Sinc Convolution Overview

• Sinc convolution is a numerical technique that applies the sinc function in convolution operations to reconstruct bandlimited signals based on the Shannon sampling theorem.
• It underpins methods in signal processing, numerical integration, and deep learning by enabling precise function approximation and efficient handling of derivatives.
• Recent advances, including double-exponential mappings and compressed-domain methods, have enhanced its convergence rates and applicability in high-dimensional, irregular sampling contexts.

Sinc convolution refers to a family of approximation, numerical, and neural methods wherein the sinc function,

$$\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x},$$

is exploited in convolutional operations, signal representations, or as a basis for function approximation and numerical integration. The sinc function is central in bandlimited signal reconstruction (as per the Shannon sampling theorem) and underpins a spectrum of numerical and computational strategies in applied mathematics, physics, and signal processing, as well as in machine learning architectures. Below, key developments, theoretical foundations, application domains, and recent refinements of sinc convolution are systematically presented.

1. Theoretical Foundations and Analytic Structure

The basis for sinc convolution lies in the sampling theorem: any function $z(x)$ bandlimited to $(-\pi/\Delta, \pi/\Delta)$ can be reconstructed from samples $z(n\Delta)$ via a sinc expansion: $$z(x) = \sum_n z(n\Delta) \, \operatorname{sinc}\left(\frac{x-n\Delta}{\Delta}\right).$$ For periodic functions, the expansion generalizes to periodic repetitions of shifted sinc functions and leads to an orthonormal set of basis functions for representing the signal over a finite interval (Marconcini et al., 2013). When the function is periodic with period $L$, and $N$ samples are taken per period ($L = N\Delta$), periodic sinc basis functions $g_{\ell,\Delta}(x)$ are constructed as

$$g_{\ell,\Delta}(x) = \sum_{\eta=-\infty}^{+\infty} \operatorname{sinc}\left(\frac{x - (\ell + \eta N) \Delta}{\Delta}\right),$$

forming an orthonormal basis under the appropriate scalar product.

These constructions support "exact" differentiation in the discrete representation for signals with spectra containing only the first $N$ Fourier components; the derivative is computed by differentiating the sinc expansion term by term, resulting in explicit, nonlocal differentiation matrices that implement the same action as Fourier derivatives, but in real space (Marconcini et al., 2013).

2. Sinc Convolution in Numerical Analysis and Scientific Computing

Sinc Convolution for Indefinite Integrals

The sinc convolution method, originally developed by Stenger, provides a highly accurate formula for indefinite convolution integrals: $p(x) = \int_a^x f(x - t) g(t) dt.$ In the classical framework, $p(x)$ is represented in operator form as $p(x) = (F(\mathcal{J})g)(x)$, with $\mathcal{J}$ the indefinite integration operator and $F$ determined via the Laplace transform of $f$. Sinc-based discretizations employ the Sinc indefinite integration formula, typically with a single-exponential (SE) change of variable (Nedaiasl, 2019).

Recent theoretical advances partially resolve open problems in the foundation of the method ("Stenger’s conjecture") by showing that the analytic requirements on $f$ can be substantially weakened. Rather than requiring the Laplace transform $\hat{f}(s)$ to be analytic in the entire right-half plane and the discretized integration matrix to have spectrum strictly inside this domain, it suffices for $F(s)$ to be analytic within a sufficiently large contour, and the spectrum of the discrete integration operator converges to zero as the number of collocation points increases (Okayama, 16 Jul 2025). This makes the convergence and applicability of the method robust in broader circumstances.

Double-Exponential Transformation for Accelerated Convergence

A major refinement replaces the SE transformation with a double-exponential (DE) mapping in constructing the discrete approximation. The DE-Sinc convolution formula exhibits error bounds

$$\max_{x \in [a,b]} |p(x) - (F(J)g)(x)| = O\left(\log(n+1) \exp\left(-\frac{d n}{\log(2 d n)}\right)\right),$$

which approaches exponential convergence in $n$ (number of collocation points), vastly outperforming the root-exponential convergence $O(\sqrt{n} \exp(-\sqrt{d n}))$ of the SE-Sinc formula (Okayama, 16 Jul 2025). This improvement is especially impactful for high-precision computations of convolution-type integrals and integral equations.

3. Sinc Convolution in Applied Physics and Engineering

Quantum and Electromagnetic Wave Equations

Sinc convolution techniques are employed to represent quantum wave equations—such as the Dirac or Schrödinger equations with periodic boundary conditions—entirely in real space (Marconcini et al., 2013). By expanding the solution in (periodic) sinc bases, all derivatives are handled nonlocally but exactly (for band-limited functions), yielding a formulation equivalent to truncated Fourier-Galerkin methods. This ensures exact treatment of spatial derivatives, exact equivalence to their reciprocal-space counterparts, and facilitates stable numerical simulations even in challenging settings (e.g., Dirac equation for armchair graphene nanoribbons).

A crucial aspect in such applications is handling the product of functions with different bandwidths (e.g., potential and wave function), which requires projecting the product onto the appropriate space. This is performed via sinc convolution (sampling on a refined grid and analytic calculation of convolution coefficients), thus preserving spectral fidelity and avoiding errors from naive pointwise multiplication (Marconcini et al., 2013).

Computational Optics

Sinc convolution techniques are used in computational optics to evaluate diffraction integrals—Rayleigh-Sommerfeld and Fresnel—in convolutional form (Cubillos et al., 2021). Discretizing the source field using sinc series guarantees exact preservation of spatial bandwidth, avoiding artifacts induced by the Fourier series periodicity enforced in FFT-based approaches. This results in error bounds and convergence that depend solely on the approximation quality of the source, independent of propagation distance, wavelength, or grid resolution, contrasting sharply with the growing errors observed in angular spectrum methods as distance increases.

4. Sinc Convolution in Modern Signal Processing and Machine Learning

Learnable Sinc-based Convolution in Deep Networks

Recent advances in deep learning for speech, EEG, and emotion recognition feature the Sinc convolution layer as the primary front-end for raw waveform processing (Ravanelli et al., 2018, Ravanelli et al., 2018, Kürzinger et al., 2020, Bria et al., 2021, Zhang et al., 19 Feb 2024, Ho et al., 4 Mar 2024). Instead of learning arbitrary filter coefficients (as in standard CNNs), sinc-convolutional filters are parameterized as band-pass filters with only their low and high cutoff frequencies as learnable parameters: $$g[n; f_1, f_2] = 2f_2 \operatorname{sinc}(2\pi f_2 n) - 2f_1 \operatorname{sinc}(2\pi f_1 n).$$ This inductive bias reduces parameter count, accelerates convergence, and confers physical interpretability. Performance advantages, such as lower error rates and improved robustness to noise, have been demonstrated in speaker recognition (Ravanelli et al., 2018), automatic speech recognition (Kürzinger et al., 2020), EEG-based motor imagery classification (Bria et al., 2021), and emotion recognition (Zhang et al., 19 Feb 2024). The parameterization also introduces filter diversity and enables explicit tracking of spectral regions prioritized by the network (Ho et al., 4 Mar 2024).

Sinc Kernel and Sinc Convolution in Gaussian Processes

In Gaussian process (GP) modeling, sinc kernels are constructed by defining the spectral density as a rectangular ("brick-wall") function, leading (via inverse Fourier transform) to a sinc-based covariance kernel: $$K(t, t') = \sigma^2 \operatorname{sinc}(\Delta (t - t')) \cos(2\pi \xi_0 (t - t')),$$ guaranteeing that sample paths are bandlimited (Tobar, 2019). The sinc kernel in GPs provides an explicit Bayesian interpretation of Shannon-Nyquist reconstruction and supports structured regression, filtering, and demodulation tasks, robust to noise and irregular sampling.

5. Sinc Convolution, Basis Stability, and Approximation Theory

Stability under Nonuniform Sampling

The robustness of sinc-based reconstruction under nonuniform or perturbed sampling grids is underpinned by the theory of Riesz bases. If the deviations from integer-spaced nodes in $\{\lambda_n\}$ are suitably bounded (e.g., $|\lambda_n - n| < 1/4$), the system $\{\operatorname{sinc}(\lambda_n - t)\}$ retains the Riesz basis property for the Paley–Wiener space, ensuring stable and convergent sinc convolution even in practical sampling scenarios (Avantaggiati et al., 2016). This result is crucial for applications such as irregular sampling in hardware, adaptive quadrature, and robust digital-to-analog conversion.

Rational Approximations and Efficient Quadrature

Practical implementation of sinc convolution (especially for integral operators) often uses rational approximations of the sinc kernel, derived from truncated cosine product expansions and Fourier analysis (Abrarov et al., 2018). Such rational approximations allow high accuracy with very few terms, enabling fast evaluation of convolutions and related transforms in numerical simulations.

Improved Sinc Approximation via Conformal Maps

Performance improvements in Sinc approximation and the associated convolution can be achieved by employing optimal conformal mappings, as shown by replacing standard single-exponential maps with carefully chosen transformations (e.g., $\phi(x) = \log(1 + e^x)$) that enlarge the region of analyticity and accelerate convergence for exponentially decaying functions on semi-infinite domains (Okayama et al., 2018). This leads to sharper error bounds and faster rates in Sinc-based numerical schemes.

6. Advanced Algorithmic Developments and Applications

High-Dimensional and Compressed-Domain Sinc Convolution

For applications involving very large data arrays (e.g., synthetic aperture radar—SAR), convolutions with sinc kernels can be rendered computationally efficient by representing both data and kernel in the quantized tensor train (QTT) format, and carrying out all calculations—including Fourier transforms—in the compressed domain (Chertock et al., 2023). The QTT approach not only reduces storage and computation but also acts as a denoiser, suppressing noise-dominated singular values in the low-rank approximation and producing more accurate convolution results.

Option Pricing and Fourier-Based Financial Computation

In mathematical finance, the “SINC approach” computes option prices by expanding expectations as convolutions and reconstructing Fourier transforms of truncated densities via sinc expansions. By leveraging the sampling theorem and only odd-frequency moments, this approach accelerates computation, achieves high accuracy, and supports efficient calibration (e.g., FFT-concurrent pricing across an entire volatility smile) (Baschetti et al., 2020).

7. Summary of Impact and Future Directions

Sinc convolution methods constitute a fundamental toolkit for numerical analysis, computational physics, applied mathematics, and increasingly, data-driven learning. Their rigorous theoretical underpinnings—spanning basis stability, convergence theory, and analytic function approximation—are matched by practical efficiency and interpretability in computational implementations. The latest refinements—such as the adoption of double-exponential mappings for accelerated convergence (Okayama, 16 Jul 2025), adaptive parameterization for deep learning (Ho et al., 4 Mar 2024), and robust compressed-domain convolution (Chertock et al., 2023)—demonstrate that sinc convolution remains a vibrant area of research with significant practical impact across disciplines.

Ongoing research targets further generalizations in high dimensions, adaptive and automatic parameter selection in neural architectures, and numerical schemes for challenging, singular, or non-smooth integrands. The flexibility of sinc convolution to integrate analytic theory, approximation, and computational practice ensures its continued relevance and evolution in both established and emerging application domains.