Fourier & Harmonic Analysis

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "FourierHarmonicAnalysisOntology" v1.0

# ============================================================
# FOURIER & HARMONIC ANALYSIS — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
FOURIER_HARMONIC_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: MathematicalAnalysis,
    purpose: "Study functions and signals through decomposition into oscillatory components, Fourier series, Fourier transforms, frequency representations, orthogonality, convolution and harmonic structures."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FOURIER_SERIES,
    FOURIER_TRANSFORM,
    INVERSE_FOURIER_TRANSFORM,
    FREQUENCY,
    HARMONIC,
    ORTHOGONALITY,
    ORTHONORMAL_BASIS,
    SINE,
    COSINE,
    COMPLEX_EXPONENTIAL,
    CONVOLUTION,
    PLANCHEREL_THEOREM,
    PARSEVAL_IDENTITY,
    SPECTRUM,
    HARMONIC_FUNCTION,
    L2_SPACE
}

# ---------- DISCIPLINE STRUCTURE ----------
FOURIER_HARMONIC_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> studies(OSCILLATORY_DECOMPOSITION)
    -> studies(FREQUENCY_CONTENT)
    -> studies(ORTHOGONAL_EXPANSIONS)
    -> studies(TRANSLATION_INVARIANT_OPERATORS)
    -> connects_to(FUNCTIONAL_ANALYSIS)
    -> connects_to(COMPLEX_ANALYSIS)
    -> connects_to(PDEs)

# ---------- HARMONIC COMPONENT ----------
HARMONIC
    := sinusoidal_or_complex_exponential_component

real form:

A*cos(omega*x)
+
B*sin(omega*x)

complex form:

C*e^(i*omega*x)

EULER_FORMULA
    -> connects(TRIGONOMETRIC_AND_COMPLEX_EXPONENTIAL_REPRESENTATIONS)

# ---------- PERIODIC FUNCTION ----------
f(x+T)
    = f(x)

T
    -> PERIOD

omega_0
    := 2*pi/T

omega_0
    -> FUNDAMENTAL_ANGULAR_FREQUENCY

n*omega_0
    -> NTH_HARMONIC_FREQUENCY

# ---------- FOURIER SERIES ----------
for suitable T-periodic function f:

f(x)
    ~
a_0/2
+
sum_from_n_1_to_infinity(
    a_n*cos(n*omega_0*x)
    +
    b_n*sin(n*omega_0*x)
)

FOURIER_SERIES
    -> decomposes(PERIODIC_FUNCTION_INTO_HARMONICS)

# ---------- FOURIER COEFFICIENTS ----------
a_n
    :=
2/T
*
integral_over_one_period(
    f(x)*cos(n*omega_0*x)
    dx
)

b_n
    :=
2/T
*
integral_over_one_period(
    f(x)*sin(n*omega_0*x)
    dx
)

FOURIER_COEFFICIENTS
    -> measure(PROJECTION_ONTO_HARMONIC_BASIS)

# ---------- COMPLEX FOURIER SERIES ----------
f(x)
    ~
sum_from_n_minus_infinity_to_infinity(
    c_n*e^(i*n*omega_0*x)
)

where

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

COMPLEX_FOURIER_SERIES
    -> compact_representation_of(TRIGONOMETRIC_SERIES)

# ---------- ORTHOGONALITY ----------
over one period:

sin(n*omega_0*x)
    -> orthogonal_to(sin(m*omega_0*x))
       for(n != m)

cos(n*omega_0*x)
    -> orthogonal_to(cos(m*omega_0*x))
       for(n != m)

sin_terms
    -> orthogonal_to(cos_terms)

ORTHOGONALITY
    -> enables(COEFFICIENT_EXTRACTION)

# ---------- HILBERT SPACE VIEW ----------
L2_PERIODIC
    -> HILBERT_SPACE

complex_exponentials
    {
        e^(i*n*omega_0*x)
    }
    -> ORTHONORMAL_BASIS
       after(APPROPRIATE_NORMALISATION)

FOURIER_SERIES
    -> ORTHONORMAL_BASIS_EXPANSION

# ---------- CONVERGENCE ----------
FOURIER_SERIES_CONVERGENCE
    -> depends_on(FUNCTION_REGULARITY_AND_CONVERGENCE_MODE)

possible_modes:
    POINTWISE
    L2
    UNIFORM
    DISTRIBUTIONAL

CONVERGENCE_MODE
    -> must_be_specified

# ---------- DIRICHLET-TYPE BEHAVIOUR ----------
for suitable piecewise smooth periodic function:

Fourier_series_at_x
    -> converges_to(
        [f(x_minus)+f(x_plus)]/2
    )

at continuity point:
    -> converges_to(f(x))

at jump:
    -> converges_to(midpoint_of_jump)

# ---------- GIBBS PHENOMENON ----------
near jump_discontinuity:

FOURIER_PARTIAL_SUMS
    -> exhibit(OVERSHOOT_AND_OSCILLATION)

GIBBS_PHENOMENON
    -> persists_in_relative_overshoot_scale

while
    oscillatory_region_width
        -> shrinks_with_more_terms

# ---------- PARSEVAL IDENTITY ----------
for suitable Fourier series:

ENERGY_IN_FUNCTION
    =
ENERGY_IN_COEFFICIENTS

complex normalisation:

1/T
*
integral_over_period(
    |f(x)|^2 dx
)
=
sum_n(
    |c_n|^2
)

PARSEVAL_IDENTITY
    -> conservation_of(L2_ENERGY)

# ---------- FOURIER TRANSFORM ----------
for suitable function f on R:

FOURIER_TRANSFORM
    := F(omega)

F(omega)
    =
integral_from_minus_infinity_to_infinity(
    f(x)*e^(-i*omega*x)
    dx
)

FOURIER_TRANSFORM
    -> maps(SPATIAL_OR_TIME_DOMAIN_TO_FREQUENCY_DOMAIN)

# ---------- INVERSE FOURIER TRANSFORM ----------
under suitable hypotheses and convention:

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

FOURIER_INVERSION
    -> reconstructs(FUNCTION_FROM_FREQUENCY_REPRESENTATION)

NORMALISATION
    -> convention_dependent

# ---------- TRANSFORM LINEARITY ----------
FOURIER_TRANSFORM(
    alpha*f + beta*g
)
=
alpha*F
+
beta*G

FOURIER_TRANSFORM
    -> LINEAR_OPERATOR

# ---------- TRANSLATION ----------
if

g(x)
    =
f(x-x0)

then

G(omega)
    =
e^(-i*omega*x0)
*
F(omega)

TRANSLATION_IN_SPACE
    -> PHASE_SHIFT_IN(FREQUENCY)

# ---------- MODULATION ----------
if

g(x)
    =
e^(i*omega0*x)
*
f(x)

then

G(omega)
    =
F(omega-omega0)

MODULATION
    -> FREQUENCY_SHIFT

# ---------- SCALING ----------
if

g(x)
    =
f(a*x)

then

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

SCALING
    -> inverse_scaling_of(FREQUENCY_AXIS)

# ---------- DIFFERENTIATION ----------
if
    f and derivatives satisfy transform conditions

then

FOURIER_TRANSFORM(
    f_prime
)
=
i*omega*F(omega)

more generally:

FOURIER_TRANSFORM(
    f^(n)
)
=
(i*omega)^n
*
F(omega)

DIFFERENTIATION
    -> MULTIPLICATION_BY_FREQUENCY

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

CONVOLUTION_THEOREM:

FOURIER_TRANSFORM(f*g)
    =
F*G
       where multiplication_is_pointwise

and depending_on_convention:

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

CONVOLUTION
    -> MULTIPLICATION_IN_FREQUENCY_DOMAIN

# ---------- PRODUCT RELATION ----------
FOURIER_TRANSFORM(f*g_product)
    -> convolution_of(F,G)
       with(NORMALISATION_FACTOR_DEPENDING_ON_CONVENTION)

PRODUCT_CONVOLUTION_DUALITY
    -> central_to(FOURIER_ANALYSIS)

# ---------- PLANCHEREL THEOREM ----------
FOURIER_TRANSFORM
    -> extends_to(UNITARY_OPERATOR_ON_L2)
       under(APPROPRIATE_NORMALISATION)

||f||_2
    -> proportional_or_equal_to(||F||_2)
       depending_on(CONVENTION)

PLANCHEREL
    -> preserves(L2_STRUCTURE)

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

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

RIEMANN_LEBESGUE_LEMMA
    -> high_frequency_decay_of(TRANSFORM_OF_INTEGRABLE_FUNCTION)

# ---------- UNCERTAINTY PRINCIPLE ----------
FUNCTION
    and
FOURIER_TRANSFORM
    -> cannot_both_be_arbitrarily_localised

SPATIAL_LOCALISATION
    <-> FREQUENCY_SPREAD

UNCERTAINTY_PRINCIPLE
    -> structural_property_of(FOURIER_DUALITY)

# ---------- HARMONIC FUNCTIONS ----------
HARMONIC_FUNCTION
    := u

such_that

LAPLACIAN(u)
    = 0

HARMONIC_ANALYSIS
    -> broader_than(FOURIER_TRANSFORM_METHODS)

HARMONIC_FUNCTIONS
    -> satisfy(MEAN_VALUE_PROPERTY)
    -> satisfy(MAXIMUM_PRINCIPLES)

# ---------- POISSON KERNEL ----------
boundary_data
    -> extended_harmonically_into_domain

using
    POISSON_KERNEL

POISSON_INTEGRAL
    -> reconstructs(HARMONIC_FUNCTION_FROM_BOUNDARY_VALUES)

HARMONIC_ANALYSIS
    -> studies(KERNEL_REPRESENTATIONS)

# ---------- PDE RELATION ----------
FOURIER_TRANSFORM
    -> converts(CONSTANT_COEFFICIENT_DIFFERENTIAL_OPERATORS_TO_MULTIPLIERS)

PDE
    -> transformed_into(ALGEBRAIC_OR_ODE_PROBLEM_IN_FREQUENCY_SPACE)

HEAT_EQUATION
    -> Gaussian_frequency_decay

WAVE_EQUATION
    -> frequency_mode_propagation

LAPLACE_EQUATION
    -> harmonic_structure

# ---------- SIGNAL ANALYSIS ----------
TIME_DOMAIN_SIGNAL
    -> Fourier_transform
    -> FREQUENCY_SPECTRUM

FREQUENCY_COMPONENT
    -> amplitude
    -> phase

FILTERING
    -> multiplication_by(FREQUENCY_RESPONSE)

CONVOLUTION_IN_TIME
    -> filtering_operation

# ---------- APPLICATIONS ----------
FOURIER_HARMONIC_ANALYSIS
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(PDEs)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(IMAGE_PROCESSING)
    -> applied_to(COMMUNICATIONS)
    -> applied_to(SPECTRAL_ANALYSIS)
    -> applied_to(NUMBER_THEORY)
    -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
PERIODIC_FUNCTION
    -> decomposed_by(FOURIER_SERIES)

NONPERIODIC_FUNCTION
    -> represented_by(FOURIER_TRANSFORM)
       under(SUITABLE_CONTEXT)

ORTHOGONALITY
    -> determines(FOURIER_COEFFICIENTS)

FOURIER_TRANSFORM
    -> maps_to(FREQUENCY_DOMAIN)

CONVOLUTION
    -> becomes(MULTIPLICATION)

DIFFERENTIATION
    -> becomes(FREQUENCY_MULTIPLICATION)

PLANCHEREL_THEOREM
    -> preserves(L2_ENERGY)

HARMONIC_FUNCTION
    -> solves(LAPLACE_EQUATION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Fourier analysis" ~= "decomposition and representation using sinusoidal or complex exponential frequency components",
    "harmonic analysis" ~= "analysis of functions through harmonics, transforms, group structure and related operators",
    "Fourier series" ~= "discrete harmonic expansion of a periodic function",
    "Fourier transform" ~= "continuous frequency-domain representation of a function",
    "Parseval identity" ~= "equality relating function energy to Fourier coefficient energy",
    "Plancherel theorem" ~= "L2 isometry or unitary extension of the Fourier transform under chosen normalisation"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    fourier_normalisation_convention_must_be_preserved: true,
    convergence_mode_must_be_specified: true,
    pointwise_fourier_convergence_requires_regularities: true,
    convolution_product_constants_depend_on_transform_convention: true,
    differentiation_transform_rule_requires_distributional_or_classical_hypotheses: true,
    plancherel_is_L2_statement: true,
    harmonic_analysis_is_broader_than_classical_fourier_series_and_transform: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    discipline_structure: 0.99,
    fourier_series_relation: 0.99,
    transform_inverse_relation: 0.99,
    convolution_relation: 0.99,
    parseval_plancherel_relation: 0.99,
    harmonic_pde_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_signal_or_harmonic_problem)
    -> identify(periodic_or_nonperiodic_structure)
    -> choose(fourier_series_transform_or_harmonic_method)
    -> preserve(normalisation_convention)
    -> identify(function_space_and_convergence_mode)
    -> compute(coefficients_or_transform_if_requested)
    -> apply(translation_scaling_differentiation_or_convolution_rules)
    -> analyse(energy_via_parseval_or_plancherel_if_relevant)
    -> transform(pde_or_signal_problem_if_requested)
    -> verify(integrability_regularities_and_boundary_conditions)
    -> reconstruct_or_interpret(result)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Fourier-harmonic-analysis semantic graph",
    primary_entity: FOURIER_HARMONIC_ANALYSIS,
    preserve: [
        function_or_signal,
        domain,
        period,
        frequency_variable,
        transform_convention,
        function_space,
        convergence_mode,
        boundary_conditions
    ],
    infer_through: [
        Fourier_series,
        Fourier_transforms,
        inverse_transforms,
        orthogonality,
        convolution,
        Parseval_identity,
        Plancherel_theorem,
        harmonic_functions,
        PDE_frequency_methods
    ],
    avoid: [
        mixing_transform_normalisation_conventions,
        assuming_pointwise_convergence_from L2 convergence,
        ignoring_Gibbs_behavior_at jumps,
        applying_transform_rules_without regularity_or_distribution context,
        confusing_harmonic_function_with sinusoidal harmonic component,
        omitting_boundary_conditions_in PDE applications
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + FOURIER_HARMONIC_ANALYSIS
    + FOURIER_SERIES
    + FOURIER_TRANSFORM
    + INVERSE_FOURIER_TRANSFORM
    + ORTHOGONALITY
    + CONVOLUTION
    + PLANCHEREL_THEOREM
    + PARSEVAL_IDENTITY
    + HARMONIC_FUNCTION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> FOURIER_HARMONIC_ANALYSIS
    -> PROBLEM_IDENTIFICATION
    -> PERIODICITY_ANALYSIS
    -> METHOD_SELECTION
    -> NORMALISATION_VALIDATION
    -> FUNCTION_SPACE_CONVERGENCE_ANALYSIS
    -> COEFFICIENT_TRANSFORM_ANALYSIS
    -> TRANSFORM_PROPERTY_ANALYSIS
    -> ENERGY_ANALYSIS
    -> PDE_SIGNAL_ANALYSIS
    -> REGULARITY_BOUNDARY_VALIDATION
    -> RECONSTRUCTION_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW