The N-th partial Fourier sum of a function f : AddCircle T → ℂ, defined as
S_N f(x) = ∑_{n = -N}^{N} ĉ_n(f) · e^{2πi n x / T}, where ĉ_n(f) = fourierCoeff f n.
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The Fejér kernel K_N, defined here as the Cesàro mean of the Dirichlet kernels:
K_N(x) = (N+1)⁻¹ · ∑_{k=0}^{N} ∑_{n=-k}^{k} e^{2πi n x / T}. Equivalently (after
algebra), K_N(x) = (N+1)⁻¹ · (sin((N+1)x/2) / sin(x/2))² away from the origin.
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Each Fourier mode t ↦ e^{2πi n t / T} is integrable on the circle (in fact
bounded by 1); used repeatedly when integrating Fourier sums termwise.
Orthogonality of the Fourier basis with respect to the Haar measure: the integral
of e^{2πi n t / T} over the circle equals 1 if n = 0 and 0 otherwise.
The integral of the k-th Dirichlet kernel ∑_{n=-k}^{k} e^{2πi n t / T} over
the circle equals 1; only the n = 0 term contributes by orthogonality.
The Fejér kernel integrates to 1 on the circle:
∫_{AddCircle T} K_N(t) dt = 1. This is one of the defining properties of an
approximation to the identity.
The Fejér kernel K_N is integrable on the circle; this follows from
continuity together with the pointwise bound ‖K_N‖ ≤ N + 1.
Convolution identity for Fourier modes: e^{-2πi n t / T} · e^{2πi n x / T} = e^{2πi n (x - t) / T}. The basic shift identity behind expressing partial Fourier
sums as convolutions with the Dirichlet kernel.
Re-indexing lemma: the sum ∑_{k=0}^{N} e^{2πi (k - (N+1)) x / T} equals the
sum over the negative range n ∈ [-(N+1), -1]. Used to expand (∑ z^k) · z̄^{N+1}
in the proof of the Fejér kernel identity.
Companion to sum_range_fourier_shift_neg: the sum
∑_{k=0}^{N} e^{2πi ((N+1) - k) x / T} equals the sum over the positive range
n ∈ [1, N+1].
Key algebraic identity: the iterated Dirichlet sum defining K_N equals
|∑_{k=0}^{N} e^{2πi k x / T}|², i.e. a sum-of-Dirichlet kernels is the squared
modulus of a geometric sum. This is what makes the Fejér kernel nonnegative.
Bundled properties of the Fejér kernel K_N on AddCircle T:
(1) K_N(x) ≥ 0 for all x,
(2) K_N is symmetric: K_N(-x) = K_N(x),
(3) K_N is T-periodic on ℝ,
(4) ∫ K_N = 1, and
(5) the decay estimate: for any δ > 0 and any x with
δ ≤ ‖1 - e^{2πi x / T}‖, we have ‖K_N(x)‖ ≤ 4 / ((N+1) δ²).
Reversed form of fourier_neg_mul_fourier_eq:
e^{2πi n (x - t) / T} = e^{-2πi n t / T} · e^{2πi n x / T}. Convenient orientation
for the convolution computation.
If f is integrable on AddCircle T, then t ↦ e^{2πi n t / T} · f(x - t) is
also integrable; a routine integrability lemma needed to interchange sums and
integrals in the convolution computation.
Convolution-with-a-mode identity: for integrable f,
∫ e^{2πi n t / T} · f(x - t) dt = ĉ_n(f) · e^{2πi n x / T}. This says that the
n-th term of the partial Fourier sum at x is the convolution of f with the
mode e^{2πi n · / T}.
The Cesàro–Fourier mean as a convolution against the Fejér kernel:
σ_N f(x) = ∫_{AddCircle T} K_N(t) · f(x - t) dt,
valid for any integrable f. This is the bridge from the abstract definition of
σ_N f to the kernel-theoretic analysis underlying Fejér's theorem.
The Fejér kernel is in fact a nonnegative real number (coerced into ℂ):
K_N(x) = (N+1)⁻¹ · |∑_{k=0}^{N} e^{2πi k x / T}|², written with normSq. This
makes ‖K_N(x)‖ = Re K_N(x).
Since K_N ≥ 0, its L¹ norm coincides with its integral:
∫ ‖K_N(t)‖ dt = 1. Used to control the convolution ‖K_N * (f - f(x - ·))‖ in
the proof of Fejér's theorem.
Quantitative form of Fejér's theorem: given ε > 0 and a modulus of continuity
δ > 0 for f ∈ C(AddCircle T, ℂ) at scale ε/2, there exists N₀ such that
∀ N ≥ N₀, ∀ x, dist (f x) (σ_N f x) < ε. The proof splits the convolution integral
into the region ‖t‖ < δ (where f(x) − f(x − t) is small) and its complement
(where the Fejér kernel decays as O(1/((N+1)δ²))).
Fejér's theorem on AddCircle T: for every continuous function
f : AddCircle T → ℂ (equivalently, every continuous T-periodic function on ℝ),
the Cesàro–Fourier means σ_N f converge uniformly to f as N → ∞.