Harmonic Analysis Ontology

flow "HarmonicAnalysisOntology" v1.0

# ============================================================
# HARMONIC ANALYSIS ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
HARMONIC_ANALYSIS := {
    type: MathematicalAnalysisFramework,
    domain: MathematicalAnalysis,
    purpose: "Study functions, signals and operators through decomposition into elementary waves, frequencies, characters and harmonic structures, extending classical Fourier analysis to broader spaces and groups."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    HARMONIC_ANALYSIS,
    FOURIER_ANALYSIS,
    FOURIER_SERIES,
    FOURIER_TRANSFORM,
    HARMONIC_FUNCTION,
    CHARACTER,
    LOCALLY_COMPACT_ABELIAN_GROUP,
    CONVOLUTION,
    TRANSLATION,
    FREQUENCY,
    ORTHOGONALITY,
    Lp_SPACE,
    MAXIMAL_OPERATOR,
    SINGULAR_INTEGRAL,
    POISSON_KERNEL,
    LAPLACIAN
}

# ---------- DISCIPLINE STRUCTURE ----------
HARMONIC_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> generalises(FOURIER_ANALYSIS)
    -> studies(FREQUENCY_DECOMPOSITION)
    -> studies(TRANSLATION_INVARIANT_STRUCTURES)
    -> studies(CONVOLUTION_OPERATORS)
    -> studies(HARMONIC_FUNCTIONS)
    -> connects_to(FUNCTIONAL_ANALYSIS)
    -> connects_to(PDEs)
    -> connects_to(GROUP_THEORY)

# ---------- CLASSICAL FOURIER FOUNDATION ----------
PERIODIC_FUNCTION
    -> FOURIER_SERIES

FUNCTION_ON_R
    -> FOURIER_TRANSFORM
       under(SUITABLE_FUNCTION_SPACE_CONTEXT)

FOURIER_ANALYSIS
    -> foundational_subfield_of(HARMONIC_ANALYSIS)

# ---------- ELEMENTARY HARMONICS ----------
HARMONIC_COMPONENT
    := exp(i*omega*x)

using

exp(i*omega*x)
    =
cos(omega*x)
+
i*sin(omega*x)

HARMONIC_COMPONENT
    -> eigenfunction_of(TRANSLATION)

# ---------- TRANSLATION ----------
TRANSLATION_OPERATOR
    := T_y

(T_y f)(x)
    :=
f(x-y)

COMPLEX_EXPONENTIAL
    -> eigenfunction_of(T_y)

T_y(
    exp(i*omega*x)
)
=
exp(-i*omega*y)
*
exp(i*omega*x)

TRANSLATION
    -> diagonalised_by(FOURIER_TRANSFORM)

# ---------- CONVOLUTION ----------
(f*g)(x)
    :=
integral(
    f(y)
    *
    g(x-y)
    dy
)

CONVOLUTION
    -> combines(FUNCTIONS_USING_TRANSLATION)

FOURIER_TRANSFORM(f*g)
    =
FOURIER_TRANSFORM(f)
*
FOURIER_TRANSFORM(g)

CONVOLUTION
    -> multiplication_in(FREQUENCY_DOMAIN)

# ---------- CHARACTERS ----------
for abelian group G:

CHARACTER
    := continuous_homomorphism(
        G -> UNIT_CIRCLE
    )

chi(x+y)
    =
chi(x)*chi(y)

CHARACTERS
    -> generalise(COMPLEX_EXPONENTIAL_FREQUENCIES)

# ---------- DUAL GROUP ----------
DUAL_GROUP
    := G_hat

G_hat
    -> set_of(CONTINUOUS_CHARACTERS_OF_G)

FREQUENCY_VARIABLE
    -> element_of(DUAL_GROUP)

PONTRYAGIN_DUALITY
    -> relates(G, DOUBLE_DUAL_GROUP)

# ---------- LOCALLY COMPACT ABELIAN GROUP ----------
LCA_GROUP
    -> supports(HAAR_MEASURE)
    -> supports(FOURIER_TRANSFORM_ON_GROUP)

for
    f in suitable_function_space_on(G)

Fourier_transform:

f_hat(chi)
    :=
integral_G(
    f(x)
    *
    conjugate(chi(x))
    d_mu(x)
)

HARMONIC_ANALYSIS
    -> extends(FOURIER_TRANSFORM_TO_LCA_GROUPS)

# ---------- HAAR MEASURE ----------
HAAR_MEASURE
    -> translation_invariant_measure_on(LOCALLY_COMPACT_GROUP)

mu(E+x)
    =
mu(E)

TRANSLATION_INVARIANCE
    -> fundamental_to(GROUP_HARMONIC_ANALYSIS)

# ---------- Lp SPACES ----------
L1
    -> natural_space_for(CLASSICAL_FOURIER_TRANSFORM)

L2
    -> Hilbert_space_for(PLANCHEREL_THEORY)

Lp
    -> supports(HARMONIC_ANALYSIS_OPERATORS)

HARMONIC_ANALYSIS
    -> studies(OPERATOR_BOUNDEDNESS_BETWEEN_Lp_SPACES)

# ---------- PLANCHEREL ----------
FOURIER_TRANSFORM
    -> extends_to(L2)

under suitable normalisation:

||f_hat||_2
    =
||f||_2

PLANCHEREL_THEOREM
    -> preserves(HILBERT_SPACE_STRUCTURE)

# ---------- FOURIER INVERSION ----------
under suitable hypotheses:

f
    -> Fourier_transform
    -> inverse_transform
    -> f

FOURIER_INVERSION
    -> reconstruction_from(FREQUENCY_DATA)

INVERSION_FORM
    -> depends_on(GROUP_AND_NORMALISATION)

# ---------- POISSON KERNEL ----------
POISSON_KERNEL
    -> approximate_identity
    -> harmonic_extension_kernel

for unit disk:

P_r(theta)
    :=
(1-r^2)
/
(
    1
    -
    2*r*cos(theta)
    +
    r^2
)

BOUNDARY_FUNCTION
    convolution_with
POISSON_KERNEL
    -> HARMONIC_EXTENSION

# ---------- HARMONIC FUNCTION ----------
u
    -> HARMONIC

iff

LAPLACIAN(u)
    = 0

HARMONIC_FUNCTION
    -> satisfies(MEAN_VALUE_PROPERTY)
    -> satisfies(MAXIMUM_PRINCIPLE)
    -> smooth_under(STANDARD_DOMAIN_CONDITIONS)

# ---------- MEAN VALUE PROPERTY ----------
if
    u harmonic
    on neighbourhood_of(closed_ball)

then

u(x0)
    =
average_of_u_over_sphere_or_ball

MEAN_VALUE_PROPERTY
    -> characterises(HARMONICITY_UNDER_SUITABLE_REGULARITY)

# ---------- MAXIMUM PRINCIPLE ----------
if
    u harmonic
    and nonconstant
    on connected_domain

then

u
    -> cannot_attain_strict_interior_maximum_or_minimum

MAXIMUM_PRINCIPLE
    -> controls(HARMONIC_BOUNDARY_VALUE_PROBLEMS)

# ---------- APPROXIMATE IDENTITY ----------
family
    K_epsilon

-> APPROXIMATE_IDENTITY

if

K_epsilon
    -> concentrates_near(identity)

and

integral(K_epsilon)
    = 1

then

f*K_epsilon
    -> f

in appropriate convergence mode

APPROXIMATE_IDENTITIES
    -> recover(FUNCTIONS_FROM_SMOOTHED_VERSIONS)

# ---------- MAXIMAL FUNCTION ----------
HARDY_LITTLEWOOD_MAXIMAL_FUNCTION:

Mf(x)
    :=
sup_over_r>0(
    average_over_ball_B(x,r)(
        |f(y)|
    )
)

MAXIMAL_OPERATOR
    -> controls(LOCAL_AVERAGES)

MAXIMAL_THEOREM
    -> central_to(DIFFERENTIATION_AND_CONVERGENCE_RESULTS)

# ---------- SINGULAR INTEGRALS ----------
SINGULAR_INTEGRAL_OPERATOR
    -> integral_operator_with(SINGULAR_KERNEL)

example:
    HILBERT_TRANSFORM

Hf(x)
    :=
principal_value(
        1/pi
        *
        integral(
            f(y)/(x-y)
            dy
        )
    )

SINGULAR_INTEGRALS
    -> central_to(MODERN_HARMONIC_ANALYSIS)

# ---------- HILBERT TRANSFORM ----------
FOURIER_MULTIPLIER_OF_HILBERT_TRANSFORM
    -> proportional_to(-i*sign(xi))

HILBERT_TRANSFORM
    -> phase_shift_operator
    -> singular_integral_operator

# ---------- FOURIER MULTIPLIERS ----------
operator T

defined by

Fourier_transform(Tf)(xi)
    =
m(xi)
*
f_hat(xi)

m
    -> MULTIPLIER_SYMBOL

FOURIER_MULTIPLIER
    -> diagonal_operator_in(FREQUENCY_DOMAIN)

MULTIPLIER_THEOREMS
    -> determine(Lp_BOUNDEDNESS)

# ---------- CALDERON-ZYGMUND STRUCTURE ----------
CALDERON_ZYGMUND_OPERATOR
    -> singular_integral_with(
        SIZE_AND_SMOOTHNESS_KERNEL_CONDITIONS
    )

CALDERON_ZYGMUND_THEORY
    -> establishes(Lp_BOUNDEDNESS)
       for(1 < p < infinity)
       under(STANDARD_HYPOTHESES)

# ---------- HARDY SPACES ----------
HARDY_SPACE
    -> function_space_used_when(L1_THEORY_IS_TOO_WEAK_FOR_CERTAIN_OPERATORS)

H1
    -> endpoint_replacement_for(L1_IN_SINGULAR_INTEGRAL_THEORY)

HARDY_SPACES
    -> linked_to(MAXIMAL_FUNCTIONS_AND_ATOMIC_DECOMPOSITIONS)

# ---------- BMO ----------
BMO
    := BOUNDED_MEAN_OSCILLATION

BMO
    -> endpoint_space_related_to(L_INFINITY)

DUALITY:
    H1_star
        ~= BMO
       under(STANDARD_EUCLIDEAN_CONTEXT)

BMO
    -> central_to(MODERN_HARMONIC_ANALYSIS)

# ---------- LITTLEWOOD-PALEY ----------
FUNCTION
    -> decomposed_into(FREQUENCY_BANDS)

LITTLEWOOD_PALEY_DECOMPOSITION
    -> multiscale_frequency_analysis

SQUARE_FUNCTION
    -> measures(ENERGY_ACROSS_SCALES)

LITTLEWOOD_PALEY_THEORY
    -> connects(FREQUENCY_LOCALISATION_AND_Lp_NORMS)

# ---------- UNCERTAINTY ----------
LOCALISATION_IN_SPACE
    -> tradeoff_with(LOCALISATION_IN_FREQUENCY)

UNCERTAINTY_PRINCIPLE
    -> fundamental_to(HARMONIC_ANALYSIS)

TIME_FREQUENCY_STRUCTURE
    -> motivates(WAVELETS_AND_WINDOWED_TRANSFORMS)

# ---------- PDE RELATION ----------
FOURIER_MULTIPLIERS
    -> represent(CONSTANT_COEFFICIENT_DIFFERENTIAL_OPERATORS)

LAPLACIAN
    -> multiplier(-|xi|^2)

PDE_SOLUTIONS
    -> analysed_using(FREQUENCY_DECOMPOSITION)

HARMONIC_ANALYSIS
    -> provides(ESTIMATES_FOR_PDEs)

# ---------- APPLICATIONS ----------
HARMONIC_ANALYSIS
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(PDEs)
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(IMAGE_ANALYSIS)
    -> applied_to(NUMBER_THEORY)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(GROUP_REPRESENTATIONS)
    -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
FOURIER_ANALYSIS
    -> subset_or_foundation_of(HARMONIC_ANALYSIS)

TRANSLATION
    -> diagonalised_by(CHARACTERS)

CHARACTERS
    -> form(DUAL_GROUP)

CONVOLUTION
    -> diagonalised_by(FOURIER_TRANSFORM)

HARMONIC_FUNCTION
    -> solves(LAPLACE_EQUATION)

MAXIMAL_OPERATOR
    -> controls(LOCAL_AVERAGES)

SINGULAR_INTEGRALS
    -> studied_via(FOURIER_AND_REAL_VARIABLE_METHODS)

FREQUENCY_DECOMPOSITION
    -> supports(MULTISCALE_ANALYSIS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "harmonic analysis" ~= "analysis of functions through frequencies, characters, convolution and harmonic structures",
    "character" ~= "continuous group homomorphism into the unit circle",
    "dual group" ~= "group of continuous characters of an abelian topological group",
    "harmonic function" ~= "function annihilated by the Laplacian",
    "Fourier multiplier" ~= "operator acting by multiplication in Fourier space",
    "approximate identity" ~= "family of kernels converging toward the identity action under convolution"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    harmonic_analysis_is_broader_than_classical_fourier_analysis: true,
    group_fourier_transform_requires_measure_and_group_structure: true,
    transform_normalisation_must_be_consistent: true,
    operator_boundedness_depends_on_function_spaces: true,
    singular_integrals_require_principal_value_or_distributional_handling_when_needed: true,
    harmonic_function_and_harmonic_frequency_component_are_distinct_concepts: true,
    convergence_and_reconstruction_results_require_theorem_specific_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    fourier_generalisation_relation: 0.99,
    group_character_relation: 0.99,
    harmonic_function_relation: 0.99,
    convolution_multiplier_relation: 0.99,
    maximal_singular_integral_relation: 0.98,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_space_group_or_operator_context)
    -> identify(translation_and_measure_structure)
    -> determine(classical_fourier_or_group_harmonic_framework)
    -> choose(character_transform_convolution_or_kernel_representation)
    -> preserve(normalisation_and_function_space)
    -> analyse(Lp_boundedness_or_convergence_if_relevant)
    -> apply(maximal_singular_integral_or_multiplier_theory_if_requested)
    -> analyse(harmonic_function_or_PDE_structure_if_present)
    -> inspect(multiscale_or_frequency_localisation_if_relevant)
    -> validate(group_measure_operator_and_regularities)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed harmonic-analysis semantic graph",
    primary_entity: HARMONIC_ANALYSIS,
    preserve: [
        function,
        underlying_space_or_group,
        measure,
        translation_structure,
        frequency_or_character_variable,
        function_space,
        operator,
        convergence_mode
    ],
    infer_through: [
        Fourier_analysis,
        characters,
        dual_groups,
        convolutions,
        harmonic_functions,
        maximal_operators,
        singular_integrals,
        Fourier_multipliers,
        multiscale_decompositions
    ],
    avoid: [
        reducing_all_harmonic_analysis_to Fourier series,
        confusing_harmonic_function_with sinusoidal harmonic,
        ignoring_Haar_measure_in group transforms,
        asserting_Lp_boundedness_without operator hypotheses,
        treating_singular_integrals_as ordinary integrals at singularities,
        mixing_frequency_normalisations
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + HARMONIC_ANALYSIS
    + FOURIER_ANALYSIS
    + CHARACTER
    + DUAL_GROUP
    + CONVOLUTION
    + HARMONIC_FUNCTION
    + MAXIMAL_OPERATOR
    + SINGULAR_INTEGRAL
    + POISSON_KERNEL
    + LAPLACIAN
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> HARMONIC_ANALYSIS
    -> CONTEXT_IDENTIFICATION
    -> GROUP_MEASURE_TRANSLATION_ANALYSIS
    -> FRAMEWORK_SELECTION
    -> TRANSFORM_CONVOLUTION_KERNEL_ANALYSIS
    -> FUNCTION_SPACE_ANALYSIS
    -> OPERATOR_BOUNDEDNESS_ANALYSIS
    -> MAXIMAL_SINGULAR_MULTIPLIER_ANALYSIS
    -> HARMONIC_PDE_ANALYSIS
    -> MULTISCALE_FREQUENCY_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW