Fourier Series Ontology

flow "FourierSeriesOntology" v1.0

# ============================================================
# FOURIER SERIES ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
FOURIER_SERIES := {
    type: HarmonicSeriesRepresentation,
    domain: FourierAnalysis,
    purpose: "Represent a periodic function as an infinite sum of sinusoidal or complex exponential harmonics with coefficients determined by orthogonal projection."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FOURIER_SERIES,
    PERIODIC_FUNCTION,
    PERIOD,
    FUNDAMENTAL_FREQUENCY,
    HARMONIC,
    FOURIER_COEFFICIENT,
    SINE,
    COSINE,
    COMPLEX_EXPONENTIAL,
    ORTHOGONALITY,
    PARTIAL_SUM,
    CONVERGENCE,
    PARSEVAL_IDENTITY,
    GIBBS_PHENOMENON,
    L2_SPACE
}

# ---------- 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

# ---------- REAL FOURIER SERIES ----------
for suitable T-periodic 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)

# ---------- REAL COEFFICIENTS ----------
a_0
    :=
2/T
*
integral_over_one_period(
    f(x) dx
)

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_COEFFICIENT
    -> projection_onto(TRIGONOMETRIC_BASIS)

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

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

COMPLEX_FORM
    -> equivalent_to(REAL_TRIGONOMETRIC_FORM)

# ---------- EULER RELATION ----------
e^(i*theta)
    =
cos(theta)
+
i*sin(theta)

COMPLEX_EXPONENTIAL
    -> unifies(SINE_AND_COSINE_COMPONENTS)

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

integral(
    e^(i*n*omega_0*x)
    *
    e^(-i*m*omega_0*x)
    dx
)

=
0
    when n != m

ORTHOGONALITY
    -> enables(COEFFICIENT_EXTRACTION)

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

FOURIER_BASIS
    -> ORTHONORMAL_BASIS
       under(APPROPRIATE_NORMALISATION)

c_n
    -> INNER_PRODUCT_OF(f, BASIS_ELEMENT_n)

FOURIER_SERIES
    -> HILBERT_SPACE_EXPANSION

# ---------- PARTIAL SUM ----------
S_N(x)
    :=
a_0/2
+
sum_from_n_1_to_N(
    a_n*cos(n*omega_0*x)
    +
    b_n*sin(n*omega_0*x)
)

PARTIAL_SUM
    -> finite_harmonic_approximation

S_N
    -> approximates(f)

# ---------- DIRICHLET KERNEL ----------
for standard 2*pi-periodic complex formulation:

D_N(x)
    :=
sum_from_n_minusN_to_N(
    e^(i*n*x)
)

D_N(x)
    =
sin((N+1/2)x)
/
sin(x/2)

FOURIER_PARTIAL_SUM
    -> expressible_as(CONVOLUTION_WITH_DIRICHLET_KERNEL)

# ---------- CONVERGENCE MODES ----------
FOURIER_SERIES
    -> may_converge(POINTWISE)
    -> may_converge(UNIFORMLY)
    -> may_converge(IN_L2)
    -> may_converge(DISTRIBUTIONALLY)

CONVERGENCE_MODE
    -> must_be_specified

# ---------- POINTWISE CONVERGENCE ----------
under suitable Dirichlet-type conditions:

S_N(x)
    ->
[
    f(x_minus)
    +
    f(x_plus)
]/2

at continuity point:

S_N(x)
    -> f(x)

at jump discontinuity:

S_N(x)
    -> midpoint_of_jump

# ---------- L2 CONVERGENCE ----------
for
    f in L2_periodic

Fourier_partial_sums
    -> converge_to(f)
       in(L2_NORM)

L2_CONVERGENCE
    -/> necessarily(POINTWISE_CONVERGENCE_EVERYWHERE)

# ---------- UNIFORM CONVERGENCE ----------
UNIFORM_CONVERGENCE
    -> requires(STRONGER_REGULARITY)

if Fourier series converges absolutely and appropriately:
    -> often_uniform_convergence

UNIFORM_CONVERGENCE
    -> permits(STRONGER_LIMIT_INTERCHANGES)

# ---------- GIBBS PHENOMENON ----------
if
    f has(JUMP_DISCONTINUITY)

then

FOURIER_PARTIAL_SUMS
    -> show(OSCILLATORY_OVERSHOOT_NEAR_JUMP)

as N increases:
    overshoot_region_width
        -> shrinks

relative_peak_overshoot
    -> persists

GIBBS_PHENOMENON
    -> characteristic_of(TRUNCATED_FOURIER_SERIES_NEAR_JUMPS)

# ---------- PARSEVAL IDENTITY ----------
under suitable L2 conditions:

1/T
*
integral_over_one_period(
    |f(x)|^2 dx
)
=
sum_from_n_minus_infinity_to_infinity(
    |c_n|^2
)

PARSEVAL_IDENTITY
    -> equates(FUNCTION_ENERGY, COEFFICIENT_ENERGY)

# ---------- BESSEL INEQUALITY ----------
for orthonormal Fourier system:

sum(
    |c_n|^2
)
    <=
||f||_2^2

BESSEL_INEQUALITY
    -> coefficient_energy_bound

COMPLETENESS_OF_BASIS
    -> upgrades_to(PARSEVAL_EQUALITY)

# ---------- SYMMETRY ----------
if
    f is EVEN

then

b_n
    = 0

FOURIER_SERIES
    -> COSINE_SERIES

if
    f is ODD

then

a_n
    = 0
    including(a_0)

FOURIER_SERIES
    -> SINE_SERIES

SYMMETRY
    -> simplifies(COEFFICIENT_STRUCTURE)

# ---------- HALF-RANGE EXPANSIONS ----------
function_defined_on([0,L])

-> may_be_extended_evenly
   to produce(COSINE_SERIES)

-> may_be_extended_oddly
   to produce(SINE_SERIES)

HALF_RANGE_SERIES
    -> useful_for(BOUNDARY_VALUE_PROBLEMS)

# ---------- DIFFERENTIATION ----------
TERM_BY_TERM_DIFFERENTIATION
    -> requires(SUFFICIENT_REGULARITY_AND_CONVERGENCE)

d/dx[
    c_n*e^(i*n*omega_0*x)
]
=
i*n*omega_0
*
c_n*e^(i*n*omega_0*x)

DIFFERENTIATION
    -> amplifies(HIGH_FREQUENCY_COEFFICIENTS)

# ---------- INTEGRATION ----------
under appropriate convergence conditions:

FOURIER_SERIES
    -> may_be_integrated_TERM_BY_TERM

INTEGRATION
    -> divides_nonzero_frequency_coefficients_by(i*n*omega_0)

ZERO_FREQUENCY_TERM
    -> handled_separately

# ---------- COEFFICIENT DECAY ----------
SMOOTHER_FUNCTION
    -> generally_faster_decay_of(FOURIER_COEFFICIENTS)

DISCONTINUITIES
    -> slower_coefficient_decay

REGULARITY
    -> encoded_in(FREQUENCY_COEFFICIENT_DECAY)

# ---------- CONVOLUTION ----------
for periodic functions:

(f*g)(x)
    -> PERIODIC_CONVOLUTION

FOURIER_COEFFICIENTS_OF_CONVOLUTION
    -> product_of(FOURIER_COEFFICIENTS)
       under(CONSISTENT_NORMALISATION)

CONVOLUTION
    -> diagonalised_by(FOURIER_BASIS)

# ---------- PDE APPLICATION ----------
SEPARATION_OF_VARIABLES
    -> produces(FOURIER_SERIES_MODES)

HEAT_EQUATION
    -> each_harmonic_mode_decays_at_rate_related_to_frequency_squared

WAVE_EQUATION
    -> each_harmonic_mode_evolves_independently

LAPLACE_EQUATION
    -> boundary_data_expanded_in(FOURIER_SERIES)

# ---------- SIGNAL INTERPRETATION ----------
PERIODIC_SIGNAL
    -> FOURIER_SERIES
    -> DISCRETE_LINE_SPECTRUM

c_n
    -> complex_amplitude_and_phase_of(nth_harmonic)

FOURIER_SERIES
    -> frequency_description_of(PERIODIC_SIGNAL)

# ---------- APPLICATIONS ----------
FOURIER_SERIES
    -> applied_to(PERIODIC_SIGNAL_ANALYSIS)
    -> applied_to(PDEs)
    -> applied_to(BOUNDARY_VALUE_PROBLEMS)
    -> applied_to(VIBRATION_ANALYSIS)
    -> applied_to(ACOUSTICS)
    -> applied_to(ELECTRICAL_ENGINEERING)
    -> applied_to(NUMERICAL_APPROXIMATION)

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

FOURIER_COEFFICIENT
    -> determined_by(ORTHOGONAL_PROJECTION)

FOURIER_SERIES
    -> approximated_by(PARTIAL_SUM)

PARTIAL_SUM
    -> may_exhibit(GIBBS_PHENOMENON)

L2_STRUCTURE
    -> supports(PARSEVAL_IDENTITY)

FUNCTION_SYMMETRY
    -> determines(SINE_OR_COSINE_REDUCTION)

COEFFICIENT_DECAY
    -> reflects(FUNCTION_REGULARITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Fourier series" ~= "infinite harmonic expansion of a periodic function",
    "Fourier coefficient" ~= "orthogonal projection coefficient onto a Fourier basis function",
    "Fourier partial sum" ~= "finite harmonic approximation using the first Fourier modes",
    "complex Fourier series" ~= "Fourier expansion using integer-frequency complex exponentials",
    "Gibbs phenomenon" ~= "persistent overshoot near jump discontinuities in truncated Fourier expansions",
    "Parseval identity" ~= "equality between function L2 energy and Fourier coefficient energy"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    period_and_frequency_normalisation_must_be_preserved: true,
    coefficient_formula_depends_on_chosen_normalisation: true,
    convergence_mode_must_be_explicit: true,
    pointwise_convergence_requires_function_regularities: true,
    L2_convergence_does_not_imply_pointwise_convergence_everywhere: true,
    termwise_differentiation_requires_stronger_conditions_than_L2_convergence: true,
    gibbs_phenomenon_must_not_be_misclassified_as_failure_of_L2_convergence: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    coefficient_relation: 0.99,
    orthogonality_relation: 0.99,
    convergence_relation: 0.99,
    parseval_relation: 0.99,
    symmetry_pde_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(periodic_function_and_period)
    -> determine(fundamental_frequency)
    -> choose(real_or_complex_series_form)
    -> identify(symmetry)
    -> compute(fourier_coefficients)
    -> construct(partial_sum_or_full_series)
    -> specify(convergence_mode)
    -> analyse(pointwise_L2_or_uniform_convergence)
    -> inspect(gibbs_behavior_if_discontinuous)
    -> apply(parseval_or_coefficient_decay_analysis_if_requested)
    -> connect_to(pde_or_signal_context_if_relevant)
    -> validate(normalisation_and_regularities)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Fourier-series semantic graph",
    primary_entity: FOURIER_SERIES,
    preserve: [
        function,
        period,
        fundamental_frequency,
        coefficient_normalisation,
        coefficients,
        symmetry,
        partial_sum_order,
        convergence_mode
    ],
    infer_through: [
        harmonic_components,
        orthogonality,
        complex_exponentials,
        partial_sums,
        Dirichlet_kernel,
        Gibbs_phenomenon,
        Parseval_identity,
        coefficient_decay
    ],
    avoid: [
        applying_nonperiodic_Fourier_transform_as Fourier series without periodisation,
        mixing_coefficient_normalisations,
        assuming_pointwise_convergence_from L2 convergence,
        ignoring_jump_midpoint convergence,
        claiming_Gibbs_overshoot_disappears in relative height,
        differentiating_series_termwise_without sufficient regularity
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + FOURIER_SERIES
    + PERIODIC_FUNCTION
    + FOURIER_COEFFICIENT
    + ORTHOGONALITY
    + PARTIAL_SUM
    + CONVERGENCE
    + GIBBS_PHENOMENON
    + PARSEVAL_IDENTITY
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> FOURIER_SERIES
    -> PERIOD_FUNCTION_IDENTIFICATION
    -> FUNDAMENTAL_FREQUENCY_ANALYSIS
    -> SERIES_FORM_SELECTION
    -> SYMMETRY_ANALYSIS
    -> COEFFICIENT_COMPUTATION
    -> PARTIAL_SUM_CONSTRUCTION
    -> CONVERGENCE_MODE_ANALYSIS
    -> GIBBS_ANALYSIS
    -> PARSEVAL_REGULARITY_ANALYSIS
    -> PDE_SIGNAL_RELATION_ANALYSIS
    -> NORMALISATION_HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW