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trigonometric series

vr.tr.trigonometric-series · thing-q1149336

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

Thing Registry Cross-cutting context XCT.REL
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Series Form.

Coefficients

Computation.

Coefficients

Coefficients.

  1. What are the Fourier coefficients of this function? measurement
  2. Over which interval? definition

Form

Sine or cosine series.

Form

Series form.

  1. Is a full, sine or cosine series appropriate? definition
  2. Why? definition
Convergence Behaviour.

Conditions

Theorems.

Conditions

Convergence conditions.

  1. Under which conditions does the series converge to the function? provenance
  2. Which theorem applies? provenance

Gibbs

Overshoot.

Gibbs

Gibbs phenomenon.

  1. Does the Gibbs phenomenon appear near discontinuities? boundary
  2. How large is the overshoot? measurement
Applications Uses.

Signals

Spectra.

Signals

Signal analysis.

  1. How does the series describe the harmonics of a signal? definition
  2. How many terms are needed? measurement

PDEs

Heat and waves.

PDEs

Differential equations.

  1. How is the series used to solve the heat or wave equation? action
  2. What boundary conditions apply? definition
Computing Numerics.

FFT

Discrete version.

FFT

Discrete Fourier transform.

  1. How does the discrete Fourier transform relate to the series? definition
  2. Which library computes it? provenance

Teaching

Visualisation.

Teaching

Teaching.

  1. How can partial sums be visualised for learners? action
  2. Which examples help? action

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.REL
Other names and narrower kinds
Ramanujan's sum

What it is Filled

A series of the form a0/2 + sum over n of (an cos nx + bn sin nx); when coefficients come from a function by integration, it is a Fourier series, used to represent periodic functions.

Why it exists Filled

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

Distinguishing features Filled

  • Sum of sines and cosines
  • Represents periodic functions
  • Convergence can be subtle
  • Foundation of harmonic analysis

What robots and AI may and may not do Filled

Must not

  • Claim convergence where it does not hold or has not been shown.
  • Present truncated series as exact results without stating the error.
  • Use signal analysis to extract private speech or data without consent.

Only with a human decision

  • Relying on series approximations in safety-critical engineering.

May

  • Compute and explain trigonometric and Fourier series, stating convergence conditions.
  • Use them to analyse signals the agent is authorized to process.

Moral aspects Filled

  • Errors in approximations can matter in engineering and medicine.
  • Signal analysis can expose private information.

Who is affected

  • Students and engineers
  • People whose signals are analysed

Owners Filled

Steward

Nobody: a mathematical concept held in common.

Links to other meta-models Filled

parent

  • Q1474614 - registry parent class

related

  • function series - category
  • Fourier series - kind
  • harmonic analysis - field
  • Fourier transform - continuous analogue

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.trigonometric-series
  • Wikidata: Q1149336 (https://www.wikidata.org/wiki/Q1149336)

Direct properties not applicable Not applicable

Not applicable

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Sums of sin nx and cos nx
  • Fourier coefficients an and bn
  • Power series use powers of x instead
  • Written as sums of sines and cosines.

Capabilities and actions required Filled

  • compute Fourier coefficients
  • discuss convergence
  • apply to signals and heat equations
  • relate to Fourier transforms

Hazards and failure modes required Filled

  • Assuming pointwise convergence
  • Gibbs phenomenon surprises

Standards and interfaces required Filled

  • ISO 80000-2 notation

Context of use required Filled

  • Representing periodic functions in mathematics, physics and engineering.
  • Fourier series
  • general trigonometric series
  • Fourier sine and cosine series
  • Ramanujan sums (related)

Sources Filled

  1. Wikidata item Q1149336: trigonometric series - identity and sense of the item
  2. Wikipedia: Trigonometric series - general description of the item

Open questions

  • Should Fourier series be a separate entry?
  • How should convergence theorems be linked?
  • How should numerical methods be linked?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q1149336/spec.json