Sinh regularized Lagrangian nonuniform sampling series

Recently, some window functions have been introduced into the nonuniform fast Fourier transform and the regularized Shannon sampling. Inspired by these works, we utilize a sinh-type function to accelerate the convergence of the Lagrangian nonuniform sampling series. Our theoretical error estimates and numerical experiments demonstrate that the sinh regularized nonuniform sampling series achieves a superior convergence rate compared to the fastest existing Gaussian regularized nonuniform sampling series.

💡 Research Summary

The paper introduces a sinh‑type regularization window into the Lagrangian nonuniform sampling series, aiming to accelerate convergence beyond what Gaussian regularization can achieve. After recalling the classical Shannon sampling theorem and its limitation to uniform sampling, the authors motivate the need for nonuniform schemes that can adapt to practical constraints such as hardware limitations and irregular acquisition patterns.

In the theoretical section, the authors work within the Paley‑Wiener space B_δ(ℝ) of band‑limited functions with bandwidth δ<π. Lemma 2.1 splits the reconstruction error of any regularized sampling series into two components: E₁,N, which depends on the frequency‑domain integral of the regularizer’s Fourier transform, and E₂,N, which accounts for contributions from sample points beyond the truncation index N. This decomposition is standard in the analysis of regularized series and provides a clear pathway for estimating convergence rates.

The core contribution is the definition of the sinh‑type regularization function
ϕ_{β,m}(x) = { sinh β sinh(π x/(2m)) / (β sinh(π x/(2m))) for |x|≤m, 0 otherwise },
where β = (N−1)(π−δ) and m = N−1. This function is compactly supported, equals 1 at the origin, and decays rapidly toward the edges of its support. The authors note that similar sinh‑type windows have been successfully employed in regularized Shannon sampling and in the nonuniform fast Fourier transform (NUFFT), but its application to Lagrangian nonuniform sampling is novel.

Using properties of sine‑type entire functions (functions whose zeros form a separated real sequence and whose growth is controlled by an exponential of the imaginary part), the authors derive Lagrange interpolation formulas of the form
f(x) = Σ_j f(λ_j) F(x) F′(λ_j)(x−λ_j),
where {λ_j} are the sampling nodes and F is a sine‑type function with bandwidth π. They then define two concrete sampling series: a non‑periodic series based on a finite set Λ = {λ_j}{j=−N}^N, and a periodic series built from a lattice τ{mn}=t_m + nM with 0≤t₁<…<t_M<M.

The main theoretical result, Theorem 2.2, states that for any f∈B_δ(ℝ) and β as above, the sinh‑regularized non‑periodic series S_{f,Q,N,ϕ} satisfies
‖f−S_{f,Q,N,ϕ}‖{L^∞(−1,1)} ≤ C_Λ β exp(−(N−1)(π−δ)) ‖f‖{L²},
and the sinh‑regularized periodic series S_{f,ψ,N,ϕ} satisfies a similar bound with an extra factor M in the exponent. The constants C_Λ and C_{F_{per}} depend only on the geometry of the sampling set and the underlying sine‑type function, not on N.

The proof hinges on an explicit expression for the Fourier transform of ϕ_{β,N−1}, which involves the Bessel function J₁. By employing known asymptotics of J₁ (oscillatory decay for large arguments and polynomial decay near zero), the authors bound the integral of |ϕ̂| over the “gap” region