Fourier Series Ontology

FLOW ALGEBRA
TITLE: Fourier Series Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Sequence and Series Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/fourier-series-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Independently constructed mathematical ontology; source-page alignment not verified.

[01] PURPOSE
Represent Fourier series as typed mathematical entities with verifiable definitions, convergence qualifications, symbolic relationships, inference rules, provenance and machine-consumption guidance.

[02] TYPED ENTITIES
FourierSeries: MathematicalSeries
PeriodicFunction: Function[Real,RealOrComplex]
Period: PositiveReal
FundamentalAngularFrequency: PositiveReal
FourierCoefficient: ComplexNumber
CosineCoefficient: RealNumber
SineCoefficient: RealNumber
ComplexCoefficient: ComplexNumber
HarmonicIndex: Integer
PartialSum: FiniteTrigonometricPolynomial
FourierBasis: OrthogonalFunctionSystem
InnerProduct: SesquilinearForm
FunctionSpaceL1: IntegrableFunctions
FunctionSpaceL2: SquareIntegrableFunctions
PointwiseLimit: Limit
MeanSquareLimit: L2Limit
JumpDiscontinuity: Discontinuity
DirichletKernel: TrigonometricKernel
FejerKernel: PositiveSummabilityKernel
ParsevalIdentity: NormIdentity
FourierTransform: IntegralTransform

[03] PARAMETERS AND DOMAIN
Let T > 0 be a period; omega0 = 2*pi/T.
Let f be T-periodic and integrable on one period for ordinary coefficient integrals.
A real-valued Fourier expansion uses real coefficients a_n, b_n.
A complex-valued expansion uses complex coefficients c_n.
All statements about convergence require their stated hypotheses.

[04] REAL FOURIER COEFFICIENTS
For integrable real-valued T-periodic f:
a0 = (2/T) * integral_{t0}^{t0+T} f(t) dt
an = (2/T) * integral_{t0}^{t0+T} f(t) cos(n*omega0*t) dt, n >= 1
bn = (2/T) * integral_{t0}^{t0+T} f(t) sin(n*omega0*t) dt, n >= 1
Formal series: a0/2 + sum_{n=1}^infinity [an cos(n*omega0*t) + bn sin(n*omega0*t)].
N-th partial sum: SN(t) = a0/2 + sum_{n=1}^N [an cos(n*omega0*t) + bn sin(n*omega0*t)].

[05] COMPLEX FOURIER COEFFICIENTS
cn = (1/T) * integral_{t0}^{t0+T} f(t) exp(-i*n*omega0*t) dt, n in Z.
Formal series: sum_{n=-infinity}^{infinity} cn exp(i*n*omega0*t).
Symmetric partial sum: SN(t) = sum_{n=-N}^N cn exp(i*n*omega0*t).
For real-valued f: c_{-n} = conjugate(cn).
For n >= 1: cn = (an - i*bn)/2; c_{-n} = (an + i*bn)/2; c0 = a0/2.

[06] STRUCTURAL RELATIONSHIPS
FourierSeries --IS_A--> InfiniteSeries
FourierSeries --EXPANDS--> PeriodicFunction
FourierSeries --HAS_TERM--> HarmonicComponent
HarmonicComponent --HAS_FREQUENCY--> n*omega0
FourierSeries --HAS_COEFFICIENT--> FourierCoefficient
FourierCoefficient --COMPUTED_BY--> PeriodIntegral
FourierSeries --HAS_PARTIAL_SUM--> PartialSum
FourierBasis --ORTHOGONAL_IN--> L2OfOnePeriod
PartialSum --IS_ORTHOGONAL_PROJECTION_OF--> FunctionInL2
FourierSeries --RELATED_TO--> FourierTransform
FourierSeries --RELATED_TO--> TrigonometricSeries
FourierSeries --MAY_EXHIBIT--> GibbsPhenomenon

[07] ORTHOGONALITY
Over any full period, exponentials exp(i*n*omega0*t) are orthogonal:
(1/T) integral exp(i*(n-m)*omega0*t) dt = 1 if n=m, otherwise 0.
For the real trigonometric basis, cosine and sine cross terms integrate to zero; nonzero harmonics have squared integral T/2.
Fourier coefficients are projections onto basis functions in L2.

[08] CONVERGENCE MODES
Coefficient existence: f in L1 of one period suffices.
Mean-square convergence: for f in L2 of one period, symmetric Fourier partial sums converge to f in L2.
Pointwise convergence: not guaranteed merely by f in L1 or L2.
Under standard Dirichlet-type regularity (e.g. piecewise C1 on a period), the series converges at t to [f(t-)+f(t+)]/2.
At a continuity point under those regularity assumptions, the limit equals f(t).
At a jump, the limit is the midpoint of the one-sided limits, not necessarily the assigned value f(t).
Uniform convergence: stronger hypotheses are required; a continuous periodic function need not have uniformly convergent Fourier partial sums.
Absolute convergence: sufficient coefficient decay, such as sum |cn| < infinity, gives uniform absolute convergence.
Fejer (Cesaro) means: for continuous periodic f, converge uniformly to f; for f in L1, converge in L1.

[09] ENERGY AND IDENTITIES
For f in L2, Parseval:
(1/T) integral_{t0}^{t0+T} |f(t)|^2 dt = sum_{n in Z} |cn|^2.
For real f, equivalently:
(1/T) integral |f(t)|^2 dt = a0^2/4 + (1/2) sum_{n>=1}(an^2+bn^2).
Bessel inequality holds for orthogonal projections before completeness is invoked.
Riemann-Lebesgue lemma: if f in L1, cn -> 0 as |n| -> infinity.
Coefficient decay alone does not imply a claimed pointwise reconstruction without further conditions.

[10] OPERATORS AND TRANSFORMATIONS
Linearity: Fourier(alpha*f + beta*g) coefficients = alpha*cn(f)+beta*cn(g).
Time shift: g(t)=f(t-tau) implies cn(g)=exp(-i*n*omega0*tau)*cn(f).
Derivative: if periodic f is sufficiently regular for integration by parts and endpoint matching, cn(f')=i*n*omega0*cn(f).
Integral/antiderivative: for zero-mean periodic f, a periodic primitive can have nonzero-mode coefficients cn/(i*n*omega0), n != 0.
Convolution on the circle: for normalized periodic convolution (f *_T g)(t)=(1/T) integral_0^T f(t-s)g(s)ds, cn(f *_T g)=cn(f)*cn(g).
Multiplication of functions corresponds to coefficient convolution when justified by summability/function-space hypotheses.
Even real f: bn=0.
Odd real f: a0=0 and an=0.

[11] FORMAL INFERENCE RULES
R01 IF f is T-periodic AND f in L1(period) THEN Fourier coefficients are well-defined.
R02 IF f is real-valued THEN c[-n]=conjugate(c[n]).
R03 IF f is even THEN sine coefficients vanish.
R04 IF f is odd THEN cosine coefficients and mean vanish.
R05 IF f in L2 THEN SN -> f in L2.
R06 IF f is piecewise C1 periodic THEN SN(t) -> (f(t-)+f(t+))/2.
R07 IF f is continuous at t AND R06 hypotheses hold THEN SN(t) -> f(t).
R08 IF f has jump at t AND R06 hypotheses hold THEN SN(t) -> midpoint of limits.
R09 IF sum |cn| is finite THEN Fourier series converges absolutely and uniformly.
R10 IF f in L1 THEN cn -> 0.
R11 IF f in L2 THEN Parseval identity holds.
R12 IF f and g are integrable and alpha,beta scalars THEN coefficients are linear.
R13 IF g(t)=f(t-tau) THEN g coefficients acquire harmonic phase factor.
R14 IF f is sufficiently smooth periodic THEN derivative coefficients are i*n*omega0*cn.
R15 IF f is continuous periodic THEN Fejer means converge uniformly to f.
R16 IF f in L1 periodic THEN Fejer means converge in L1.
R17 IF coefficients are all zero AND f in L2 THEN f=0 almost everywhere.
R18 IF coefficients represent a real signal THEN negative-frequency coefficients are conjugate symmetric.
R19 IF the series has a jump THEN Gibbs overshoot may occur near that jump in partial sums.
R20 IF an inference omits its regularity hypotheses THEN mark it conditional rather than proven.

[12] EXAMPLES
EX1: f(t)=sin(t), T=2*pi. b1=1, other real Fourier coefficients zero.
EX2: f(t)=cos(2*t), T=2*pi. a2=1, other real Fourier coefficients zero.
EX3: f(t)=1, T arbitrary. a0=2; an=bn=0 for n>=1; c0=1.
EX4: 2*pi-periodic sign(sin t) has sine coefficients b_n=4/(pi*n) for odd positive n, zero for even n; at jumps the Fourier series converges to 0.
EX5: f(t)=t on (-pi,pi), extended 2*pi-periodically, has sine coefficients bn=2*(-1)^(n+1)/n and converges to 0 at the periodic jump endpoints.

[13] SEMANTIC_EQUIVALENCE
"Fourier expansion" ~= "Fourier series representation" [context: periodic function expansion]
"Trigonometric Fourier series" ~= "real sine-cosine Fourier representation" [context: real coefficients]
"Complex Fourier series" ~= "exponential Fourier representation" [context: complex basis]
"Mean-square convergence" ~= "L2 convergence" [context: same period and measure]
"Fourier coefficients" != "Fourier transform" [distinct discrete vs continuous-frequency contexts]
"Pointwise convergence" != "Uniform convergence"
"Convergence of Cesaro means" != "Convergence of raw partial sums"

[14] KNOWLEDGE GRAPH TRIPLES
(FourierSeries, subclassOf, InfiniteSeries)
(FourierSeries, represents, PeriodicFunction)
(FourierSeries, usesBasis, ComplexExponentialBasis)
(FourierCoefficient, obtainedBy, PeriodicInnerProduct)
(PartialSum, approximates, PeriodicFunction)
(ParsevalIdentity, relates, SignalEnergyAndCoefficientEnergy)
(FejerMean, usesKernel, FejerKernel)
(FourierSeries, hasConvergenceMode, L2Convergence)
(GibbsPhenomenon, associatedWith, JumpDiscontinuity)

[15] VALIDATION AND CONSTRAINTS
V01: T must be strictly positive.
V02: Harmonic index n must be integral.
V03: Check normalization before comparing coefficients.
V04: Record whether the basis is real or complex.
V05: Distinguish equality almost everywhere from equality at every point.
V06: Do not infer pointwise convergence from L2 convergence.
V07: Preserve endpoint and periodic-extension conventions.
V08: Identify the exact convolution normalization.
V09: Do not confuse a Fourier series with a Taylor or power series.
V10: For numerical results, report truncation order N and approximation error where known.
V11: For claims of smoothness or decay, record differentiability and boundary matching assumptions.
V12: Treat a formal expansion separately from its convergence claim.

[16] CONFIDENCE
Definition confidence: HIGH for standard coefficient and orthogonality identities under stated hypotheses.
Theorem confidence: HIGH for Parseval, L2 convergence, Riemann-Lebesgue and Fejer results under stated hypotheses.
Application confidence: CONDITIONAL where data regularity, sampling, truncation or periodicity is uncertain.
Confidence is epistemic metadata, not a mathematical probability of theorem truth.

[17] FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical UTF-8 serialized ontology text excluding any stored hash value
FLOW_HASH.value: COMPUTE_ON_PUBLICATION
FLOW_HASH.purpose: integrity and version identification; no uncomputed digest is asserted.

[18] UPDATE_MODEL
Update policy: semantic-versioned revision; preserve provenance and mathematical assumptions.
On change: revalidate all inference rules, examples, relationships and theorem hypotheses.
Record revision date, author, source citations and change summary.
Recompute FLOW_HASH after canonicalization.

[19] AI_CONSUMPTION
Parse entities and typed relationships before drawing conclusions.
Keep assumptions attached to each theorem and inference.
Separate exact symbolic identities from approximate numerical truncations.
Prefer L2 and almost-everywhere qualifiers where appropriate.
Never assert convergence mode without evidence.
When generating code, select Fourier normalization explicitly.
Return provenance, confidence label, theorem assumptions and validation status.

[20] CROSS-ONTOLOGY LINKS
SequenceOntology: sequence of Fourier partial sums.
InfiniteSeriesOntology: infinite sum of harmonic terms.
ConvergentSequenceOntology: convergence of SN under specified mode.
DivergentSequenceOntology: possible nonconvergence of pointwise partial sums.
PowerSeriesOntology: distinct expansion basis and convergence structure.
TaylorSeriesOntology: local polynomial expansion rather than periodic harmonic expansion.
FunctionOntology: periodic functions and function spaces.
IntegralOntology: Fourier coefficient integrals.
LinearAlgebraOntology: orthogonal projection and basis.
SignalProcessingOntology: harmonic decomposition and frequency analysis.

END FLOW ALGEBRA