---
title: "The Fourier Transform on the Line"
description: "Extend Fourier series from an interval to the whole line, where the sum over discrete frequencies becomes an integral over a continuum."
canonical: https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U
source: https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U.md
retrieved: 2026-09-12
---

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# The Fourier Transform on the Line

Extend Fourier series from an interval to the whole line, where the sum over discrete frequencies becomes an integral over a continuum.

Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 23
Page: https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U

## Ready when they can

- Derive the transform as a limit of Fourier series on intervals of growing length
- Compute the transform of a Gaussian and of an indicator function
- State the inversion theorem and the hypotheses it needs

## Lesson: From repeating tones to a continuum

A Fourier series on an interval of length L uses discrete tones spaced 1 over L apart. As L grows without bound, the tones pack closer together, and in the limit the sum over discrete tones becomes an integral over all frequencies. That is how a Fourier series turns into the Fourier transform: the period tends to infinity and the frequency ladder becomes a continuum.

**Example.** Two landmark pairs show what the transform does: it rewrites a function as a recipe of frequencies. The transform of a Gaussian bell curve is another Gaussian, with narrow in time meaning wide in frequency and back again. The transform of an indicator function of an interval is a sinc shaped ripple, sin of x over x: sharp cutoffs in one domain ring in the other.

The inversion theorem says that applying the transform twice, with matching normalization, recovers the original function. It needs hypotheses to hold: enough decay and smoothness, with absolutely integrable plus bounded variation as the classic safe zone. Kim's claim that inversion holds for every function with no hypotheses at all is false, and wild functions are where it breaks.

**Tip.** Differential equations love the transform because differentiation becomes multiplication by frequency, turning calculus into algebra. When a problem differentiates a signal repeatedly, try transforming first, solving the algebra, and inverting at the end.

**Recap.** Stretch the period to infinity, trade tones for a continuum, and invert only where the hypotheses hold.

## Practice

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

## Needs first

- [Fourier Series](https://lightmysky.com/learn/mathematics/fourier-series-mt_DJ3iwfj7NK)
- [Lp Spaces and the Inequalities They Rest On](https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE)

## Opens up

- [Convolution, Plancherel and Transforming a Derivative](https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt)