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Research draft

Fourier series

vr.tr.fourier-series · XCT.REL

Let an agent handle Fourier series by function, coefficients, convergence and applications.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent handle Fourier series by function, coefficients, convergence and applications.

A representation of a periodic function as a sum of sines and cosines (or complex exponentials) with frequencies that are integer multiples of a fundamental frequency.

What it is for: Analysing periodic signals, heat and wave equations, and vibrations.

It can be compute Fourier coefficients; approximate functions with partial sums; analyse signal harmonics; solve differential equations.

Distinguishing features

Periodic functions as sums of harmonics

Coefficients from integrals

Convergence depends on smoothness

Related to the Fourier transform

What it looks like

A sum of sine and cosine terms; spectra of harmonics.

How it is recognised

Coefficients a_n and b_n or c_n

Harmonic spectrum

Gibbs phenomenon near jumps

Related models

is a kind of - category

function series

is generalised by - extension

Fourier transform

is used in - application

signal processing

is related to - series

Taylor series

In practice

Families and kinds

trigonometric Fourier series

complex exponential form

sine and cosine series

multidimensional Fourier series

Failure modes and hazards

Gibbs overshoot misread as signal

Assuming convergence everywhere

Sign and normalisation conventions

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Series Which expansion.

Function and period.

Function

What is expanded.

Function

Function and period.

  1. Which periodic function is expanded, with what period? definition
  2. Is it even, odd or neither? definition

Coefficients

Computation.

Coefficients

Coefficients.

  1. What are the Fourier coefficients? measurement
  2. Which normalisation convention is used? definition
Convergence Where it works.

Convergence depends on smoothness.

Pointwise

Conditions.

Pointwise

Convergence.

  1. Does the series converge to the function at each point? boundary
  2. What happens at discontinuities? definition

Gibbs

Overshoot.

Gibbs

Gibbs phenomenon.

  1. How large is the Gibbs overshoot? measurement
  2. How can it be reduced? action
Applications Uses.

Fourier series are practical.

Signals

Harmonics.

Signals

Signal analysis.

  1. What harmonics does the signal contain? measurement
  2. How is that used in audio or power systems? definition

Equations

Heat and waves.

Equations

Differential equations.

  1. How does a Fourier series solve the heat or wave equation? definition
  2. Which boundary conditions apply? boundary
Learning Teaching.

Visuals help.

Visual

Partial sums.

Visual

Visualisation.

  1. How do partial sums approach the function? definition
  2. Which examples help, such as square waves? action

Transform

Link to FT.

Transform

Fourier transform.

  1. How does the series relate to the Fourier transform? definition
  2. When is each used? boundary

What the second pass must settle

  • Should forms be separate entries?
  • How should conventions be recorded?
  • How should the transform be linked?
Draft as JSON