---
title: Sinh Regularized Nonuniform Sampling Series
url: https://www.emergentmind.com/topics/sinh-regularized-nonuniform-sampling-series
type: topic
---

# Sinh Regularized Nonuniform Sampling Series

The sinh regularized nonuniform sampling series is a family of high-convergence reconstruction formulas for bandlimited functions sampled at nonuniform nodes. Its central innovation is the use of a compactly supported, even window function of the form $\varphi_{\sinh}(x) = (1/\sinh\beta)\sinh(\beta\sqrt{1 - x^2/m^2})$ for $|x|\le m$ and $0$ otherwise, which regularizes classical kernels such as the Lagrange and Shannon sinc basis. When applied to Lagrangian and Shannon sampling frameworks, this regularization yields approximation errors that decay exponentially in $m$ and the regularization parameter $\beta$—often with convergence exponents nearly twice as strong as those yielded by Gaussian windowing, and with robust numerical properties in the presence of sampling noise [2601.17685, 2203.09973].

## 1. Mathematical Definition and Kernel Construction

Given a finite sampling set $\Lambda = \{\lambda_j\}_j$, the classical Lagrange basis for entire sine-type generating function $F_\Lambda(z)$ is
$$
Q_{\Lambda,j}(x) = \frac{F_\Lambda(x)}{F'_\Lambda(\lambda_j)(x-\lambda_j)},
$$
and reconstruction is via $f(x) \approx \sum_j f(\lambda_j) Q_{\Lambda,j}(x)$.

The sinh-regularized kernel augments this basis by multiplication with the window $\varphi_{\beta, m}$:
$$
K_{\sinh}(x, \lambda_j) = Q_{\Lambda,j}(x)\;\varphi_{\beta,m}(x-\lambda_j)
$$
where $\varphi_{\beta,m}(x)$ is defined as
$$
\varphi_{\beta,m}(x) =
\begin{cases}
\frac{1}{\sinh\beta}\sinh\left(\beta\sqrt{1-\frac{x^2}{m^2}}\right), & |x|\le m,\\
0, & |x|>m.
\end{cases}
$$
For the regularized Shannon series with uniform grid, the kernel becomes
$$
\psi_{\sinh}(x) = \mathrm{sinc}(L\pi x)\;\varphi_{\sinh}(x),
$$
and the reconstruction formula is
$$
(R_{\sinh,m} f)(t) = \sum_{k} f\left(\frac{k}{L}\right)\;\mathrm{sinc}(L\pi t-\pi k)\;\varphi_{\sinh}\left(t-\frac{k}{L}\right)
$$
with window support $|t-k/L| \le m/L$ and oversampling $L = N(1+\lambda)$ [2306.17594, 2203.09973].

## 2. Error Bounds and Convergence Analysis

The central analytical advantage of the sinh regularizer is exponential error decay, with main error terms proportional to $C\,\beta\,e^{-\beta}$, where $\beta$ is typically chosen as $(N-1)(\pi-\delta)$ for bandwidth $\delta$ [2601.17685]:
$$
\sup_{x \in [-1,1]} | f(x) - S_{f,Q,N}^{\sinh}(x) | \le C_\Lambda\,\beta\,e^{-(N-1)(\pi-\delta)} \|f\|_{L^2(\mathbb{R})}.
$$
For the Shannon setting, the error fulfills
$$
\|f - R_{\sinh,m}f\|_{C(\mathbb{R})} \le 3\sqrt{2\delta}\,e^{-\beta}\,\|f\|_{L^2(\mathbb{R})}
$$
subject to $\beta = \pi m(1+\lambda-2\tau)/(1+\lambda)$, $\tau = \delta/N$ [2203.09973].

In the nonuniform case, provided the nodes are $D$-dense (no gaps larger than $T = 1/L$) and separated, identical exponential bounds hold for
$$
(R_{\sinh, m}^{\text{non}} f)(t) = \sum_{n: |t-x_n|\le m T} f(x_n)\;\mathrm{sinc}(\pi (t-x_n)/T)\;\varphi_{\sinh}(t-x_n).
$$

The improved rate—$e^{-(\pi-\delta)N}$ for sinh versus $e^{-(\pi-\delta)N/2}$ for Gaussian—implies errors $2$–$5$ orders of magnitude smaller for moderate $N$.

## 3. Parameter Selection and Localization

Optimal performance requires tuning of $\beta$ and support $m$:
- **Choice of $\beta$:** Set $\beta = (N-1)(\pi-\delta)$, leading to window support $m = N-1$. In practical contexts, further increase beyond this offer marginal returns [2601.17685].
- **Localization:** At any evaluation point $t$, the series requires only those nodes with $|t-\lambda_j| \le m$ for Lagrangian or $|k-Lt|\le m$ for Shannon series, restricting summation to $2m+1$ terms.

This compact support directly localizes computation and reduces truncation error (no tail term), as proven in multiple error analyses [2306.17594, 2203.09973].

## 4. Numerical Stability and Robustness to Noise

Numerical experiments demonstrate that sinh-regularized series maintain exponential error decay even in the presence of sample perturbations. For noise of amplitude $\epsilon$ at each sample, the output perturbation is bounded by $O(\epsilon \sqrt m)$ [2203.09973]. The localized summation and rapid error decay confer robustness for large-scale practical implementations, including the nonuniform fast Fourier transform (NNFFT) and fast sinc transforms [2107.02671].

## 5. Comparative Performance: Numerical Evidence

Empirical tests reconstruct bandlimited signals under both uniform and nonuniform node distributions. For the canonical example function, the following table exhibits reconstruction errors for no regularization, Gaussian, and sinh-type regularization ($\delta = \pi/2$):

Non-periodic Case (excerpt from [2601.17685]):  
| $N$ | No Reg.        | Gaussian         | Sinh              |
|-----|---------------|------------------|-------------------|
| 6   | $1.85\times10^{-1}$ | $9.25\times10^{-2}$   | $2.67\times10^{-3}$   |
| 12  | $1.64\times10^{-2}$ | $1.53\times10^{-3}$   | $1.75\times10^{-7}$   |
| 18  | $5.90\times10^{-3}$ | $1.79\times10^{-5}$   | $1.75\times10^{-11}$  |

Periodic Case ($M=3$ channels):
| $N$ | No Reg.        | Gaussian         | Sinh              |
|-----|---------------|------------------|-------------------|
| 2   | $2.68\times10^{-3}$ | $2.99\times10^{-3}$   | $7.54\times10^{-5}$   |
| 4   | $7.77\times10^{-4}$ | $6.07\times10^{-6}$   | $2.64\times10^{-9}$   |
| 6   | $3.68\times10^{-4}$ | $2.64\times10^{-8}$   | $6.08\times10^{-14}$  |

Observed convergence matches $e^{-(\pi-\delta)N}$ for sinh and $e^{-(\pi-\delta)N/2}$ for Gaussian regularization.

## 6. Applications to Fast Algorithms

Sinh windows regularize transform kernels in the NNFFT, producing error terms that decay exponentially as $m_j \to \infty$, specifically as $\exp(-2\pi m_j \sqrt{1-1/\sigma_j})$ for oversampling factor $\sigma_j$ [2107.02671]. For the fast sinc transform, these windows are reused to achieve overall error $O(\varepsilon + E_{\text{win}})$, with $E_{\text{win}}$ the compounded window error. The entire process is computationally efficient, with $O((N+L_1+L_2)\log(N+L_1+L_2))$ complexity.

## 7. Practical Recommendations and Generalization

For high-accuracy reconstruction of bandlimited signals from nonuniform samples, the recommended configuration is:
- Use oversampling $L = N(1+\lambda)$ with moderate $\lambda$.
- Set regularization $\beta = (N-1)(\pi-\delta)$ and support $m = N-1$ for direct coverage of interpolation nodes.
- Restrict summation to localized nodes, yielding exponential decay in approximation error and robustness to outlier noise.

The methodology extends naturally to general separated, $D$-dense node sets; all key features—exponential decay, numerical stability, and computational efficiency—persist under nonuniform sampling regimens [2203.09973, 2601.17685]. The sinh-type window is thus the preferred regularizer for practical and theoretical applications demanding minimal reconstruction error from finite, possibly irregularly spaced samples.

Source: https://www.emergentmind.com/topics/sinh-regularized-nonuniform-sampling-series