, energy is preserved, and derivatives become multipliers, which turns equations into algebra.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [The Fourier Transform on the Line](https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U)
- [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx)
- [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL)