Sampling Operators and Approximation Theory

Sampling operators constitute a class of linear and nonlinear mappings designed to reconstruct or approximate functions from discrete data. Rooted in classical Shannon theory, modern developments extend to Kantorovich and Durrmeyer variants, which integrate kernel functions and moments to control smoothness and convergence. Approximation theory for these operators centres on quantifying the discrepancy between an original signal or function and its sampled–reconstructed counterpart. Key metrics include rates of convergence in spaces such as Lp, Sobolev and Orlicz, often expressed via moduli of smoothness or K‐functionals. Voronovskaja‐type theorems provide asymptotic expansions that characterise the leading error term, while inverse theorems link error decay to intrinsic smoothness classes like Lipschitz or Besov. Recent advances have explored weighted settings, multivariate extensions and shape‐preserving properties, with applications spanning signal processing, image super‐resolution and numerical solutions of partial differential equations. The interplay between kernel regularity, sampling density and function space yields a rich tapestry of theoretical results that guide practical algorithm design and error control in data reconstruction tasks.

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Sampling Operators and Approximation Theory publication trend

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Technical terms

Sampling operator: A mapping that reconstructs a continuous function from discrete samples, often via convolution with a kernel or by integral means.

Kantorovich operator: A sampling operator variant incorporating integral averaging over sampling intervals, which enhances stability and convergence in Lp‐spaces.

Durrmeyer operator: A modification that integrates kernel functions against the target function, preserving moments and enabling convergence in modular spaces.

Kernel function: A weighting function—often bandlimited or satisfying Strang–Fix conditions—that determines the shape and support of the reconstruction operator.

Modulus of smoothness: A quantitative measure of a function’s smoothness used to bound approximation errors and derive convergence rates.

Voronovskaja theorem: An asymptotic result describing the leading term of the approximation error for certain operators, revealing the influence of second‐order derivatives.

References

1. Sampling by Difference as a Method of Applying the Sampling Kantorovich Model in Digital Image Processing. Applied Sciences (2023).
2. Convergence of generalized sampling series in weighted spaces. Demonstratio Mathematica (2022).
3. Approximation Properties of the Sampling Kantorovich Operators: Regularization, Saturation, Inverse Results and Favard Classes in Lp-Spaces. Journal of Fourier Analysis and Applications (2022).
4. On the convergence properties of sampling Durrmeyer‐type operators in Orlicz spaces. Mathematische Nachrichten (2022).
5. Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces. Sampling Theory, Signal Processing, and Data Analysis (2022).
6. Variation diminishing-type properties for multivariate sampling Kantorovich operators. Bollettino dell'Unione Matematica Italiana (2020).
7. Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals. Entropy (2012).