Fourier Analysis Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Fourier Analysis Ontology within the Kireth Research Semantic Library. Fourier analysis studies the representation of functions and signals as superpositions of sinusoidal or complex exponential components. This ontology supports Fourier series, Fourier transforms, discrete-time Fourier transforms, discrete Fourier transforms, spectral representations, orthogonality, frequency content, amplitude and phase spectra, Parseval and Plancherel relations, convolution and multiplication duality, shift, scaling and differentiation properties, convergence, windowing, spectral leakage, time-frequency extensions and relationships with signal processing, harmonic analysis, differential equations, numerical analysis and communications mathematics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Fourier Analysis Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Signal Processing Mathematics / Fourier Analysis"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/signal-processing-mathematics/fourier-analysis-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = FourierAnalysis
 SYMBOLIC_FORMS = [
  "X(omega)=integral x(t)e^{-j omega t}dt",
  "x(t)=sum_k c_k e^{j k omega_0 t}",
  "X[k]=sum_n x[n]e^{-j 2 pi kn/N}"
 ]
 ENTITY_CLASS = FrequencyDecompositionFramework
 PRIMARY_CONTEXT = SignalProcessingMathematics
 DEFINITION = "Fourier analysis is the mathematical decomposition and representation of functions or signals in terms of sinusoidal or complex exponential frequency components."
 SEMANTIC_ROLE = "Provides the machine-readable framework for Fourier series, continuous and discrete Fourier transforms, spectral coefficients, frequency-domain dualities, energy relations, convergence and numerical spectral analysis."
}

SCOPE {
 INCLUDE = [
  FourierAnalysis,
  HarmonicComponent,
  FrequencyComponent,
  ComplexExponential,
  Sinusoid,
  AngularFrequency,
  FundamentalFrequency,
  HarmonicFrequency,
  FourierSeries,
  TrigonometricFourierSeries,
  ComplexFourierSeries,
  FourierCoefficient,
  ContinuousFourierTransform,
  FourierTransform,
  InverseFourierTransform,
  DiscreteTimeFourierTransform,
  DTFT,
  DiscreteFourierTransform,
  DFT,
  InverseDFT,
  FastFourierTransform,
  FFT,
  Spectrum,
  MagnitudeSpectrum,
  PhaseSpectrum,
  PowerSpectrum,
  EnergySpectrum,
  SpectralDensity,
  Orthogonality,
  BasisFunction,
  ParsevalTheorem,
  PlancherelTheorem,
  ConvolutionTheorem,
  ModulationProperty,
  TimeShiftProperty,
  FrequencyShiftProperty,
  ScalingProperty,
  DifferentiationProperty,
  IntegrationProperty,
  DualityProperty,
  HermitianSymmetry,
  EvenSymmetry,
  OddSymmetry,
  Convergence,
  PointwiseConvergence,
  MeanSquareConvergence,
  GibbsPhenomenon,
  WindowFunction,
  SpectralLeakage,
  FrequencyResolution,
  TimeFrequencyAnalysis,
  ShortTimeFourierTransform,
  STFT
 ]

 EXCLUDE_AS_PRIMARY = [
  LaplaceTransformWithoutFourierContext,
  ZTransformWithoutUnitCircleOrFrequencyInterpretation,
  WaveletAnalysisWithoutFourierRelation,
  GenericSpectrumDiscussionWithoutTransformSemantics
 ]

 DISAMBIGUATION_RULE =
  "When a function or signal is represented, analysed or reconstructed using sinusoidal or complex exponential frequency components, interpret the context as Fourier analysis."
}

CORE_CONCEPTS {

 FourierAnalysis {
  definition = "The analysis of functions and signals through frequency-domain decomposition."
 }

 FourierSeries {
  definition = "A representation of a periodic function as a sum of harmonically related sinusoids or complex exponentials."
 }

 FourierCoefficient {
  definition = "A scalar coefficient describing the contribution of a Fourier basis component."
 }

 FourierTransform {
  definition = "An integral transform mapping a nonperiodic or suitably general signal from time or spatial domain to continuous frequency domain."
 }

 DiscreteTimeFourierTransform {
  definition = "A frequency representation of a discrete-time sequence that is periodic in frequency."
 }

 DiscreteFourierTransform {
  definition = "A finite-dimensional transform mapping a finite discrete sequence to discrete frequency bins."
 }

 Spectrum {
  definition = "A representation of signal content as a function of frequency."
 }

 MagnitudeSpectrum {
  definition = "The magnitude of Fourier coefficients or transform values as a function of frequency."
 }

 PhaseSpectrum {
  definition = "The phase angle of Fourier coefficients or transform values."
 }

 ParsevalTheorem {
  definition = "An equality relating signal energy or power in one domain to an equivalent sum or integral in another."
 }

 GibbsPhenomenon {
  definition = "Persistent oscillatory overshoot near jump discontinuities in truncated Fourier series."
 }

 SpectralLeakage {
  definition = "Spreading of spectral energy across bins caused by finite observation or non-bin-centred frequency content."
 }
}

RELATIONS {
 FourierAnalysis --is_child_of--> SignalProcessingMathematics
 FourierSeries --represents--> PeriodicSignal
 FourierTransform --represents--> AperiodicOrGeneralSignal
 FourierCoefficient --weights--> BasisFunction
 BasisFunction --may_be--> ComplexExponential
 DFT --samples--> DTFT
 FFT --computes--> DFT
 Spectrum --has--> MagnitudeSpectrum
 Spectrum --has--> PhaseSpectrum
 ParsevalTheorem --relates--> TimeDomainEnergy
 ParsevalTheorem --relates--> FrequencyDomainEnergy
 ConvolutionTheorem --maps--> ConvolutionToMultiplication
 TimeShiftProperty --changes--> PhaseSpectrum
 FrequencyShiftProperty --shifts--> Spectrum
 WindowFunction --reduces_or_redistributes--> SpectralLeakage
 GibbsPhenomenon --occurs_near--> Discontinuity
 STFT --extends--> FourierAnalysis
}

OPERATORS {
 COMPLEX_FOURIER_SERIES_COEFFICIENT(x,T0,k) :=
  (1/T0)*integral_over_one_period x(t)*exp(-j*k*omega0*t) dt

 COMPLEX_FOURIER_SERIES(x,c_k,omega0,t) :=
  sum_{k=-infinity}^{infinity} c_k*exp(j*k*omega0*t)

 FOURIER_TRANSFORM(x,t,omega) :=
  integral_-infinity^infinity x(t)*exp(-j*omega*t) dt

 INVERSE_FOURIER_TRANSFORM(X,omega,t) :=
  (1/(2*pi))*integral_-infinity^infinity X(omega)*exp(j*omega*t) d omega

 DTFT(x,n,omega) :=
  sum_{n=-infinity}^{infinity} x[n]*exp(-j*omega*n)

 DFT(x,N,k) :=
  sum_{n=0}^{N-1} x[n]*exp(-j*2*pi*k*n/N)

 IDFT(X,N,n) :=
  (1/N)*sum_{k=0}^{N-1} X[k]*exp(j*2*pi*k*n/N)

 PARSEVAL_CONTINUOUS(x,X) :=
  integral |x(t)|^2 dt
  = (1/(2*pi))*integral |X(omega)|^2 d omega

 PARSEVAL_DFT(x,X,N) :=
  sum_n |x[n]|^2
  = (1/N)*sum_k |X[k]|^2

 CONVOLUTION_THEOREM_TIME(x,h) :=
  F{x*h}=X*H

 MULTIPLICATION_THEOREM_TIME(x,h) :=
  F{x*h_pointwise}=(1/(2*pi))*(X convolution H)

 TIME_SHIFT(x,t0) :=
  x(t-t0) <-> exp(-j*omega*t0)*X(omega)

 FREQUENCY_SHIFT(x,omega0) :=
  exp(j*omega0*t)*x(t) <-> X(omega-omega0)

 DIFFERENTIATION(x) :=
  d^n x/dt^n <-> (j*omega)^n X(omega)
}

DEFINITIONS {
 D1 = "Fourier analysis decomposes functions into frequency components."
 D2 = "Fourier series apply naturally to periodic functions."
 D3 = "Fourier transforms extend frequency decomposition to aperiodic and general signals."
 D4 = "The DTFT is continuous in frequency but periodic with period 2pi."
 D5 = "The DFT produces a finite set of frequency bins."
 D6 = "The FFT is an efficient algorithm for computing the DFT."
 D7 = "Magnitude and phase together determine the complex spectrum."
 D8 = "Parseval relations connect energy between time and frequency domains."
 D9 = "Convolution in time corresponds to multiplication in frequency."
 D10 = "Finite observation windows produce spectral leakage and resolution trade-offs."
}

AXIOMS {
 A1_ORTHOGONAL_EXPANSION:
  Fourier representations use orthogonal or orthonormal harmonic basis functions under suitable spaces

 A2_PERIODIC_REPRESENTATION:
  Periodic signals admit Fourier-series representations under appropriate convergence conditions

 A3_TRANSFORM_INVERSION:
  Fourier transform and inverse transform recover each other under declared regularity conditions

 A4_ENERGY_PRESERVATION:
  Parseval or Plancherel relations preserve signal energy under the corresponding normalisation convention

 A5_CONVOLUTION_DUALITY:
  Convolution in one domain corresponds to multiplication in the transformed domain

 A6_SHIFT:
  Time and frequency shifts introduce predictable phase or spectral translations

 A7_SYMMETRY:
  Real-valued signals produce conjugate-symmetric Fourier spectra

 A8_FINITE_DATA:
  DFT results represent finite or periodic-extension data and are affected by windowing and resolution
}

REASONING {

 R1_FOURIER_CONTEXT_EXTRACTION {
  INPUT = signal_or_function
  IDENTIFY = periodicity
  IDENTIFY = continuity_or_discreteness
  IDENTIFY = finite_or_infinite observation
  IDENTIFY = desired spectral quantity
  RETURN = Fourier_context
 }

 R2_METHOD_SELECTION {
  INPUT = Fourier_context
  IF periodic_continuous
   SELECT = FourierSeries
  ELSE IF continuous_aperiodic
   SELECT = FourierTransform
  ELSE IF discrete_infinite
   SELECT = DTFT
  ELSE IF finite_discrete
   SELECT = DFT_or_FFT
  RETURN = Fourier_method
 }

 R3_SERIES_REASONING {
  INPUT = periodic_signal
  COMPUTE = fundamental_frequency
  COMPUTE = Fourier_coefficients
  RECONSTRUCT = partial_or_full series
  RETURN = series_representation
 }

 R4_TRANSFORM_REASONING {
  INPUT = continuous_signal
  COMPUTE = Fourier_transform
  EXTRACT = magnitude_and_phase
  RETURN = continuous_spectrum
 }

 R5_DFT_REASONING {
  INPUT = finite_sample_sequence_and_sample_rate
  COMPUTE = DFT_or_FFT
  MAP = bins_to_physical frequencies
  RETURN = discrete_spectrum
 }

 R6_PROPERTY_REASONING {
  INPUT = transformed_or_untransformed expression
  APPLY = linearity_shift_scaling_modulation_differentiation_or_convolution property
  RETURN = simplified_transform
 }

 R7_ENERGY_REASONING {
  INPUT = signal_and_transform
  APPLY = Parseval_or_Plancherel
  RETURN = energy_equivalence
 }

 R8_CONVERGENCE_REASONING {
  INPUT = Fourier_series_partial sums
  IDENTIFY = smoothness_and_discontinuities
  ASSESS = pointwise_or_meanSquare convergence
  IDENTIFY = Gibbs phenomenon
  RETURN = convergence_profile
 }

 R9_WINDOW_AND_LEAKAGE_REASONING {
  INPUT = finite_record
  IDENTIFY = window
  COMPUTE = spectral estimate
  ASSESS = leakage_resolution_tradeoff
  RETURN = windowed_spectrum profile
 }

 R10_FOURIER_VALIDATION {
  INPUT = Fourier_result
  CHECK = convention_scaling_units_frequency axis_symmetry_and_inverse reconstruction
  RETURN = validated_Fourier_result
 }
}

CONSTRAINTS {
 C1 = "Fourier normalisation conventions must be explicit."
 C2 = "Continuous and discrete Fourier representations must not be conflated."
 C3 = "DFT bin frequencies depend on sample rate and transform length."
 C4 = "Finite data imply windowing or periodic-extension assumptions."
 C5 = "Magnitude-only spectra are insufficient for exact reconstruction without phase information or extra constraints."
 C6 = "Discontinuities can produce Gibbs oscillations."
 C7 = "Spectral leakage can obscure nearby components."
 C8 = "Zero-padding increases display interpolation but does not create new physical frequency resolution."
 C9 = "Transform inversion requires compatible regularity or distributional interpretation."
 C10 = "Real-signal conjugate symmetry must respect transform convention."
}

SEMANTIC_EQUIVALENCE {
 "FourierAnalysis" <=> "HarmonicFrequencyAnalysis"
 "ContinuousFourierTransform" <=> "FourierTransform"
 "DiscreteTimeFourierTransform" <=> "DTFT"
 "DiscreteFourierTransform" <=> "DFT"
 "FastFourierTransform" <=> "FFT"
 "MagnitudeSpectrum" <=> "AmplitudeSpectrum"
}

VALIDATION {
 V1 = "Signal type and domain are explicit."
 V2 = "Periodicity is correctly classified."
 V3 = "Transform convention is documented."
 V4 = "Sampling rate and DFT length are known for discrete spectra."
 V5 = "Frequency-axis mapping is correct."
 V6 = "Magnitude and phase are interpreted consistently."
 V7 = "Parseval scaling matches the chosen convention."
 V8 = "Windowing and leakage are documented."
 V9 = "Inverse reconstruction is checked where applicable."
 V10 = "Real-signal spectral symmetry is verified where applicable."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FOURIER_ANALYSIS_DEFINITION = 0.99
 METHOD_SELECTION_REASONING = 0.99
 FOURIER_SERIES_REASONING = 0.99
 FOURIER_TRANSFORM_REASONING = 0.99
 DFT_REASONING = 0.99
 PROPERTY_REASONING = 0.99
 CONVERGENCE_REASONING = 0.98
 WINDOWING_REASONING = 0.98
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when periodicity, sampling rate, transform convention, finite-record length, window function, scaling or phase information are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FourierAnalysisReasoning,
  FourierSeriesReasoning,
  FourierTransformReasoning,
  DTFTReasoning,
  DFTReasoning,
  FFTReasoning,
  SpectrumReasoning,
  MagnitudePhaseReasoning,
  ParsevalReasoning,
  PlancherelReasoning,
  ConvolutionTheoremReasoning,
  ShiftPropertyReasoning,
  ModulationReasoning,
  GibbsPhenomenonReasoning,
  WindowingReasoning,
  SpectralLeakageReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "Fourier analysis" -> FourierAnalysis
  "Fourier series" -> FourierSeries
  "Fourier transform" -> FourierTransform
  "continuous Fourier transform" -> FourierTransform
  "DTFT" -> DiscreteTimeFourierTransform
  "DFT" -> DiscreteFourierTransform
  "FFT" -> FastFourierTransform
  "magnitude spectrum" -> MagnitudeSpectrum
  "phase spectrum" -> PhaseSpectrum
  "spectral leakage" -> SpectralLeakage
 }

 INTERPRETATION_ORDER = [
  detect_Fourier_analysis_context,
  identify periodicity_continuity_discreteness_sample_rate_record_length_and_required spectral representation,
  select FourierSeries_FourierTransform_DTFT_DFT_or_FFT pathway,
  compute coefficients_or_transform,
  apply shift_scaling_modulation_convolution_differentiation_or_energy properties where useful,
  assess convergence_leakage_windowing_and_resolution,
  verify scaling_frequency axis_symmetry_and_inverse reconstruction,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("Fourier analysis")
  OR asks_for_frequency_decomposition
  THEN activate(FourierAnalysisOntology)

 E2:
  IF signal_is_periodic_and_continuous
  THEN use FourierSeries

 E3:
  IF signal_is_aperiodic_and_continuous
  THEN use FourierTransform

 E4:
  IF signal_is_discrete_and_infinite_or_symbolic
  THEN use DTFT

 E5:
  IF signal_is_finite_discrete
  THEN use DFT_or_FFT

 E6:
  IF convolution_present
  THEN consider frequency-domain multiplication

 E7:
  IF finite_record
  THEN assess windowing_and_spectral leakage

 E8:
  IF real_signal
  THEN check conjugate_symmetry

 E9:
  IF reconstruction_required
  THEN preserve phase_and_scaling information
}

RECURSION {
 ENTRY = FourierAnalysis

 EXPAND(FourierAnalysis) -> [
  HarmonicComponent,
  FrequencyComponent,
  ComplexExponential,
  Sinusoid,
  AngularFrequency,
  FundamentalFrequency,
  HarmonicFrequency,
  FourierSeries,
  TrigonometricFourierSeries,
  ComplexFourierSeries,
  FourierCoefficient,
  ContinuousFourierTransform,
  FourierTransform,
  InverseFourierTransform,
  DiscreteTimeFourierTransform,
  DTFT,
  DiscreteFourierTransform,
  DFT,
  InverseDFT,
  FastFourierTransform,
  FFT,
  Spectrum,
  MagnitudeSpectrum,
  PhaseSpectrum,
  PowerSpectrum,
  EnergySpectrum,
  SpectralDensity,
  Orthogonality,
  BasisFunction,
  ParsevalTheorem,
  PlancherelTheorem,
  ConvolutionTheorem,
  ModulationProperty,
  TimeShiftProperty,
  FrequencyShiftProperty,
  ScalingProperty,
  DifferentiationProperty,
  IntegrationProperty,
  DualityProperty,
  HermitianSymmetry,
  EvenSymmetry,
  OddSymmetry,
  Convergence,
  PointwiseConvergence,
  MeanSquareConvergence,
  GibbsPhenomenon,
  WindowFunction,
  SpectralLeakage,
  FrequencyResolution,
  TimeFrequencyAnalysis,
  ShortTimeFourierTransform,
  STFT
 ]

 EXPAND(FourierRepresentations) -> [
  FourierSeries,
  FourierTransform,
  DTFT,
  DFT,
  FFT,
  STFT
 ]

 EXPAND(FourierProperties) -> [
  Linearity,
  TimeShiftProperty,
  FrequencyShiftProperty,
  ScalingProperty,
  ConvolutionTheorem,
  ModulationProperty,
  DifferentiationProperty,
  IntegrationProperty,
  DualityProperty
 ]

 EXPAND(SpectralEffects) -> [
  WindowFunction,
  SpectralLeakage,
  FrequencyResolution,
  GibbsPhenomenon,
  ZeroPadding
 ]

 STOP_CONDITION =
  "Stop when the appropriate Fourier representation, coefficients or spectrum, transform properties, convergence, windowing, leakage, resolution and reconstruction conditions are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Signal Processing Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Signal Processing Ontology",
  "Signal Processing Mathematics",
  "Harmonic Analysis Ontologies",
  "Numerical Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Signal Processing Ontology",
  "Signal Processing Mathematics",
  "Harmonic Analysis Ontologies",
  "Linear Algebra Ontologies",
  "Complex Analysis Ontologies",
  "Differential Equations Ontologies",
  "Numerical Analysis Ontologies",
  "Probability & Statistics Ontologies",
  "Communications Mathematics Ontologies",
  "Spectral Analysis Ontologies",
  "Wavelet Analysis Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Fourier Analysis -> Signal Processing Ontology for signal semantics",
  "Fourier Analysis -> Signal Processing Mathematics for parent framework",
  "Fourier Analysis -> Harmonic Analysis for abstract orthogonal frequency representations",
  "Fourier Analysis -> Linear Algebra for finite-dimensional DFT structure",
  "Fourier Analysis -> Complex Analysis for complex exponential and contour-related methods",
  "Fourier Analysis -> Differential Equations for transform solution methods",
  "Fourier Analysis -> Numerical Analysis for FFT and discretisation",
  "Fourier Analysis -> Probability & Statistics for random spectra and PSD",
  "Fourier Analysis -> Communications Mathematics for modulation and bandwidth",
  "Fourier Analysis -> Wavelet Analysis for time-frequency localisation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Fourier representations, properties or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve periodic-versus-aperiodic distinction.",
  "Preserve continuous-versus-discrete Fourier distinction.",
  "Preserve DFT-versus-FFT distinction.",
  "Preserve magnitude-and-phase semantics.",
  "Preserve Parseval and convolution dualities.",
  "Preserve convergence and Gibbs semantics.",
  "Preserve finite-record and leakage semantics.",
  "Preserve machine-readable Fourier-analysis semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Fourier analysis represents functions and signals as combinations of sinusoidal or complex exponential frequency components."

 SERIES_SUMMARY =
  "Fourier series represent periodic signals using harmonically related frequency components."

 TRANSFORM_SUMMARY =
  "The Fourier transform represents continuous aperiodic signals in a continuous frequency domain."

 DTFT_SUMMARY =
  "The DTFT represents discrete-time sequences in a continuous periodic frequency variable."

 DFT_SUMMARY =
  "The DFT maps a finite sequence to discrete frequency bins."

 FFT_SUMMARY =
  "The FFT is an efficient algorithm for computing the DFT."

 ENERGY_SUMMARY =
  "Parseval and Plancherel relations connect energy or norm across time and frequency domains."

 CONVOLUTION_SUMMARY =
  "Convolution in time corresponds to multiplication in frequency."

 LEAKAGE_SUMMARY =
  "Finite observation windows create spectral leakage and resolution trade-offs."
}

FLOW {
 INPUT
 -> detect_Fourier_analysis_context
 -> identify signal_periodicity_continuity_discreteness_sampling_rate_record length_and_desired spectral representation
 -> select FourierSeries_FourierTransform_DTFT_DFT_FFT_or_STFT method
 -> compute Fourier_coefficients_or_transform
 -> derive magnitude_phase_power_or_energy spectrum
 -> apply linearity_shift_scaling_modulation_convolution_differentiation_integration_duality_or_Parseval relations as relevant
 -> assess convergence_Gibbs_windowing_spectral leakage_and_frequency resolution
 -> verify transform convention_sampling_frequency axis_symmetry_scaling_and_inverse reconstruction
 -> route to signal_processing_ontology signal_processing_mathematics harmonic_analysis linear_algebra complex_analysis differential_equations numerical_analysis probability_statistics communications_mathematics spectral_analysis wavelet_analysis_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Fourier Analysis Ontology defines frequency-domain decomposition of functions and signals. It supports Fourier series, trigonometric and complex forms, Fourier coefficients, continuous Fourier transforms, DTFT, DFT, FFT, spectra, magnitude and phase, orthogonality, Parseval and Plancherel relations, convolution and modulation theorems, time and frequency shifts, scaling, differentiation and integration properties, conjugate symmetry, pointwise and mean-square convergence, Gibbs phenomenon, windowing, spectral leakage, frequency resolution and short-time Fourier analysis."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-FOURIER-ANALYSIS-ONTOLOGY-V1"
}

}

END_FLOW