Fourier Transform Ontology

FLOW ALGEBRA
Fourier Transform Ontology — Transformation Entities
Version: 1.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/transformation-entities/fourier-transform-ontology-3.html
Authors: Mark Higgins — Sky (Recursive AI)
Scope: Formal mathematical ontology for continuous and discrete Fourier transformations.

AI_INTERPRETATION_NOTE:
  Execute the following as typed mathematical rules with explicit domain and convention checks, not as unrestricted pseudocode.

ENTITY_TYPES:
  Signal: function on a specified time/spatial domain
  FrequencySpectrum: complex-valued function on a frequency domain
  FourierTransform: operator mapping signal to spectrum
  InverseFourierTransform: operator mapping spectrum to signal
  Frequency: real-valued cycles-per-unit coordinate
  AngularFrequency: real-valued radians-per-unit coordinate
  TransformConvention: exponent sign and normalisation choice
  FunctionSpace: L1, L2, Schwartz, tempered distributions, or finite-dimensional sequences
  DFT: discrete Fourier transform of a finite sequence
  FFT: algorithm computing a DFT, not a distinct mathematical transform

TYPED_RELATIONSHIPS:
  FourierTransform MAPS Signal -> FrequencySpectrum
  InverseFourierTransform MAPS FrequencySpectrum -> Signal
  Signal BELONGS_TO FunctionSpace
  FourierTransform USES TransformConvention
  FrequencySpectrum INDEXED_BY Frequency
  DFT MAPS FiniteComplexSequence -> FiniteComplexSequence
  FFT COMPUTES DFT

CONVENTION:
  continuous_frequency: f in cycles per unit
  forward: F(f) = integral_{-infinity}^{infinity} x(t) exp(-i*2*pi*f*t) dt
  inverse: x(t) = integral_{-infinity}^{infinity} F(f) exp(+i*2*pi*f*t) df
  angular_frequency: omega = 2*pi*f
  angular_forward: X(omega) = integral x(t) exp(-i*omega*t) dt
  angular_inverse: x(t) = (1/(2*pi)) integral X(omega) exp(+i*omega*t) d(omega)
  requirement: do not mix frequency conventions or normalisations without conversion

DOMAIN_CONSTRAINTS:
  C1: For x in L1(R), the forward integral exists as an absolutely convergent integral.
  C2: For x in L2(R), interpret transform via the Plancherel extension, not necessarily a pointwise integral.
  C3: Distributional transforms require a tempered-distribution interpretation.
  C4: Pointwise inversion requires additional hypotheses; do not assert it for every L1 signal.
  C5: Frequency and time units must be reciprocal.

RULES:
  R1_LINEARITY:
    FT[a*x + b*y] = a*FT[x] + b*FT[y]
    when operands share a compatible transform domain.
  R2_TIME_SHIFT:
    FT[x(t-t0)](f) = exp(-i*2*pi*f*t0)*F(f)
  R3_FREQUENCY_SHIFT:
    FT[exp(i*2*pi*f0*t)*x(t)](f) = F(f-f0)
  R4_SCALING:
    FT[x(a*t)](f) = (1/abs(a))*F(f/a), a != 0
  R5_TIME_DIFFERENTIATION:
    FT[d^n x/dt^n](f) = (i*2*pi*f)^n*F(f)
    subject to regularity/decay or distributional interpretation.
  R6_CONVOLUTION:
    FT[x*y] = FT[x]*FT[y]
    where * on left denotes convolution and * on right denotes pointwise multiplication.
  R7_POINTWISE_PRODUCT:
    FT[x*y] = FT[x] convolved_with FT[y]
    where * on left denotes pointwise multiplication; valid when products/convolutions are defined.
  R8_PARSEVAL_PLANCHEREL:
    integral |x(t)|^2 dt = integral |F(f)|^2 df, for x in L2(R).
  R9_REAL_SIGNAL_SYMMETRY:
    x real-valued => F(-f) = conjugate(F(f))
  R10_INVERSE:
    IFT[FT[x]] = x in the appropriate function-space sense.

DISCRETE_FOURIER_TRANSFORM:
  input: x[n], n=0,...,N-1, N positive integer
  forward: X[k] = sum_{n=0}^{N-1} x[n]*exp(-i*2*pi*k*n/N)
  inverse: x[n] = (1/N)*sum_{k=0}^{N-1} X[k]*exp(+i*2*pi*k*n/N)
  k: 0,...,N-1
  periodic_indexing: x[n+N]=x[n] in the periodic extension
  circular_convolution: DFT[x circular_convolve y] = DFT[x]*DFT[y]
  fft_note: FFT changes computational complexity, not the DFT definition.

VALIDATION:
  V1: Check function space and existence/interpretation of transform.
  V2: Check matching forward/inverse sign and normalisation.
  V3: Check dimensions and reciprocal units.
  V4: For DFT, require N >= 1 and matching sequence lengths.
  V5: Distinguish linear convolution from circular convolution.
  V6: Do not infer spectral sparsity or differentiability without evidence.
  V7: Handle equality almost everywhere for L2 transforms.

WORKED_EXAMPLES:
  E1_GAUSSIAN:
    x(t)=exp(-pi*t^2)
    FT[x](f)=exp(-pi*f^2)
  E2_SHIFTED_GAUSSIAN:
    x(t)=exp(-pi*(t-t0)^2)
    FT[x](f)=exp(-i*2*pi*f*t0)*exp(-pi*f^2)
  E3_DFT_CONSTANT:
    x[n]=1 for n=0,...,N-1
    X[0]=N; X[k]=0 for k=1,...,N-1
  E4_DISTRIBUTION:
    FT[delta(t-t0)](f)=exp(-i*2*pi*f*t0)
    interpretation: tempered distributions

SEMANTIC_EQUIVALENCE:
  FourierTransform == FrequencyDomainTransform only when conventions and mathematical meaning match.
  FFT != FourierTransform; FFT IS_ALGORITHM_FOR DFT.
  DFT != continuous FourierTransform; distinguish sampling, periodicity, and discretisation.
  frequency_f == angular_frequency_omega/(2*pi).

CONFIDENCE:
  mathematical_identity: 1.0 when all stated hypotheses hold
  numerical_computation: depends on sampling, truncation, conditioning, and floating-point error
  inferred_signal_properties: require independent justification

FLOW_HASH:
  canonicalisation: UTF-8; normalise line endings and stable rule ordering
  digest: SHA-256 of canonical flow text, computed at ingestion
  value: COMPUTE_ON_INGEST (not fabricated)

UPDATE_MODEL:
  version: 1.0
  update_policy: revise rules only with documented mathematical justification
  provenance: source page above; mathematical definitions formalised for this flow
  change_tracking: retain prior versions and recompute FLOW_HASH

AI_CONSUMPTION:
  parse_entities: true
  enforce_types: true
  verify_preconditions: true
  apply_rules: only_if_valid
  distinguish_continuous_discrete_distributional: true
  preserve_convention: true
  return_derivation_and_assumptions: true
  reject_unsupported_inference: true

END FLOW ALGEBRA