Fourier Transform Ontology

flow "FourierTransformOntology" v1.0

# ============================================================
# FOURIER TRANSFORM ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/fourier-harmonic-analysis/fourier-transform-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Fourier and Harmonic Analysis"
ENTITY := "Fourier Transform"

# ---------- CORE ENTITY ----------
FOURIER_TRANSFORM := {
    type: IntegralTransform,
    domain: FourierAnalysis,
    purpose: "Transform a function or signal from its original domain into a continuous frequency-domain representation using complex exponential basis functions."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FOURIER_TRANSFORM,
    INVERSE_FOURIER_TRANSFORM,
    FUNCTION,
    SIGNAL,
    FREQUENCY,
    ANGULAR_FREQUENCY,
    COMPLEX_EXPONENTIAL,
    FREQUENCY_DOMAIN,
    AMPLITUDE_SPECTRUM,
    PHASE_SPECTRUM,
    CONVOLUTION,
    PLANCHEREL_THEOREM,
    PARSEVAL_IDENTITY,
    DISTRIBUTION,
    SPECTRUM
}

# ---------- TRANSFORM CONVENTION ----------
let
    f: R -> C

define

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

FOURIER_TRANSFORM(f)
    := F

omega
    -> ANGULAR_FREQUENCY

TRANSFORM_CONVENTION
    -> must_be_preserved

# ---------- INVERSE TRANSFORM ----------
under this convention:

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

INVERSE_FOURIER_TRANSFORM
    -> reconstructs(ORIGINAL_FUNCTION)
       under(SUITABLE_HYPOTHESES)

# ---------- FREQUENCY CONVENTION ----------
alternative frequency variable
    nu

kernel:
    exp(-2*pi*i*nu*x)

NORMALISATION_AND_FREQUENCY_VARIABLE
    -> convention_dependent

MIXING_CONVENTIONS
    -> INVALID_TRANSFORM_RELATIONS

# ---------- COMPLEX EXPONENTIAL BASIS ----------
exp(i*omega*x)
    =
cos(omega*x)
+
i*sin(omega*x)

COMPLEX_EXPONENTIAL
    -> oscillatory_basis_function

FOURIER_TRANSFORM
    -> measures(CONTINUOUS_FREQUENCY_CONTENT)

# ---------- LINEARITY ----------
FT(
    alpha*f
    +
    beta*g
)
=
alpha*FT(f)
+
beta*FT(g)

FOURIER_TRANSFORM
    -> LINEAR_OPERATOR

# ---------- TIME / SPACE SHIFT ----------
if
    g(x)
    =
f(x-x0)

then

G(omega)
    =
exp(-i*omega*x0)
*
F(omega)

TRANSLATION
    -> PHASE_MULTIPLICATION_IN_FREQUENCY_DOMAIN

# ---------- FREQUENCY SHIFT ----------
if
    g(x)
    =
exp(i*omega0*x)
*
f(x)

then

G(omega)
    =
F(omega-omega0)

MODULATION
    -> FREQUENCY_TRANSLATION

# ---------- SCALING ----------
if
    g(x)
    =
f(a*x)

with
    a != 0

then

G(omega)
    =
1/abs(a)
*
F(omega/a)

SCALING
    -> reciprocal_domain_frequency_relation

# ---------- REFLECTION ----------
if
    g(x)
    =
f(-x)

then

G(omega)
    =
F(-omega)

REFLECTION
    -> reflects(FREQUENCY_DOMAIN)

# ---------- COMPLEX CONJUGATION ----------
if
    g(x)
    =
conjugate(f(x))

then

G(omega)
    =
conjugate(F(-omega))

CONJUGATION
    -> frequency_reflection_and_conjugation

# ---------- REAL SIGNAL SYMMETRY ----------
if
    f(x)
        -> REAL_VALUED

then

F(-omega)
    =
conjugate(F(omega))

REAL_SIGNAL
    -> HERMITIAN_SYMMETRY_OF_SPECTRUM

# ---------- EVEN / ODD ----------
if
    f real_and_even
then

F
    -> real_and_even
       under(STANDARD_CONVENTION)

if
    f real_and_odd
then

F
    -> imaginary_and_odd
       under(STANDARD_CONVENTION)

SYMMETRY
    -> simplifies(FOURIER_TRANSFORM)

# ---------- DIFFERENTIATION ----------
under suitable decay or distributional interpretation:

FT(
    f_prime(x)
)
=
i*omega*F(omega)

more generally:

FT(
    f^(n)(x)
)
=
(i*omega)^n
*
F(omega)

DIFFERENTIATION_IN_ORIGINAL_DOMAIN
    -> multiplication_by_frequency

# ---------- MULTIPLICATION BY x ----------
under suitable conditions:

FT(
    x*f(x)
)
=
i
*
dF/domega

more generally:

FT(
    x^n*f(x)
)
=
i^n
*
d^nF/domega^n

ORIGINAL_DOMAIN_MULTIPLICATION
    -> FREQUENCY_DIFFERENTIATION

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

CONVOLUTION_THEOREM:

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

CONVOLUTION
    -> POINTWISE_MULTIPLICATION_IN_FREQUENCY_DOMAIN

# ---------- PRODUCT THEOREM ----------
under this transform convention:

FT(f*g_pointwise)
    =
1/(2*pi)
*
(F convolution G)

PRODUCT
    -> FREQUENCY_CONVOLUTION

CONVENTION
    -> controls(CONSTANT_FACTOR)

# ---------- PARSEVAL RELATION ----------
under stated convention:

integral(
    f(x)*conjugate(g(x))
    dx
)
=
1/(2*pi)
*
integral(
    F(omega)*conjugate(G(omega))
    d_omega
)

PARSEVAL_RELATION
    -> preserves(INNER_PRODUCT_UP_TO_NORMALISATION)

# ---------- PLANCHEREL THEOREM ----------
FOURIER_TRANSFORM
    -> extends_from(SUITABLE_DENSE_CLASS)
       to(L2)

under this convention:

||F||_2^2
    =
2*pi
*
||f||_2^2

under unitary convention:

||F||_2
    =
||f||_2

PLANCHEREL
    -> L2_ISOMETRY_UP_TO_NORMALISATION

# ---------- RIEMANN-LEBESGUE ----------
if
    f in L1(R)

then

F
    -> CONTINUOUS

and

F(omega)
    -> 0
       as(abs(omega) -> infinity)

RIEMANN_LEBESGUE_LEMMA
    -> high_frequency_decay

# ---------- FOURIER INVERSION ----------
if
    f and F satisfy appropriate integrability_or_regularities

then

INVERSE_FOURIER_TRANSFORM(
    FOURIER_TRANSFORM(f)
)
    = f

at suitable points or in suitable function-space sense

FOURIER_INVERSION
    -> context_dependent

# ---------- GAUSSIAN ----------
f(x)
    :=
exp(-a*x^2)

for
    a > 0

FOURIER_TRANSFORM(f)
    -> GAUSSIAN_IN_FREQUENCY_DOMAIN

GAUSSIAN
    -> eigenfamily_of(FOURIER_TRANSFORM_UP_TO_SCALING)

# ---------- DIRAC DELTA ----------
in distributional sense:

FT(
    delta(x-x0)
)
=
exp(-i*omega*x0)

and

FT(1)
    =
2*pi*delta(omega)

DISTRIBUTION_THEORY
    -> extends(FOURIER_TRANSFORM_TO_NONINTEGRABLE_OBJECTS)

# ---------- COMPLEX EXPONENTIAL ----------
distributionally:

FT(
    exp(i*omega0*x)
)
=
2*pi
*
delta(omega-omega0)

PURE_FREQUENCY
    -> POINT_MASS_IN_FREQUENCY_DOMAIN

# ---------- RECTANGLE / SINC ----------
RECTANGULAR_PULSE
    -> transforms_to(SINC_LIKE_FUNCTION)

TIME_OR_SPACE_LOCALISATION
    -> broad_frequency_distribution

FOURIER_DUALITY
    -> illustrates(UNCERTAINTY_RELATION)

# ---------- UNCERTAINTY PRINCIPLE ----------
function_localisation
    and
transform_localisation
    -> trade_off

HEISENBERG_TYPE_INEQUALITY
    -> lower_bound_on(PRODUCT_OF_SPREADS)

FOURIER_TRANSFORM
    -> connects(DUAL_LOCALISATION_SCALES)

# ---------- PDE TRANSFORMATION ----------
if
    PDE has constant_coefficient_derivatives

then

FOURIER_TRANSFORM
    -> converts(DERIVATIVES_TO_ALGEBRAIC_MULTIPLIERS)

example:

FT(
    partial^2 u / partial_x^2
)
=
-omega^2
*
U(omega)

PDE
    -> simplified_in(FREQUENCY_DOMAIN)

# ---------- FILTERING ----------
LINEAR_TIME_INVARIANT_SYSTEM
    -> impulse_response(h)

output
    y
    =
h*x

then

Y(omega)
    =
H(omega)*X(omega)

H(omega)
    -> FREQUENCY_RESPONSE

FOURIER_TRANSFORM
    -> diagonalises(CONVOLUTION_OPERATORS)

# ---------- MULTIDIMENSIONAL TRANSFORM ----------
for
    f: R^n -> C

F(xi)
    :=
integral_over_Rn(
    f(x)
    * exp(-i*x_dot_xi)
    dx
)

MULTIDIMENSIONAL_FOURIER_TRANSFORM
    -> maps(SPATIAL_FREQUENCY_VECTOR)

used_in:
    IMAGE_PROCESSING
    PDEs
    TOMOGRAPHY
    QUANTUM_MECHANICS

# ---------- APPLICATIONS ----------
FOURIER_TRANSFORM
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(IMAGE_PROCESSING)
    -> applied_to(PDEs)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(COMMUNICATIONS)
    -> applied_to(SPECTRAL_ANALYSIS)
    -> applied_to(NUMERICAL_METHODS)

# ---------- RELATION GRAPH ----------
FUNCTION
    -> FOURIER_TRANSFORM
    -> FREQUENCY_REPRESENTATION

FOURIER_TRANSFORM
    -> inverse_by(INVERSE_FOURIER_TRANSFORM)

TRANSLATION
    -> PHASE_FACTOR

MODULATION
    -> FREQUENCY_SHIFT

DIFFERENTIATION
    -> FREQUENCY_MULTIPLICATION

CONVOLUTION
    -> FREQUENCY_MULTIPLICATION

PLANCHEREL
    -> preserves(L2_STRUCTURE)

DISTRIBUTIONS
    -> extend(TRANSFORM_DOMAIN)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Fourier transform" ~= "continuous frequency-domain integral transform",
    "inverse Fourier transform" ~= "reconstruction of a function from its Fourier-frequency representation",
    "frequency spectrum" ~= "Fourier-domain amplitude and phase representation",
    "convolution theorem" ~= "Fourier transformation converts convolution into multiplication",
    "Plancherel theorem" ~= "L2 norm preservation up to transform normalisation",
    "Fourier inversion" ~= "recovery of the original function from its Fourier transform under appropriate conditions"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    transform_and_inverse_normalisations_must_match: true,
    frequency_variable_convention_must_be_preserved: true,
    classical_integral_transform_requires_integrability_or_extension_framework: true,
    transform_rules_require_their_regularities_or_distributional_interpretation: true,
    product_convolution_constants_depend_on_normalisation: true,
    inversion_statement_depends_on_function_space_and_pointwise_context: true,
    distributional_transforms_must_not_be_treated_as_ordinary_functions_without_context: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    inverse_relation: 0.99,
    shift_scaling_relation: 0.99,
    differentiation_convolution_relation: 0.99,
    parseval_plancherel_relation: 0.99,
    distribution_pde_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_signal_or_distribution)
    -> identify(original_and_frequency_variables)
    -> establish(transform_normalisation)
    -> identify(function_space_or_distribution_context)
    -> compute_or_recognise(transform)
    -> apply(shift_scaling_symmetry_or_modulation_rules)
    -> apply(differentiation_convolution_or_product_rules_if_relevant)
    -> analyse(amplitude_phase_energy_or_localisation)
    -> perform(inverse_transform_if_requested)
    -> connect_to(pde_or_filtering_context_if_relevant)
    -> validate(all_regularities_and_normalisation)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Fourier-transform semantic graph",
    primary_entity: FOURIER_TRANSFORM,
    preserve: [
        original_function,
        original_variable,
        frequency_variable,
        transform_convention,
        normalisation,
        function_space,
        transform,
        distributional_status
    ],
    infer_through: [
        inverse_transforms,
        translations,
        modulations,
        scaling,
        differentiation,
        convolution,
        Plancherel_theorem,
        distributional_transforms,
        frequency_response
    ],
    avoid: [
        mixing_omega_and_cycle_frequency_conventions,
        mixing_normalisation_constants,
        assuming_every_transform_is ordinary integrable function,
        applying_derivative_rule_without valid decay_or_distribution context,
        omitting_product_convolution factor,
        claiming_pointwise_inversion_without hypotheses
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(fourier_transform_context)
    -> compare(existing_relations)
    -> preserve(variable_normalisation_space_and_transform_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + FOURIER_TRANSFORM
    + INVERSE_FOURIER_TRANSFORM
    + FREQUENCY_DOMAIN
    + CONVOLUTION
    + PLANCHEREL_THEOREM
    + PARSEVAL_IDENTITY
    + DISTRIBUTION
    + SPECTRUM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> FOURIER_TRANSFORM
    -> OBJECT_VARIABLE_IDENTIFICATION
    -> CONVENTION_NORMALISATION_ANALYSIS
    -> FUNCTION_SPACE_DISTRIBUTION_ANALYSIS
    -> TRANSFORM_COMPUTATION
    -> SHIFT_SCALING_SYMMETRY_ANALYSIS
    -> DIFFERENTIATION_CONVOLUTION_ANALYSIS
    -> SPECTRUM_ENERGY_ANALYSIS
    -> INVERSE_TRANSFORM_ANALYSIS
    -> PDE_FILTERING_RELATION_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW