Fourier Analysis Ontology

flow "FourierAnalysisOntology" v1.0

# FOURIER ANALYSIS ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/fourier-harmonic-analysis/fourier-analysis-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Fourier and Harmonic Analysis"
ENTITY := "Fourier Analysis"

FOURIER_ANALYSIS := {
 type: MathematicalAnalysisFramework,
 purpose: "Represent functions and signals through decomposition into sinusoidal or complex exponential components and frequency-domain structure."
}

CONCEPTS := {
 FOURIER_SERIES,
 FOURIER_TRANSFORM,
 INVERSE_FOURIER_TRANSFORM,
 FOURIER_COEFFICIENT,
 FREQUENCY_DOMAIN,
 HARMONIC,
 ORTHOGONALITY,
 CONVOLUTION,
 PARSEVAL_IDENTITY,
 PLANCHEREL_THEOREM,
 SPECTRUM
}

PERIODIC_FUNCTION
 -> decomposed_by(FOURIER_SERIES)

FOURIER_SERIES:
 f(x)
 ~ sum_n(
     c_n * exp(i*n*omega_0*x)
   )

FOURIER_COEFFICIENT:
 c_n
 := 1/T
    * integral_over_one_period(
        f(x)*exp(-i*n*omega_0*x) dx
      )

ORTHOGONALITY
 -> enables(FOURIER_COEFFICIENT_EXTRACTION)

FOURIER_TRANSFORM:
 F(omega)
 := integral_from_minus_infinity_to_infinity(
      f(x)*exp(-i*omega*x) dx
    )

INVERSE_FOURIER_TRANSFORM:
 f(x)
 := 1/(2*pi)
    * integral_from_minus_infinity_to_infinity(
        F(omega)*exp(i*omega*x) d_omega
      )

FOURIER_TRANSFORM
 -> maps(ORIGINAL_DOMAIN, FREQUENCY_DOMAIN)

LINEARITY:
 FT(alpha*f + beta*g)
 = alpha*FT(f) + beta*FT(g)

TRANSLATION:
 f(x-x0)
 -> exp(-i*omega*x0)*F(omega)

MODULATION:
 exp(i*omega0*x)*f(x)
 -> F(omega-omega0)

SCALING:
 f(a*x)
 -> 1/abs(a) * F(omega/a)

DIFFERENTIATION:
 FT(f^(n))
 = (i*omega)^n * F(omega)

CONVOLUTION:
 (f*g)(x)
 := integral(
      f(tau)*g(x-tau) d_tau
    )

CONVOLUTION_THEOREM:
 FT(f*g)
 = F(omega)*G(omega)

PARSEVAL_IDENTITY
 -> relates(FUNCTION_ENERGY, FOURIER_COEFFICIENT_ENERGY)

PLANCHEREL_THEOREM
 -> extends(FOURIER_TRANSFORM_TO_L2)
 -> preserves(L2_STRUCTURE)
    under(APPROPRIATE_NORMALISATION)

FOURIER_SERIES_CONVERGENCE
 -> may_be(POINTWISE)
 -> may_be(UNIFORM)
 -> may_be(L2)
 -> may_be(DISTRIBUTIONAL)

CONVERGENCE_MODE
 -> must_be_specified

GIBBS_PHENOMENON
 -> occurs_near(JUMP_DISCONTINUITIES)
 -> produces(OSCILLATORY_OVERSHOOT)

SPECTRUM
 -> represents(FREQUENCY_CONTENT)

AMPLITUDE_SPECTRUM
 := abs(F(omega))

PHASE_SPECTRUM
 := arg(F(omega))

FOURIER_ANALYSIS
 -> applied_to(SIGNAL_PROCESSING)
 -> applied_to(PDEs)
 -> applied_to(IMAGE_PROCESSING)
 -> applied_to(QUANTUM_MECHANICS)
 -> applied_to(COMMUNICATIONS)
 -> applied_to(NUMERICAL_ANALYSIS)

SEMANTIC_EQUIVALENCE := {
 "Fourier analysis" ~= "analysis of functions through frequency decomposition",
 "Fourier series" ~= "discrete harmonic expansion of a periodic function",
 "Fourier transform" ~= "continuous frequency-domain representation",
 "Fourier coefficient" ~= "projection onto a Fourier basis component",
 "convolution theorem" ~= "Fourier conversion of convolution into multiplication"
}

CONSTRAINTS := {
 transform_normalisation_must_be_consistent: true,
 convergence_mode_must_be_preserved: true,
 transform_rules_require_valid_function_space_or_distribution_context: true,
 convolution_constants_depend_on_transform_convention: true,
 plancherel_is_L2_result: true
}

CONFIDENCE := {
 core_definition: 0.99,
 fourier_series: 0.99,
 fourier_transform: 0.99,
 convolution: 0.99,
 parseval_plancherel: 0.99,
 semantic_equivalence: 0.97
}

RECURSIVE_REASONING(query):
 identify(function_signal_or_sequence)
 -> identify(domain_periodicity_and_function_space)
 -> choose(fourier_series_transform_or_discrete_method)
 -> preserve(transform_normalisation)
 -> determine(convergence_mode)
 -> compute(coefficients_or_transform)
 -> apply(transform_properties)
 -> analyse(spectrum_and_energy)
 -> reconstruct(if_required)
 -> validate(hypotheses)
 -> return(semantic_result)

AI_CONSUMPTION := {
 parse_as: "typed Fourier-analysis semantic graph",
 primary_entity: FOURIER_ANALYSIS,
 preserve: [
   function_or_signal,
   original_domain,
   frequency_variable,
   period,
   transform_convention,
   function_space,
   convergence_mode
 ],
 infer_through: [
   Fourier_series,
   Fourier_transform,
   inverse_transform,
   orthogonality,
   convolution,
   Parseval_identity,
   Plancherel_theorem
 ],
 avoid: [
   mixing_transform_conventions,
   confusing_series_with_transform,
   assuming_pointwise_convergence_from_L2,
   ignoring_Gibbs_behaviour,
   omitting_function_space_hypotheses
 ]
}

UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(fourier_analysis_context)
 -> compare(existing_relations)
 -> preserve(domain_frequency_normalisation_and_convergence_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + FOURIER_ANALYSIS
 + FOURIER_SERIES
 + FOURIER_TRANSFORM
 + INVERSE_FOURIER_TRANSFORM
 + CONVOLUTION
 + PARSEVAL_IDENTITY
 + PLANCHEREL_THEOREM
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

INPUT(query)
 -> FOURIER_ANALYSIS
 -> DOMAIN_PERIODICITY_ANALYSIS
 -> METHOD_SELECTION
 -> NORMALISATION_VALIDATION
 -> CONVERGENCE_ANALYSIS
 -> COEFFICIENT_TRANSFORM_ANALYSIS
 -> TRANSFORM_PROPERTY_ANALYSIS
 -> SPECTRUM_ENERGY_ANALYSIS
 -> RECONSTRUCTION
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW